Luc Hillairet

Chris Judge

HYPERBOLIC TRIANGLES WITHOUT EMBEDDED EIGENVALUES

We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degenerating family of triangles. In particular, we use a careful analysis of spectral projections near the crossings of these eigenvalue branches with the eigenvalue branches of a model operator.

The work of L.H.

Introduction

Though well-studied for over fifty years, the spectral theory of hyperbolic surfaces still presents basic unresolved questions [START_REF] Sarnak | Spectra of hyperbolic surfaces[END_REF]. For example, does there exist a noncompact, finite area hyperbolic surface whose Laplacian has no nonconstant square-integrable eigenfunctions? This question has been the subject of many investigations including [START_REF] De Verdiere | Pseudo-laplaciens[END_REF], [START_REF] Phillips | On cusp forms for co-finite subgroups of PSL(2, R)[END_REF], [START_REF] Deshouillers | Maass cusp forms[END_REF], [START_REF] Phillips | Perturbation theory for the Laplacian on automorphic functions[END_REF], [START_REF] Wolpert | Spectral limits for hyperbolic surfaces[END_REF], [START_REF] Wolpert | Disappearance of cusp forms in special families[END_REF], and [START_REF] Phillips | Cusp forms for character varieties[END_REF].

T t t 0 1
Figure 1. The triangle T t defined by x 2 + y 2 ≥ 1 and 0 ≤ x ≤ t.

As a model problem, Phillips and Sarnak [START_REF] Phillips | Automorphic spectrum and Fermi's golden rule[END_REF] suggested studying the Neumann eigenvalue problem on the domain T t ⊂ R × R + pictured in Figure 1. In this paper, we prove the following: Theorem 1.1. For all but at most countably many t ∈ ]0, 1[, the Neumann Laplacian on the geodesic triangle T t in the hyperbolic plane has no nonconstant (squareintegrable) eigenfunction.

The group G t of hyperbolic isometries generated by reflections in the geodesic arcs that bound T t is discrete if and only if t = cos(π/n) for n ≥ 3 an integer. For example, if t = 1/2, then G t contains an index two subgroup that is naturally isomorphic to P SL 2 (Z). It follows from the seminal work of A. Selberg [Selberg] that if n = 3, 4, or 6, then the Neumann Laplacian has infinitely many nonconstant eigenfunctions. In particular, for some special t, there do exist square-integrable solutions to the Neumann problem. 1 In [START_REF] Judge | On the existence of Maass cusp forms on hyperbolic surfaces with cone points[END_REF], Theorem 1.1 was verified under an additional-and as of yet unjustifiedassumption concerning the spectral multiplicities of the Neumann Laplacian acting on L 2 (T 1 ). The proof consisted of studying the singular perturbation problem associated with letting t tend to 1. Similar singular perturbations were studied in the context of degenerating hyperbolic surfaces [START_REF] Wolpert | Disappearance of cusp forms in special families[END_REF] and unitary characters over a fixed hyperbolic surface [START_REF] Phillips | Cusp forms for character varieties[END_REF]. In [START_REF] Wolpert | Disappearance of cusp forms in special families[END_REF], [START_REF] Phillips | Cusp forms for character varieties[END_REF], [START_REF] Judge | On the existence of Maass cusp forms on hyperbolic surfaces with cone points[END_REF], and all prior work on this problem, it was necessary to make assumptions about the multiplicities of the spectrum of the limiting surface.

The angles of a geodesic triangle in the hyperbolic plane determine the isometry class of the triangle. The angles of T t are (π/2, arccos(t), 0). It is not difficult to extend Theorem 1.1 to triangles with angles (θ 1 , θ 2 , 0) (see §4).

Theorem 1.2. The set of (θ 1 , θ 2 ) for which the hyperbolic triangle with angles (θ 1 , θ 2 , 0) admits a nonconstant Neumann Laplace eigenfunction has Lebesgue measure zero and is contained in a countable union of nowhere dense sets.

In other words, the generic hyperbolic triangle with one cusp has no non-constant Neumann eigenvalue where 'generic' can be taken in both a topological and a measurable sense. Theorem 1.1 gives the existence of a triple of angles (θ 1 , θ 2 , 0) for which there are no nonconstant Neumann eigenfunctions. Theorem 1.2 then results from applying a general and well-understood principle concerning analytic perturbations (See, for example, [START_REF] Hillairet | Generic spectral simplicity of polygons[END_REF]). On the other hand, the proof of Theorem 1.1 is much more involved. In particular, the proof will rely upon a refined analysis of 'crossings' of eigenvalue branches.

To prove Theorem 1.1, we further develop the method of asymptotic separation of variables that we introduced in [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF] to study generic simplicity of eigenvalues. This method facilitates the study of real-analytic eigenvalue branches in situations where a geometric domain degenerates onto a lower dimensional domain. There is a vast literature-for instance, [START_REF] Borisov | Asymptotics of Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin domains in R d[END_REF], [START_REF] Grieser | Asymptotics of the first nodal line of a convex domain[END_REF], [START_REF] Friedlander | On the spectrum of the Dirichlet Laplacian in a narrow strip[END_REF]concerning perturbations involving degeneration onto lower dimensional domains, but most of these studies do not address analytic eigenvalue branches. In contrast, our results depend crucially on a study of real-analytic eigenbranches and their crossings.

1.1. An outline of this paper. We now describe the content of each section.

In §2, we establish notation and recall basic features of the spectral theory of the Laplacian acting on functions on a domain in the hyperbolic plane having one cusp. We describe the Fourier decomposition associated to the cusp. The zeroth Fourier mode is responsible for an essential spectrum of [ 1 4 , ∞[. Following [LaxPhl] and 1 In the case where n ≥ 3 is an integer not equal to 3, 4, or 6, Phillips and Sarnak asked whether the domain T cos(π/n) has no nonconstant Neumann eigenfunctions [START_REF] Phillips | Automorphic spectrum and Fermi's golden rule[END_REF]. We should point out that since Theorem 1.1 allows for countably many exceptional t, it does not directly answer their question.

[ColinDeVerdière82], we will replace the Dirichlet quadratic form E(u) = ∆u, u with a modification E β obtained by 'truncating' the zeroth Fourier coefficient above y = β. An eigenfunction u of E β corresponds to an eigenfunction of E if and only if the zeroth Fourier coefficient of u vanishes identically. We will call such an eigenfunction a cusp form. 2 The operator associated to E β has compact resolvent and hence discrete spectrum. This makes E β a much better candidate for the application of methods from spectral perturbation theory.

In §3, we recall and make precise some ideas familiar in the perturbational study of cusp form existence. We consider a real-analytic family, t → q t , of quadratic forms that have the same domain as E β . We say that a real-analytic family t → u t of eigenfunctions of q t is a cusp form eigenbranch if and only u t is a cusp form for each t. We demonstrate a dichotomy: Either the family t → q t has a real-analytic cusp form eigenbranch or the set of t such that q t has a cusp form is countable.

In §4, we consider arbitrary real-analytic paths in the space of hyperbolic triangles with one cusp. We apply the results of §3 to deduce Theorem 1.2 under the assumtption that there exists a triangle with no nonconstant Neumann eigenfunction. The remainder of the paper is devoted to proving Theorem 1.1 which will give the existence of such a triangle.

In §5 we specialize to the family T t . After renormalizing by a factor of t 2 , we find that for each u, the function t → q t (u) has a Taylor expansion at t = 0. We compute the leading order terms of this expansion.

In §6 we show that the method of the asymptotic separation of variables introduced in [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF] may be used to analyse the family of quadratic forms q β,t . In particular, we define a reference quadratic form a t to which separation of variables applies and that is asymptotic to q β,t at 'first order'. By separation of variables we mean that each eigenfunction of a t is of the form v t (y)•cos(π x) with ∈ Z and v t a solution to a renormalized form of the equation for a modified Bessel function with imaginary parameter. We show that one may deduce a non-concentration estimate (Proposition 7.2).

In §7 we use the non-concentration estimate to derive information concerning the real-analytic eigenbranches (E t , u t ) of q β,t . First, we show that there exists an integer k so that E t limits to (πk) 2 as t tends to zero. Second, we find that if the spectral projection of u t onto the space V k spanned by functions of the form the ψ(y) • cos(πkx) is relatively small, then the derivative ∂ t E t is of order 1/t. Finally, we show that if (E t , u t ) is a cusp form eigenbranch, then E t can not limit to zero.

In §8 we prove Theorem 1.1. By the dichotomy of §3, it suffices to show that real-analytic cusp form eigenbranches do not exist. We suppose to the contrary that the real-analytic family q β,t has a cusp form eigenbranch (E t , u t ). By the results of §7, we have that E t limits to (πk) 2 where k ∈ Z + . By improving the analysis of [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF], we show that there exists an eigenbranch, λ * t , of a t that 'tracks' E t at order t in the sense that (1) lim sup

t→0 1 t • |E t -λ * t | < ∞.
2 For t = cos(π/n), these are 'cusp forms' in the sense of the theory of automorphic forms, but otherwise there is no discrete group, and hence they are not cusp forms in the traditional sense.

In this paper we will always be considering 'even' cusp forms, that is, eigenfunctions satisfying Neumann conditions.

We will obtain a contradiction to this estimate by estimating f (t) := d dt (E tλ * t ) from below.

Indeed, we show that when the norm of the projection w k t of u t onto V k is relatively large with respect to u t , the function f (t) is of order t -1 3 , whereas when w k t is relatively small, the function f (t) is of order t -1 . By controling the sizes of the sets where w k t is respectively small and large relative to u t and by integrating, we will contradict (1).

The key observation is the following: Since E t limits to (kπ) 2 , it has to 'cross' each of the eigenbranches of a t that limit to zero. We show that near such a crossing, the branch u t must 'interact' with the functions in V 0 to such an extent that the projection onto V k cannot be too large. The effect of each interaction is made precise by careful estimates of the off-diagonal terms in the quadratic form q ta t (Appendix A). By summing the effects of these interactions, we eventually prove that there exists c > 0 so that

E t -λ * t ≥ c • t 2 3
thus contradicting (1).

The spectrum of a domain in the hyperbolic plane with a cusp

In this section, we describe some basic spectral theory of the Neumann Laplace operator acting on the square-integrable functions on domains in the hyperbolic plane with a cusp. We define the Laplacian and associated Dirichlet quadratic form, describe the Fourier decomposition of eigenfunctions along horocycles, construct a modification of the Dirichlet form whose eigenfunctions include the eigenfunctions (cusp forms) of the standard Laplacian but whose spectrum is discrete.

2.1. The quadratic forms associated to the Neumann Laplacian. The halfplane {(x, y), y > 0} equipped with the Riemannian metric y -2 (dx 2 + dy 2 ) is the Poincaré-Lobachevsky model for the 2-dimensional hyperbolic space H 2 . The measure associated to the Riemannian metric g = y -2 (dx 2 + dy 2 ) is given by integrating dm = dxdy y 2 .

In the present context, a cusp of width w and height y 0 is the subset S w,y0 := [0, w] × [y 0 , ∞[ of the upper half plane. A domain Ω ⊂ H 2 is said to have one cusp if Ω is the union of a cusp and a compact set. We will assume that the boundary of Ω is the the union of finitely many geodesic arcs and that the interior of Ω is connected.

Let D(Ω) denote the set of functions u : Ω → C such that u is the restriction to Ω of a compactly supported smooth function defined on some neighbourhood of Ω.

The L 2 -inner product of two functions u and v in D(Ω) is defined by

(2) N (u, v) := Ω u(x, y) • v(x, y) dm.
Abusing notation slightly, we will often write N (u) in place of N (u, u). Let L 2 (Ω, dm) denote the completion of D(Ω) with respect to the norm N (u)

1 2 .
To define the Neumann Laplacian we consider the bilinear form defined on D(Ω) by

E(u, v) := Ω g(∇u, ∇v) dm
where ∇ satisfies g(∇f, X) = Xf for all vector fields X and smooth functions f . Let E(u) denote the value of the quadratic form u → E(u, u). One computes that

E(u) = Ω |∂ x u(x, y)| 2 + |∂ y u(x, y)| 2 dxdy.
Let H 1 (Ω) denote the completion of D(Ω) with respect to the norm (E(u)+N (u))

1 2 . We will consider E as a nonnegative symmetric bilinear form on L 2 (Ω) with domain H 1 (Ω). As such it is densely defined and closed, and hence there exists a unique, densely defined, self-adjoint operator ∆ on L 2 (Ω) such that for each v ∈ H 1 (Ω) and u in the domain of ∆ we have3

(3) N (∆u, v) = E(u, v).
The operator ∆ is called the Neumann Laplacian. It can be shown that u ∈ H 1 (Ω) if and only if u ∈ L 2 (Ω, dm) and E(u) < +∞ where in the definition of E the partial derivatives are to be taken in the distributional sense.

It is well-known that ∆ has essential spectrum equal to [1/4, ∞[. (For example, see [LaxPhl] or [START_REF] De Verdière | Pseudo-laplaciens[END_REF]). Apart from this essential spectrum, ∆ may also have eigenvalues either smaller than 1/4 or embedded in the continuous spectrum.

From (3) we see that u is an eigenfunction of ∆ with eigenvalue E if and only if u ∈ H 1 (Ω) and for each v ∈ H 1 (Ω)

(4) E(u, v) = E • N (u, v).
2.2. Fourier decomposition in the cusp. For each integer k = 0, define

e k (x) := 2 1 2 • cos(kπ • x)
and define e 0 ≡ 1. The collection {e k | k ∈ N} is an orthonormal basis for L 2 ([0, 1]). Hence, the functions

x → 1 √ w • e k x w provide an orthonormal basis of L 2 ([0, w]).
For positive w, y 0 , let S w,y0 = [0, w] × [y 0 , ∞[ be a cusp of width w and height y 0 .

For each u in L 2 (S w,y0 , dm) and almost every y ≥ y 0 , the function x → u(x, y) belongs to L 2 ([0, w]). Thus we can write

(5) u(x, y) = k≥0 u k (y) • e k x w where (6) u k (y) := 1 w w 0 u(x, y) • e k x w dx.
belongs to L 2 [y 0 , ∞[, y -2 dy . We refer to u k as the k th Fourier coefficient of u. More generally, if Ω is a domain with a cusp S w,y0 , then we define the k th Fourier coefficient of a function v : Ω → C to be the k th Fourier coefficient restriction of v to S w,y0 . Parseval's theorem gives

N u • 1I [y0,∞[ = k≥0 ∞ y0 u k (y) 2 dy y 2
where 1I X denotes the characteristic function of a set X.

Lemma 2.1. If u ∈ H 1 (Ω, dm) is a Neumann eigenfunction of E with eigenvalue E, then for each k ∈ N and each y > y 0 , the Fourier coefficient u k satisfies

(7) -u k + (kπ) 2 w 2 - E y 2 u k = 0.
Proof. If v is a smooth function on Ω, then since u is a Neumann eigenfunction of E, integration by parts gives

- Ω u • ∂ 2 x v + u • ∂ 2 y v dxdy = E Ω u • v dxdy y 2 .
By letting v = φ(y) • e k (x/w) where φ is a smooth function with compact support in ]y 0 , ∞[, we find that

(kπ) 2 w 2 ∞ y0 u k (y) • φ(y) dy - ∞ y0 u k (y) • φ (y) dy = E ∞ y0 u k (y) • dy y 2 .
It follows that u k satisfies (7) in the distributional sense in D ((y 0 , ∞)). By elliptic regularity, u k is actually smooth and satisfies (7) in the strong sense.

In particular, u 0 = Ay s + By 1-s for some constants A and B with E = s(1s). If E ≥ 1/4, then the real part of s equals 1/2, and hence u 0 does not belong to L 2 [y 0 , ∞[, y -2 dy unless both A and B equal zero. Therefore, we have the following.

Corollary 2.2. If u is a Neumann eigenfunction with eigenvalue E ≥ 1 4 , then the zeroth Fourier coefficient u 0 vanishes identically on [y 0 , ∞[. In the classical spectral theory of a quotient of H by a lattice in SL 2 (R), a Laplace eigenfunction u with vanishing zeroth Fourier coefficient in each cusp is called a (weight zero) Maass cusp form. Even though most of the domains that we will consider are not fundamental domains for discrete groups of isometries, we will adapt this terminology.

Definition 2.3. If u is an eigenfunction for the Neumann Laplacian on a domain with a cusp, and u • 1I [y0,∞[ 0 ≡ 0, then we will call u a cusp form.

Traditionally, the Neumann eigenfunctions for T cos(π/n) are called even cusp forms whereas the solutions to the Dirichlet eigenvalue problem are called odd cusp forms. We will not consider odd cusp forms in this paper.

2.3.

A related quadratic form. We wish to apply analytic perturbation theory to study the behavior of eigenfunctions of E on T t as we vary t. Because the eigenvalues of E might lie inside the essential spectrum, standard perturbation theory does not apply directly. Following [START_REF] De Verdière | Pseudo-laplaciens[END_REF] and [START_REF] Phillips | On cusp forms for co-finite subgroups of PSL(2, R)[END_REF], we will use a modification of E first constructed by P. Lax and R. Phillips [LaxPhl] [LaxPhl80]. 4 In this section, we recall the construction, show that the eigenvalues of the modification are isolated, and relate the eigenfunctions of the modification to those of E.

For β > y 0 , let Z β denote the set of u ∈ D(Ω) such that for each y ≥ β we have u 0 (y) = 0. Let L 2 β (Ω, dm) denote the Hilbert space completion of Z β with respect to u → N (u)

1 2 . Let N β denote the restriction of N to L 2 β (Ω). Let H 1 β (Ω) denote the Hilbert space completion of Z β with respect to the norm u → (E(u) + N (u)) 1 2 . The restriction, E β , of E to H 1 β (Ω) is a closed, densely defined quadratic form on L 2 β (Ω). A simple argument shows that L 2 β (Ω) = u ∈ L 2 (Ω, dm), | ∀y > β, u 0 (y) = 0. , and
H 1 β (Ω) = u ∈ H 1 (Ω), | ∀y > β, u 0 (y) = 0.
. In the sequel it will be more convenient to replace Z β by the following other set.

Definition 2.4. Define W β to be the set of functions u in H 1 β such that • u extends to a continuous function on the closure Ω of Ω,

• u is smooth on Ω \ {y = β}. Observe that since Z β ⊂ W β , the closure of W β with respect to the norm u → (E(u) + N (u)) 1 2 is H 1 β .
The latter assertion says that W β is a core of the quadratic form E β Let ∆ β denote the unique operator such that dom(∆ β ) ⊂ H 1 β and that satisfies

N β (∆ β u, v) = E β (u, v) for each u ∈ dom(∆ β ), v ∈ H 1 β (Ω).
Lemma 2.5 ( [LaxPhl]). For each β > y 0 , the resolvent of ∆ β is compact. Hence, the spectrum of E β with respect to N β is discrete and each eigenspace is finite dimensional.

Proof. Using the Fourier decomposition, one shows that for each b > 0, the set of

v ∈ H 1 β (Ω) such that N (v) ≤ 1 and E(v) ≤ b is compact in L 2 β (Ω) (see Lemma 8.7 [LaxPhl]
). It follows that ∆ β has compact resolvent. Hence, by standard spectral theory, the spectrum is discrete and the eigenspaces are finite dimensional.

Definition 2.6 (cusp form). We will say that an eigenfunction u of E β with respect to N β is a cusp form if and only for each y > y 0 we have u 0 (y) = 0.

Lemma 2.7. The following assertions are equivalent:

(1) u is a cusp form of E with respect to N .

(2) There exists β > y 0 such that u is a cusp form of E β with respect to N β .

(3) For each β > y 0 , the function u is a cusp form for E β with respect to N β .

Proof. (1) ⇒ (2): If u is an eigenfunction of E with eigenvalue E, then by Lemma 2.1 the zeroth Fourier coefficient u 0 satisfies the differential equation 0 = (u 0 ) + (E/y 2 ) • u 0 . Since u 0 (y) vanishes for y > y 0 , it must vanish for y > β.

(2) ⇒ (1): Fix a smooth function χ such that χ(y) = 0 for y ≤ 2y0+β 3 and χ(y) = 1 for y ≥ y0+2β

3

. If u 0 (y) = 0 for each y > y 0 , then

N β (u, v -χ • v 0 ) = N (u, v), and E(u, v) = E(u, v -χ • v 0 ) = E β (u, v -χ • v 0 ), for each v ∈ H 1 (Ω). Thus, if E β (u, v) = E • N β (u, v), then E(u, v) = E • N (u, v).
(2) ⇔ (3): Follows from the equivalence of (1) and (2).

Not every eigenfunction u of E β is an eigenfunction of E. For example, if w = cos(π/n) and E s is an Eisenstein series whose zeroth Fourier coefficient vanishes at

y = β, then E s (x, y) -E 0 s (y) • χ [β,∞[ (y
) is an eigenfunction of E β . This function is not smooth across y = β, but elliptic regularity implies that each eigenfunction of E is smooth.

Real-analyticity and generic properties of eigenfunctions

Let S = [0, 1] × [y 0 , ∞) and let β > α > y 0 . In this section, we consider a fixed domain Ω that contains the cusp S and a real-analytic family t → q t of quadratic forms defined on H 1 β (Ω) ⊂ L 2 β (S, dm) that represents the cusp of width w t for y > y 0 and some real-analytic function t → w t (see Definition 3.1 below). We prove the following dichotomy: Either there exists a real-analytic eigenfunction branch consisting of 'cusp forms' or the set of t such that q t has a 'cusp form' eigenfunction is countable. This fact is fundamental to the proofs of both Theorem 1.1 and Theorem 1.2.

Let S = [0, 1] × [y 0 , ∞[ and let β > α > y 0 . For any w > 0, we can define a transformation Φ w between L 2 β (S) and L 2 β (S w,y0 ) by asking that, for any u ∈ L 2 β (S) the function v := Φ w (u) is defined by v(x, y) = 1 √ w u( x w , y). Since, in the sequel y 0 will be fixed, we will drop the index y 0 and denote by S w the cusp of width w.

It is straighforward that Φ w is an isometry between L 2 β (S) and L 2 β (S w ). Moreover Φ w also preserves H 1 β in the sense that Φ w (u) ∈ H 1 β (S w ) if and only if u ∈ H 1 β (S). We may thus define E β,w the quadratic form obtained by pulling-back E β on S w using Φ w . This quadratic form is then closed on the domain H 1 β (S), and for each u ∈ H 1 β (S)

E β,w (u) = S w -2 • |∂ x u(x, y)| 2 + |∂ y u(x, y)| 2 dxdy. Definition 3.1.
Let Ω be a domain that has S as a cusp and β > α > y 0 . Let q be a quadratic form closed over the domain H 1 β (Ω). We will say that q represents the cusp of width w for y ≥ α if, for any u ∈ H 1 β that is supported in {y > α} we have q(u) = E β,w (u).

For such a quadratic form, we will say that an eigenfunction u is a cuspform if u 0 (y) vanishes on {y 0 ≤ y ≤ β}.

The aim of this section is to prove that being a cuspform is a real-analytic condition. To make this statement precise we have to consider a family q t of quadratic forms that satisfies the following assumptions. Assumption 3.2. Let t -< t + and β > α > y 0 . For each t ∈ I := ]t -, t + [, let w t be a positive real-analytic function on I. Let q t denote a nonnegative, closed quadratic form with domain H 1 β (S) that represents the cusp of width w t for y ≥ α. Lastly, we assume that the family t → q t is real-analytic of type (a) in the sense of [Kato]. That is, for each u ∈ H 1 β (S), the map t → q t (u) is real-analytic. A straightforward application of analytic perturbation theory- [Kato] §VIIgives the following.

Theorem 3.3 (Existence of a real-analytic eigenbasis). Let t → q t satisfy the assumptions above 3.2. Then there exist a collection of real-analytic paths {t → u j,t ∈ L 2 (Ω, dm) | j ∈ Z + } and a collection of real-analytic functions {t → λ j,t ∈ R | j ∈ Z + } so that for each t, the set {u j,t | j ∈ Z} is an orthonormal basis for L 2 β (Ω, dm), and for each (j, t), the function u j,t is an eigenfunction of q t with eigenvalue λ j,t .

Proof. Since the embedding from H 1 β into L 2 β is compact, for any t the spectrum of q t consists only in eigenvalues. The proof is then similar to the proof of Theorem 3.9 in [Kato] §VIII.3.5.

For u ∈ W β , define L(u) = lim y→β - u 0 (y) y -β .
Lemma 3.4. An eigenfunction u of q t is a cusp form if and only if L(u) = 0.

Proof. L(u) is the left-sided derivative of u 0 at β. Since u is an eigenfunction, u ∈ W β and u 0 is a solution to a second order ordinary differential equation on [y 0 , β] with u 0 (β) = 0. Thus, u 0 vanishes identically on [y 0 , β] if and only if L(u) = 0.

For real-analytic eigenbranches we have the following.

Lemma 3.5. Let (u t , λ t ) be an analytic eigenbranch of q t then the mapping t → L(u t ) is analytic on ]t -, t + [.

Proof. The zeroth mode u 0 t of u t is a solution to the ODE -u -λ t y 2 • u = 0, on [α, β] with Dirichlet boundary condition at β. Denote by G λ the unique solution to this ordinary differential equation that satisfies G λ (β) = 0, G λ (β) = 1. Since the coefficients of the ordinary differential equation depend analytically on the parameter λ, the mapping λ → G λ is analytic (with values in C 2 ([α, β]) for instance). Moreover, for each compact set K ⊂]t -, t + [ we can find α < α K < β such that, for each t ∈ K,

β α K G λ > 0. Since u t is a multiple of G λt , we then have L(u t ) = β α K G λt (y) dy -1 • β α K u 0 t (y) dy, = β α K G λt (y) dy -1 • 1 0 β α K u t (x, y) dxdy, .
Analyticity on K then follows from the analyticity of t → u t and t → G λt and the choice of α K .

We will say that real-analytic eigenfunction branch u t of q t is a cusp form eigenbranch if and only if for each t ∈ I, the eigenfunction u t is a cusp form (see Definition 3.1). Using the real-analyticity proved in Lemma 3.4 we obtain the following Corollary 3.6. If u t is a real-analytic eigenfunction branch that is not a cusp form eigenbranch, then the set of t ∈ I such that u t is a cusp form is discrete.

We now proceed to prove that if q t has no real-analytic cusp form eigenbranch then, for a generic t, the form q t has no cusp form. As it turns out, we will actually first prove that the spectrum of q t is generically simple.

Let I mult denote the set of t ∈ I such that q t has an eigenspace of dimension at least two.

Proposition 3.7. If q t does not have a real-analytic cusp form eigenbranch, then I mult is countable.

Proof. Let {u j,t | j ∈ Z + , t ∈ I} and {λ j,t | j ∈ Z + , t ∈ I} be as in Theorem 3.3. For each j, k ∈ Z + , let Z j,k = {t | λ j,t = λ k,t }. Since each eigenspace of q t is spanned by a finite collection of {u j,t }, the union j,k Z j,k equals I mult .

The function t → λ j,tλ k,t is analytic, and hence Z j,k = {t | λ j,t = λ k,t } is either countable or equals I. Thus to prove the claim, it suffices to show that it is not possible for Z j,k to equal I.

Suppose that there exists j and k so that u j,t and u k,t are real-analytic eigenbranches so that λ j,t = λ k,t for each t ∈ I. To prove the proposition, it suffices to produce a linear combination u t of u j,t and u k,t so that for each t ∈ I, the function u t is a real-analytic cusp form eikgenbranch.

By hypothesis, neither u j,t nor u k,t are cusp form eigenbranches. By Corollary 3.6, the set J of t such that either u j,t or u k,t is a cusp form is discrete.

For each t / ∈ J, define

u t = L(u k,t ) • u j,t -L(u j,t ) • u k,t L(u j,t ) 2 + L(u k,t ) 2 .
Its suffices to show that t → u t extends to a real-analytic function on I. Indeed, since L is linear, we have L(u t ) = 0 for each t / ∈ J. By Corollary 3.5, the realanalytic extension would satisfy L(u t ) ≡ 0.

The order of vanishing of t → L(u j,t ) (resp. t → L(u k,t )) is finite at each t ∈ J. If the order of vanishing of L(u k,t ) at t 0 ∈ J is at least the order of vanishing of L(u j,t ) at t 0 , then the ratio L(u k,t )/L(u j,t ) has a real-analytic extension near t 0 . Hence, factorizing L(u j,t ) we obtain that

L(u k,t ) L(u j,t ) 2 + L(u k,t ) 2 = L(u k,t ) L(u j,t ) • 1 + L(u k,t ) L(u j,t ) 2 -1 2 , and
L(u j,t ) L(u j,t ) 2 + L(u k,t ) 2 = 1 + L(u k,t ) L(u j,t ) 2 -1 2
have real-analytic extensions near t 0 . If the order of vanishing of L(u k,t ) at t 0 is at most the order of vanishing of L(u j,t ) at t 0 , then a similar argument applies by factorizing L(u k,t ) everywhere. Thus, u t extends to a real-analytic cusp form eigenbranch.

Let I cf denote the set of t ∈ I such that q t has at least one cusp form eigenfunction.

Proposition 3.8. If q t has no real-analytic cusp form eigenbranch, then I cf is countable.

Remark 3.9 (Dichotomy). If there exists a cusp form eigenbranch, then I cf = I. Therefore, we have the following dichotomy: Either the set I cf countable or the family t → q t has a real-analytic cusp form eigenbranch.

Proof of Proposition 3.8. Let {u j,t | j ∈ Z, t ∈ I} be as in Theorem 3.3. By Corollary 3.6, the zero set

Z j = {t | L(u j,t )} is countable.
If each eigenspace E of q t is one-dimensional, then there exists a unique j such that E equals the span of u j,t . Thus, if t does not belong to I mult or to any Z j , then t does not belong to I cf . In other words, I cf ⊂ ( Z j ) ∪ I mult . By Proposition 3.7, the set I mult is countable, and hence so is I cf .

Perturbation theory for hyperbolic triangles with one cusp

In this section we use the results of the previous section to explain how Theorem 1.2 can be deduced from the existence of a triangle with a cusp that has no nonconstant Neumann eigenfunctions. This fact may probably be considered as folklore and follows the general philosophy of using analyticity to prove generic spectral results (see [START_REF] Hillairet | Generic spectral simplicity of polygons[END_REF]). The main task here is thus construct a real-analytic family of quadratic forms that is associated with each real-analytic path in the moduli space of triangles. 4.1. The moduli space of triangles. First, we discuss the parametrization of the set of triangles with one cusp. The statement of Theorem 1.2 makes use of the fact that hyperbolic triangles with one cusp are parametrized by the two nonzero vertex angles. But in order to prove Theorem 1.2, it will be more convenient to use an alternate set of parameters.

For each geodesic triangle T in the hyperbolic upper half plane H 2 having (exactly) one cusp, there exists a unique c ∈ ]0, 1[ and w ∈

[2c, 1 + c[ so that T is isometric to the domain (8) T c,w = (x, y) | 0 ≤ x ≤ w, (x -c) 2 + y 2 > 1 .
See Figure 2. In this way, the set of hyperbolic triangles may be identified with the Euclidean triangle

M = {(c, w) | 0 ≤ c < 1, 2c < w < c + 1}.
We say that a subset of the set of triangles has measure zero if and only if the corresponding subset of M has measure zero. Similarly, a subset of the set of triangles is said to be a real-analytic curve if and only if the corresponding subset of M is a real-analytic curve. These notions are equivalent to those used in the statement of Theorem 1.2 because the relationship between the angles (θ 1 , θ 2 ) and the parameters (c, w) is real-analytic. Indeed, we have c = cos(θ 1 ) and cos(θ 2 ) = wc. See Figure 2.

To prove Theorem 1.2, we will apply perturbation theory. The following fact makes this approach feasible. 0 Let M cf denote the set of (c, w) ∈ M such that there exists β > 1 so that the modified quadratic from E β has a cusp form. To prove Theorem 1.2 it will suffice to show that M cf has measure zero and is a countable collection of nowhere dense sets.

✓ 1 ✓ 2 T c,w c w c + 1 c 1 Figure 2.

A family of diffeomorphisms.

To show that M cf is non-generic, we will use analytic perturbation theory and Proposition 3.8. In order to use analytic perturbation theory we will have to normalize the Hilbert space and the domains of the quadratic forms. To accomplish this, we let S = [0, 1] × [1, ∞[ and for each (c, w) we define a C 1 diffeomorphism ϕ c,w : T c,w → S such that 6

(1) The restriction of ϕ c,w is the identity for y > α = (β + 1)/2.

(2) For each path γ : I → M, the family t → ϕ c,w is a real-analytic path.

To construct ϕ c,w , we use the fact that the map (x, y) → x defines a fibration of T c,w over [0, w] and a fibration of S over [0, 1]. We define ϕ c,w by sending the fiber over {x} onto the fiber over {x/w}. Lemma 4.2. For each α ∈ ]0, α[, there exists a unique cubic polynomial B α so that

• B α (α) = 1,
5 See also [Sarnak] for the case of triangles that are fundamental domains for the Hecke groups.

6 In [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF], we considered a simpler mapping from T 0,t onto S. The mapping that we define here is more complicated because it must preserve the notion of zeroth Fourier coefficient for all y above some point. In particular, the vertical displacement of vertical lines should not depend on x for large y. In [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF], we considered Dirichlet boundary conditions, and in that context there is no need to truncate the zeroth Fourier coefficient.

• B α (0) = α. • B α (α) = α, • B α (α) = 1
The coefficients of B α are real analytic for α ∈]0, α[. Moreover if α > 2 + √ 3 then , for all α ∈ (0, 1], and y ∈ [0, α], we have B α (y) > 0.

Proof. Since α = 0, satisfying the two conditions on B α is equivalent to the existence of some A such that

B α (y) = A • y(α -y) + 1 α • y + α α • (α -y).
Denote by Q α the cubic polynomial defined by

Q α (y) = y α z(α -z) dz.
By integration, there exists some C such that

B α (y) = A • Q α (y) + 1 α • y 2 2 - α α • (α -y) 2 2 + C.
Evaluating at α and using the condition on B α (α) we find

C = 1 - α 2 2α - α(α -α) 2 2α . Observe that Q α (α) = 0 if α ∈ [0, α[
and, under this condition, we can solve the last equation on B α to find A. We obtain

Q(α)A = α - α 2 -C = 1 2α α 2 + α 2 -α(α -α) 2 -2α = 1 -α 2α α 2 + α 2 -2α(α + 1) , = (1 -α) α 2 -2αα + α 2 -2α 2α
It follows that we have a unique solution provided α = 0 and 0 < α < α, and that the coefficients are real-analytic in α.

We now check the last statement. For α = 1, we have B α (y) = y so that the claim follows. For α < 1 we observe that the numerator of A is a cubic polynomial that has three roots at 1, α ± √ 2α. Thus, if α > 2 + √ 3 then 1 is the smallest root. Since this cubic polynomial is positive for large negative α and the denominator also is positive, it follows that A is positive for 0 < α < 1. So B α is a concave function, and by construction B α (0) > 0 and B α (α) > 0. The claim follows.

Notation 4.3. We will use the notation B α (y) as well as the notation B(α, y).

Define F c,x : R → R by F c (x, y) = B (f c (x), y) if y ≤ α y if y ≥ α.
where

f c (x) = 1 -(x -c) 2 . Define ϕ c,w : T c,w → S by ϕ c,w (x, y) = (x/w, F c (x, y)) .
Observe that the conditions on B imply that F, ∂ x F c and ∂ y F c are continuous on T c,w so that ϕ c,w is C 1 . We will use this function ϕ c,w to normalize the triangle T c,w . This is made possible by the following lemma.

Lemma 4.4. Suppose that ∂ y B(f c (x), y) > 0 for each (x, y) ∈ T c,w ∩ {y ≤ α} then the map ϕ c,w is a C 1 diffeomorphism from T c,w onto S.
In particular, for each α > 2 + √ 3 and each (c, w) ∈ M, the mapping ϕ c,w is a C 1 diffeomorphism from T c,w onto S.

Proof. It suffices to show that the map

F c,x is a C 1 diffeomorphism from [f c (x), ∞[ onto [1, ∞[. By assumption ∂ y B (f c (x), y) > 0 for each x. We have B(f c (x), f c (x)) = 1, B(f c (x), α) = α and ∂ y B(f c (x), α) = 1. Since F c,x is the identity for y > α, we find that F c,x is a C 1 diffeomorphism from [f c (x), ∞[ onto [1, ∞].
For each α and each M ⊂ M, we define X α,M and A α,M by

X α,M := {(x, y, c, w) | (c, w) ∈ M, (x, y) ∈ T c,w , y ≤ α} , A α,M := {(a, b, c, w) | (c, w) ∈ M, (a, b) ∈ S, b ≤ α} .
We then have Lemma 4.5. For each α, M , each of the following maps is analytic on X α,M :

(1) (x, y, c, w) → ϕ c,w (x, y) (2) (x, y, c, w) → ∂ x ϕ c,w (x, y) (3) (x, y, c, w) → ∂ y ϕ c,w (x, y)
If, for each (c, w) ∈ M, the assumption of Lemma 4.4 holds then the map (a, b, c, w) → ϕ -1 c,w (a, b) is also analytic on A α,M . Moreover, each restriction extends analytically to an open neighbourhood.

Proof. The coefficients of the cubic polynomial B α depend analytically on α and hence (α, y) → B(α, y) is analytic. The map (c, x) → f c (x) is analytic and hence it follows that map (1) is analytic. Maps (2) and (3) are therefore analytic.

Since (α, y) → B(α, y) is analytic and ∂ y B(α, y) > 0 for y > 0, the implicit function theorem (Theorem 2.1.2 in [Hörm]) implies that there exists a function (α, b) → Y α (b) which is analytic and a solution to

B α (Y α (b)) -b = 0.
We then have

ϕ -1 c,w (a, b) = w • a, Y fc(w•a) (b) , and, since (c, x) → f c (x) is analytic, the claim follows.
In the rest of the section, α > 2 + √ 3 will be fixed so that we can use lemmas 4.4 and 4.5.

4.3.

The quadratic form with fixed domain. We use the family of diffeomorphisms ϕ c,w to define a quadratic form q t with domain

H 1 β (S) ⊂ L 2 β (S) that is unitarily equivalent to E β on H 1 β (T c,w ) ⊂ L 2 β (T c,w ). Define Φ c,w : L 2 (S, da db/b 2 ) → L 2 (T c,w , dx dy/y 2 ) by Φ c,w (u) = y • | det(Jac(ϕ c,w ))| • u b • ϕ c,w
where Jac is the operator that returns the Jacobian matrix of a map.

Lemma 4.6. The isometry Φ c,w is a unitary isomorphism from

L 2 β (S) onto L 2 β (T c,w ) and it maps H 1 β (S) onto H 1 β (T c,w ). On functions that are supported in b ≥ α, Φ c,w coincides with Φ w .
Proof. We have

Tc,w |Φ c,w (u)| 2 dx dy y 2 = Tc,w u b • ϕ c,w 2 | det(Jac(ϕ c,w ))| • y 2 • dx dy y 2 = S u b 2 da db. It follows that Φ c,w is a unitary isomorphism from L 2 β (S) onto L 2 β (T c,w ). Let u ∈ H 1 β (S). Since ϕ c,w is a C 1 diffeomorphism and | det(Jac(ϕ c,w ))| is continuous on T c,w and smooth away from y = β, then Φ c,w (u) is continuous and in H 1 β (T c,w \ {y = β}). The jump formula implies that Φ c,w (u) ∈ H 1 β (T c,w
). Since, for y > α, ϕ c,w (x, y) = ( x w , y), the last statement is a direct verification. Definition 4.7. Define the quadratic form q c,w on H 1 β (S) ⊂ L 2 (S, da db/b 2 ) by q c,w (u) := E β • Φ c,w (u).

Lemma 4.8. u is a cusp form for q c,w if and only if v = Φ c,w • u is a cusp form for E on T c,w .

Proof. If y ≥ α, then ϕ c,w (x, y) = (x/w, y). It follows that if y ≥ α, then u 0 (y) = 0 if and only if v 0 (y) = 0. For y ≥ 1, the function v 0 is a solution to a second order ordinary differential equation, and hence v 0 (y) = 0 for y ≥ α if and only if v 0 (y) = 0 for y ≥ 1.

It will be convenient to have the following alternate form for q t . Proposition 4.9. We have

(9) q c,w (u) = S ∇(ρ c,w • u) • Q c,w • ∇(ρ c,w • u) * da db
where ρ c,w : S → R is defined by

ρ c,w = y • | det(Jac(ϕ c,w )| • ϕ -1 c,w b ,
and

Q c,w : S → GL 2 (R) is defined by (10) Q c,w • ϕ c,w = 1 det (Jac(ϕ c,w )) • Jac(ϕ c,w ) • Jac(ϕ c,w ) * .
Moreover, q c,w represents the cusp of width w for y ≥ α.

Proof. This is a straightforward calculation using the chain rule and the change of variables formula.

4.4.

Analytic paths in M. Let I = ]t -, t + [ and let γ : I → M be a real-analytic path.

Theorem 4.10. The family quadratic forms t → q γ(t) is analytic of type (a) in the sense of [Kato].

Proof. For each t, the quadratic form q γ(t) = E β •Φ γ(t) is a closed form with domain H 1 β (S). It suffices to show that for each u ∈ H 1 β (S), the function t → q γ(t) (u) is real-analytic.

By Proposition 4.9, we have

(11) q c,w (u) = α 1 1 0 I t da db + ∞ α 1 0 I t da db.
where

I t = ∇(ρ γ(t) • u) • Q γ(t) • ∇(ρ γ(t) • u) * . If (a, b) ∈ [0, 1] × [α, ∞[, then the matrix Q(a, b) is given by Q = 1 w 2 t 0 0 1 . and ρ γ(t) (a, b) = 1.
Thus, the second integral on the right of (11) depends analytically on t.

It remains to consider the integral over [0, 1] × [1, α]. The integrand I t can be expanded into a finite sum of terms of the form

(12) α 1 1 0 w(a, b) • H(t, a, b)dadb,
where H is a function that is obtained by multiplying ρ, or its derivatives and the entries of Q and w is one of the L 1 functions obtained by making the product v 1 v 2 where both v i are either u or one of its partial derivatives.

By Lemma 4.5, the coordinates of ϕ c,w and ϕ -1 c,w are analytic functions of (c, w).

It follows that (t, a, b) → ρ γ(t) (a, b) and (t, a, b) → Q ij (t)(a, b) are analytic (in a neighborhood of I × [0, 1] × [1, α]
). In all possible choices, the function H then is analytic.

The analyticity of t → q γ(t) (u) follows from Lemma 4.11 below.

Lemma 4.11.

If H : I ×[0, 1]×[1, α] is analytic, then for each p ∈ L 1 ([0, 1]×[1, α]), the function (13) t -→ α 1 1 0 p(a, b) • H(t, a, b) dadb is analytic on I.
Proof.

There exists an open neighborhood

U ⊂ C 3 of I × [0, 1] × [1, α] such that the map h extends to a holomorphic function on U . Since [0, 1] × [1, α] is compact, H(t, a, b) -H(s, a, b) t -s
converges uniformly to d dt H(s, a, b) as t approaches s. It follows that the (complex) t-derivative of the map in (13) exists at each t ∈ U . 4.5. Generic absence of cusp forms. Given Theorem 4.10, we now explain why the generic triangle T c,w has no cusp forms provided that one triangle has none.

Theorem 4.12. If there exists a point (c 0 , w 0 ) ∈ M such that E on L 2 β (T c0,w0 , dm) has no nonconstant eigenfunction, then M cf has measure zero and is a countable union of nowhere dense sets.

Proof. By Proposition 4.1, the quadratic form E β on L 2 β (T c0,w0 , dm) has no cusp form, and hence by Lemma 4.8, the quadratic form q c0,w0 has no cusp form.

To show that M cf has measure zero, we apply Fubini's theorem in a fashion similar to [START_REF] Hillairet | Generic spectral simplicity of polygons[END_REF]: Let γ c0 (t) = (c 0 , w 0 +t) and apply Lemma 3.8 to find that the set B of w such that (c 0 , w) ∈ M cf is countable. For each w / ∈ B, let γ w (s) = (c 0 + s, w) and apply Lemma 3.8 to find that the intersection I w of the line {(c, w) | c ∈ R} with M cf is countable. Hence for each w / ∈ B, the set I w has measure zero with respect to the linear measure da. Hence, the measure of M cf equals the measure of w∈B I w . Since B is countable, the measure equals zero.

For N ∈ Z, let M N cf be the set of (c, w) ∈ M such that E on L 2 (T c,w , dm) has a cusp form with eigenvalue at most N . Using the continuity of (c, w) → q c,w and the continuity of linear functional L, one can show that M N cf is a closed. Thus, it suffices to show that

M N cf is nowhere dense. Given a point (c, w) ∈ M N cf , let γ : [0, 1] → M be a real-analytic path joining (c 0 , w 0 ) to (c, w). Since E β on L 2 β (T c0,w0
, dm) has no cusp forms, the family t → q γ(t) has no cusp form eigenfunction branch. It follows from Lemma 3.8, that for each open neighborhood U of (c, w), there exists t ∈ [0, 1] such that γ(t) ∈ U and q γ(t) has no cusp forms. Hence M N cf is nowhere dense.

The family T t

In the remainder of this paper we consider the specific family of triangles T t = T 0,t defined in the introduction. In particular, we will study the spectral properties of q 0,t for small t. The family q 0,t of quadratic forms does not converge as t tends to zero nor do its real-analytic eigenbranches. But a simple renormalization will give convergence.

Fix β > 1 and α such that 1 < α < β. Let B be the function defined in Lemma 4.2. When α tends to 1, the function y → ∂ y B(α, y) converges to 1 uniformly for y ∈ [0, α]. Thus, there exists η such that if 1η ≤ α ≤ 1 and 0 ≤ y ≤ α then ∂ y B(α, y) ≥ 1 2 . Choose t 0 such that 1t 2 0 < η then, for each t < t 0 and each (x, y) ∈ T t ∩{y ≤ α} f 0 (x) < η so that ∂ y B(f 0 (x), y) > 0. We may thus use Lemmas 4.4 and 4.5. The methods and results of section 4.4 then apply and we define the quadratic form q 0,t as previously.

For each t ∈ [0, t 0 [ , define the renormalized quadratic form by

q t := t 2 • q 0,t
with domain H 1 β (S). By Theorem 4.10, the family t → q t is real-analytic of type (a) for t ∈ ]0, 1[. In particular, the results of §4.5 apply.

To study the limiting properties of the family q t , we re-express q t in a more convenient form: For each

C 1 function w : S → C define (14) ∇ t w = (∂ x w, t • ∂ y w) .
Recall that Y α is the inverse of B α and set f

(x) = f 0 (x) = √ 1 -x 2 . Define (15) ρ t (a, b) = Y (f (ta)b) b • (∂ y Y (f (at), b)) 1 2 and Q t (a, b) = (16) (∂ y (Y (f (at), b)) -1 • 1 (∂ α B • Y ) • f (ta) (∂ α B • Y ) • f (ta) ((∂ α B • Y ) • f (ta)) 2 + (∂ y B) 2
where the subscript (or first argument) in each Y and B is f (t • a). When comparing ρ and ρ (Q and Q) we see that we only miss some powers of t than eventually cancel in the computation leading to Proposition 4.9.

This shows that for each u ∈ H 1 β (S)

(17) q t (u) = S- ∇( ρ t • u) • Q t • ∇( ρ t • u) * da db + S+ ∇u • ∇u * da db
where

S -= [0, 1] × [1, α] and S + = [0, 1] × [α, ∞].
By arguing as in the proof of Theorem 4.10, one can show that t → q t (u) is analytic at t = 0.7

We will now compute the first few terms in the Taylor series in t for ρ and Q. These functions are analytic on a neighbourhood of [0, t 0 [×S -. In particular, in the following, the expressions like O(t 2 ) are uniform with respect to (a, b) ∈ S -and may be differentiated with respect to t, a and b.

We first compute

(18) f (ta) = 1 - 1 2 • t 2 • a 2 + O(t 4 ),
and

(19) f (ta) = -t • a + O(t 3 ). Since α → Y α is analytic and Y 1 (b) = b, it follows from (18) that (20) Y f (ta) (b) = b + O(t 2 ).
Moreover, using analyticity, this asymptotic expansion may be differentiated with respect to (a, b). We thus obtain,

(21) Y f (ta) (b) = 1 + O(t 2 ).
Substituting these into (15), and differentiating, we find that

(22) ρ t (a, b) = 1 + O(t 2 ), ∇ a,b ρ t (a, b) = O(t 2 ).
Using (18), ( 19), (20), and (21) we find that

(23) Q t (a, b) = I + t • a • p(b) • 0 1 1 0 + O(t 2 )
where I is the identity matrix, O(t 2 ) is a matrix whose operator norm is bounded by a constant times t 2 as t tends to zero, and p is the polynomial

(24) p(b) = -∂ α B α (b)| α=1 .
To prove Theorem 1.1 we will need to know that p(1) = 0.

Lemma 5.1. p(1) = 1.

Proof. By construction we have B(α, α) = 1. By differentiating with respect to α and setting α = 1 we get

∂ α B(1, 1) + ∂ y B(1, 1) = 0. Since ∂ y B(α, y) = 1 + O((α -1)
2 ) the claim follows.

Asymptotic separation of variables

In this section we apply the method of asymptotic separation variables developed in [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF] to the family q t . Using the small t asymptotics derived in §5, we approximate q t to first order with a family of quadratic forms a t for which separation of variables apply. We also derive a non-concentration estimate for eigenfunctions of q t . Notation 6.1. In this section and the following sections, we will use (x, y) in place of (a, b) as coordinates for S = [0, 1] × [1, ∞[ and unless it is specified otherwise • is the norm in L 2 (S, y -2 dxdy).

6.1. Asymptotic approximation. We begin by using the expansions obtained in §5 to determine the forms used to approximate q t . In particular, by substituting the expansions ( 22) and ( 23) into (17) we are led to define (25)

a t (u, v) = S ∇u • ∇v dx dy = S u x • v x + t 2 • u y • v y dx dy and (26) b t (u, v) = S- ∇u • B(x, y) • ∇v dx dy
where the operator ∇ is defined by (14

), S -= [0, 1] × [1, α], and
B(x, y) = x • p(y) • 0 1 1 0 .
We wish to show that q t is asymptotic to a t at first order in the sense of [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF]. It will also be used to help derive a key estimate for crossing eigenbranches. However, although a t is a positive quadratic form, the bottom of its spectrum tends to 0 so that it is more convenient to use the quadratic form a t that we now define to control quantities. Definition 6.2. The quadratic form a t is defined on dom(a t ) by

a t (v) = a t (v) + v .
The following proposition can be seen as the beginning of an asymptotic expansion for q t . Proposition 6.3. There exists C such that for each u, v ∈ H 1 β (S)

(27) |q t (u, v) -a t (u, v) -t • b t (u, v)| ≤ C • t 2 • a t (u) 1 2 a t (v) 1 2 .
Proof. We have

(28) ∇ t ρ t • u = ρ t • ∇ t u + u • ∇ t ρ t .
If y ≥ α, then ρ t is identically equal to 1 and Q t is identically equal to I. Hence, by subsituting (28) into (17), we find that q t (u, v)a t (u, v)t • b t (u, v) is the sum of the following four terms S-

∇ t u • (ρ 2 • Q t -I -t • B) • ∇ t v dx dy , (29)
S-

ρ t • v • ( ∇ t ρ t • Q t • ∇ t u) dx dy , (30) S- ρ t • u • ∇ t ρ t • Q t • ∇ t v dx dy , (31) S- ∇ t ρ t • Q t • ∇ t ρ t • u • v dx dy. ( 32
)
In order to estimate these four terms, we use the asymptotic expansions of §5. For example, by (23) we have that ( 29) is equal to

S ∇ t u • O(t 2 ) • ∇ t v dx dy
Since the operator norm of the matrix O(t 2 ) is bounded by a constant C times t 2 , we can apply the Cauchy-Schwarz inequality to find that the norm of ( 23) is bounded by

C • t 2 • a t (u) 1 2 • a t (v) 1 2 .
Similar arguments show that there is a constant C so that

• (30) is bounded above by C • t 2 • a t (u) 1 2 • v 1 2 • (31) is bounded above by C • t 2 u 1 2 • a t (v) 1 2 • (32) is bounded above by C • t 2 • u 1 2 • v 1 2 .
The claim follows.

6.2. The spectrum of a t via separation of variables. We recall the Fourier decomposition of section 2.2. Since now w = 1 we thus have, for each u ∈ L 2 (S, dxdy y 2 )

u k (y) = 1 0 u(x, y) • e k (x) dx.
where the latter makes sense for almost every y and defines an element of L 2 ((1, ∞), dy y 2 ). As above, let D(S) denote the set of functions v : S → C such that v is the restriction of a compactly supported, smooth function defined in a neighborhood of S. If u ∈ D, then each u k is smooth, and a straightforward computation shows that 8

a t (u) = k∈N a t u k ⊗ e k = k∈N ∞ 1 t 2 • ∂ y u k (y) 2 + (kπ) 2 • u k (y) 2 dy (33)
8 Here ⊗ is the operation defined by (v ⊗ w)(x, y) = v(y) • w(x).

We define D([1, ∞[) to be the set of compactly supported, smooth functions defined on [1, ∞[. For v ∈ D([1, ∞)), each integer k, and each t > 0, we define (34)

a k t (v) = ∞ 1 t 2 • v (y) 2 + (kπ) 2 • v(y) 2 dy.
For v, w in L 2 ([1, ∞[ , y -2 dx dy), the inner product is defined by

u, v y = ∞ 1 u(y) • v(y) dy y 2 .
Let L 2 β denote the subspace consisting of those functions whose support lies in [1, β]. For each k ∈ N, the quadratic form a k t extends to a closed, densely defined form on the completion of

D([1, ∞[) with respect to v → a k t (v) 1 2 + v, v 1 2
y . For k = 0, we will restrict the domain of a k t to be the completion of those smooth functions whose support lies in [1, β].

If u is an eigenfunction of a t with eigenvalue λ, then for each v in the domain of a k t , we have

a k t (u k , v) = a t (u, v ⊗ e k ) = λ • u, v ⊗ e k = λ • u k , v y ,
and hence u k is an eigenfunction of a k t with eigenvalue λ with respect to •, • y . Thus, each eigefunction u of a t may be written uniquely as

u = k∈N u k ⊗ e k
where the k th Fourier coefficient u k is an eigenfunction of a k t . Moreover, the spectrum of a t with respect to •, • is the union of the spectra of a k t with respect to •, • y . In what follows we will often suppress the subscript y from the notation.

The next two lemmas identify the eigenfunctions of a k t for each k. We begin with the nonzero modes. Lemma 6.4. For each t and k and eigenvalue λ of a k t with respect to •, • y , the associated eigenspace consists of functions of the form y → f (πk

• y/t) such that i) f (z) = 1 - λ (t • z) 2 • f (z), ii) f ∈ L 2 [1, ∞), dy y 2 , iii) f (πk/t) = 0.
Moreover, when t varies, the spectrum is organized into eigenvalues branches λ i (t). Each of these is a growing function of t and

lim t→0 λ i (t) = k 2 π 2 .
Proof. Integrate by parts as in the proof of Lemma 6.5, make the change of variables z → πk • y/2t, and use the boundary conditions.

The second part follows from Theorem 3.3.

The zero modes are given by the following lemma.

Lemma 6.5. The spectrum of a 0 t with respect to •, • y is the set

(35) t 2 • 1 4 + r 2 r > 0 and 2r = tan(r • ln(β)) .
The eigenspace associated to t 2 • ( 1 4 + r 2 ) is spanned by the eigenvector

(36) ψ(y) = y 1 2 • cos (r ln(y)) - y 1 2 2r
• sin (r ln(y)) .

Proof. Suppose that v is an eigenfunction, that is a 0 t (v, w) = λ • v, w for all w. This implies first that

-t 2 • v (y) = - λ y 2 • v(y)
. holds in the distributional sense. Ellipticity then yield that v is smooth. Moreover, by integrating by parts against a smooth function that is identically equal to 1 near y = 1, we also find that v (0) = 0. Let s be such that s • (1s) = λ/t 2 . Then two linearly independent solutions are given by y s and y 1-s if s = 1 2 and by y 1 2 and y

1 2 • ln(y) if s = 1
2 . The condition that λ/t 2 is real and nonnegative implies either that s = 1 2 + ir with r > 0, that s ∈ [0, 1/2) or that s = 1 2 . If Re(s) = 1/2, then the boundary conditions v (1) = 0 and v(β) = 0 imply that the solutions take the form given in in (36) with 2r = tan(r • ln(β)). If s ∈ [0, 1/2], then there are no solutions that satisfy the boundary conditions.

As a consequence of the identification of the eigenfunctions we have the following Poincaré type inequality. Lemma 6.6. For each t ≤ 2π, and each u ∈ H 1 β (S) we have

(37) a t (u) ≥ t 2 4 • n(u).
Proof. We have

a t (u) = k a k t (u k ) and n(u) = K ∞ 1 u k 2 dy y 2 .
Lemmas 6.4 and 6.5 imply

a 0 t (u 0 ) ≥ t 2 4 • β 1 u 0 2 dy y 2 , and (38)
a k t (u k ) ≥ k 2 π 2 • ∞ 1 u k 2 dy y 2
for k > 0.

In the sequel we will use different kind of projections associated either with the Fourier decomposition u = u k ⊗ e k or with the spectral decomposition of a t . More precisely, for each ∈ N define the orthogonal projection Π :

L 2 β (S) → L 2 β (S) by (39) Π (v) = v (y) • e (x),
and let V denote the image of Π . For each k ∈ Z, we define

(40) Π <k (v) = <k v (y) • e (x),
the projection onto 0≤ <k V .

We also define P λ at to be the orthogonal projection onto the eigenspace of a t associated to the eigenvalue λ of a t . For a fixed interval I, define the a t -spectral projection in the energy interval I to be

P I at (v) := λ∈spec(at)∩I P λ at (v).
For each eigenvalue λ of a t , the associated a t -eigenspace W λ is the orthogonal direct sum ⊕ (W λ ⊗ vect(e )) where W λ is the λ-eigenspace of a t and vect(e ) is the span of e . It follows that

(41) Π P I at (v) := λ∈spec(a t )∩I P λ at (Π (v)).
More generally, for a quadratic form b the notation P I b will always denote the spectral projection onto the interval I.

6.

3. Asymptotic at first order. We prove that a t and q t are asymptotic at first order in the sense of [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF]. In the following, we let qt (resp. ȧt ) denote the derivative of q t (resp. a t ) in t. Proposition 6.7 (Asymptotic at first order). There exists a constant C and t 0 such that, for all u, v ∈ H 1 β and all t ≤ t 0 ,

|q t (u, v) -a t (u, v)| ≤ C • t • a t (u) 1 2 • a t (v) 1 2 (42) | qt (u) -ȧt (u)| ≤ C • a t (u). ( 43
)
Proof. One argues as in the proof of Proposition 6.3 paying a little more attention to the terms (30), (31), and (32). For example, to estimate (30), use the Cauchy-Schwarz inequality and (22) to obtain

(44) | ∇ t ρ t • ∇ t u| ≤ | ∇ t ρ t | • | ∇ t u| = O(t 2 ) • ∇ t u .
The Cauchy-Schwarz inequality and Lemma 6.5 give

(45) S |v| • | ∇ t u| dx dy ≤ 1 t • 1/4 + r 2 0 • a t (u) 1 2 • a t (v) 1 2 .
By combining (44) and (45) and using (22) we find that

S | ρ t | • |v| • | ∇ t ρ t • Q t • ∇ t u| dx dy = O(t) • a t (u) 1 2 • a t (v) 1 2 .
Switching the roles of u and v, we obtain the same bound for the expression in (31). Similar methods apply to bound the other terms.

The estimate for qȧ is obtained in a similar way.

Limits of eigenvalue branches

Since q t is asymptotic to a t at first order and a t and ȧt are nonnegative quadratic forms, each real-analytic eigenvalue branch E t of q t converges to a finite limit E 0 as t tends to zero (Theorem 3.4 of [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF]). For the Dirichlet eigenvalue problem on T t , we showed in [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF] that each limit E 0 has the form (πk) 2 where k is an integer. The methods of [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF] can be applied to show that the same fact is true in the present context. In this section we highlight the necessary modifications. We also show that if the eigenvalue branch is associated to a cusp form, then k must be positive. This latter fact will be used crucially in the proof of Theorem 1.1. 7.1. Non-concentration and first variation. The proof of convergence depends crucially on the following 'non-concentration' result proved for the Dirichlet problem in [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF].

We will denote by D the domain of the quadratic form a t and by a t the quadratic form defined for vinD by

a t (v) = a t (v) + v .
Proposition 7.1. (Compare Proposition 9.1 of [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF]) Let ∈ N, let K be a compact subset of ](π ) 2 , ∞[, and let C > 0. There exist positive constants t 0 and κ (that only depend on , K and C) so that if E ∈ K, if t < t 0 , and if for each w ∈ D , the function v ∈ D satisfies

a t (v, w) -E • v, w ≤ C • t • w • v , then (46) ∞ 1 E y 2 -( π) 2 • |v(y)| 2 dy ≥ κ • v 2 .
Proof. If > 0 this is Proposition 9.1 of [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF] with µ = (π ) 2 and σ(y) = y -2 . If = 0 and we let κ = inf(K) > 0, then (46) holds.

In the language of semi-classical analysis, Propostion 7.1 asserts that a quasimode v of order t at energy E does not concentrate at y = √ E/( π) if = 0. In §12 of [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF], we used non-concentration to derive indirect estimates for ȧ. The following proposition and corollary make these estimates more transparent and simpler to apply. Proposition 7.2. Let ∈ N and let K ⊂ ](π ) 2 , ∞[ be compact. For each > 0, there exists κ > 0 and t 0 > 0 such that for each v ∈ D and t < t 0 (47)

v 2 ≤ t κ ȧ t (v) + ε t 2 N (v, E) 2 .
where

N (v, E) = sup w∈D |a t (v, w) -E v, w | a t (w) 1 2
Proof. From (34) we find that ȧ t (v) = 2t v (y) 2 and hence

(48) t • ȧ t (v) = 2 • E y 2 -(π ) 2 • v 2 + 2 a t (v) -E • v 2 .
If the claim is not true, then for each κ > 0, there exists a sequence (t n ) n≥1 tending to zero and sequences (ṽ

n ) n≥1 , ṽn ∈ D, (E n ) n≥1 , E n ∈ K such that (49) ṽn 2 ≥ t n κ • ȧ (ṽ n ) + • N (ṽ n , E n ) 2 t 2 n .
In particular, since ȧ ≥ 0, we have

N (ṽ n , E n ) 2 ≤ t 2 • ṽn 2 . It follows that for each w ∈ D (50) | a t (ṽ n , w) -(E n + 1) • ṽn , w | ≤ t n √ • ṽn • a t (w) 1 2 .
We fix δ such that [-δ, δ] + K ⊂ (( π) 2 , ∞), and we set v n = P

[En-δ,En+δ] at (ṽ n ) = P [En+1-δ,En+1+δ] a t (ṽ n ). Reasoning as in the proof of Lemma 2.3 in [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF] we find that

a t (ṽ n -v n ) ≤ t 2 n ε ṽn 2 1 + E n δ .
Observe that the sequences (v n ) n≥1 , (t n ) n≥1 and (E n ) n≥1 depend on the initial choice of κ but the preceding estimate gives a constant C that is independent of κ such that

a t (ṽ n -v n ) ≤ C • t 2 n • ṽn 2 .
This implies in particular ṽn -

v n 2 ≤ C • t 2 n • ṽn 2 so that, for n large enough we have v n ≤ ṽn ≤ 2 v n .
In equation ( 50) we replace the test function w by P

[En-δ,En+δ] a t (w) and use that the spectral projector is self-adjoint and commutes with a t . We obtain that for each w ∈ D ,

| a t (v n , w) -(E n + 1) • v n , w | ≤ t n √ • ṽn • a t P [En-δ,En+δ] a t (w) 1 2 ≤ C • t n • v n w , (51)
where we have used that v n is controlling ṽn and that

a t P [En-δ,En+δ] a t (w) ≤ (sup(K) + δ) w 2
by definition of a spectral projector. Since ȧ t ≤ 2 t • a t and ȧ t is a non-negative quadratic form, we also have

ȧ t (ṽ n ) 1 2 -ȧ t (v n ) 1 2 ≤ ȧ t (ṽ n -v n ) 1 2 ≤ C t a t (ṽ n -v n ) 1 2 ≤ C • √ t • v n .
Equation (51) implies that we may use Proposition 7.1 to find

∞ 1 E n y 2 -( π) 2 |v n (y)| 2 dy ≥ κ v n 2 .
Since (51) also implies

a t (v n ) -E n v n 2 ≤ C • t n • v n 2 , using (48) we find that (52) t n ȧ t (v n ) ≥ (κ -C • t n ) v n 2 .
On the other hand, the contradiction assumption implies that

κ v n 2 ≥ t n ȧ t (ṽ n ) ≥ √ t n ȧ t (v n ) 1 2 -Ct n v n 2 2 ≥ (κ -C • t n ) 1 2 -Ct n 2 • v n 2 ≥ (κ -C • t n ) v n 2 . ( 53
)
The implied constant C does not depend on κ so if we take κ < κ then choosing t n small enough yields the contradiction.

This proposition yields an estimate for ȧ(w) from below in terms of the projection Π <k w.

Corollary 7.3. Let k ∈ Z + and let K ⊂ R + be a compact subset of ](πk) 2 , ∞[. For each ε > 0, there exists κ > 0 t 0 > 0 such that if E ∈ K, w ∈ dom(a t ), and t < t 0 , then

(54) ȧt (w) ≥ κ t • Π <k (w) 2 - ε t 2 • N (w, E) 2 ,
where

N (w, E) = sup v∈dom(at) |a t (w, v) -E • w, v | a t (v) 1 2 . Remark 7.4. The functional v → N (v, E) is equivalent to the H -1 -norm of (A t - E)(v)
where here A t is the operator such that A t u, v = a t (u, v) for each u, v ∈ dom(a t ).

Proof of 7.3. Since ȧt is block diagonal with respect to the sum V and ȧ ≥ 0, we have

ȧ(w) = ȧ ∞ =0 w ⊗ e = ȧ (w ) ≥ k-1 =0 ȧ (w ).
We may apply Proposition 7.1 with ε = ε /k to each term on the right hand side to find that

ȧ(w) ≥ κ t • k-1 =0 w 2 - ε k • t 2 • k-1 =0 N (w , E) 2 .
where κ is the minimum of the κ coming from Proposition 46. For each , and v ∈ D , we have

|a t (w , v) -E • w , v | a t (v) 1 2 = |a t (w ⊗ e , v ⊗ e ) -E • w ⊗ e , v ⊗ e |. a t (v ⊗ e ) 1 2 = |a t (w, v ⊗ e ) -E • w, v ⊗ e |. a t (v ⊗ e ) 1 2
and hence N (w , E) ≤ N (w, E). We also have, <k w 2 = Π <k (w) 2 , and the claim follows.

7.2. The spectral projection w t . The bounds proved in §7.1 depend on a bound on N (w, E). In this subsection, we show that if w is an a t -spectral projection of a q t -eigenfunction in an interval containing the eigenvalue E, then N (w, E) is of order t.

We start with a real-analytic eigenfunction branch u t for q t with associated realanalytic eigenvalue branch E t . We let (55) w I t := P I at (u t ) denote the associated spectral projection.

Let E 0 denote the limit of E t as t tends to zero. The following two lemmas express the fact that the projection w I t is an order t quasimode for a t at energy E t . First we have a Lemma that is comparable to Lemma 2.3 in [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF].

Lemma 7.5. If I is a compact interval whose interior contains E 0 , then there exist t 0 > 0 and C such that if t < t 0 , then

a t u t -w I t + u t -w I t 2 ≤ C • t 2 • u t 2
Proof. Using the fact that u t is an eigenfunction of q t and that a t and q t are asymptotic at first order, for each w ∈ H 1

β (56) |a t (u t , w) -E t u t , w | ≤ C • t • a t (u t ) 1 2 a t (w) 1 2 .
Observe that letting w = u t yields that a t (u t ) ≤ Et 1-Ct 2 u t 2 . Moreover the former equation can be rewritten as

a t (u t , w) -Ẽt u t , w ≤ C • t • a t (u t ) 1 2 a t (w) 1 2 ,
where Ẽt := E t + 1. We may now follow the proof of Lemma 2.3 in [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF] observing that P I at = P I+{1} at

. This yields a constant C such that

a t (u t -w I t ) + u t -w I t 2 ≤ C • t 2 • a t (u t ) ≤ C • t 2 1 -C • t 2 u t 2 , ≤ C • t 2 u t 2 .
The claim follows.

Observe that by orthogonality the preceding Lemma also imples the following estimate that roughly expresses the fact that all the mass of u t is supported by w

I t (57) (1 -Ct 2 ) u t 2 ≤ w I t 2 ≤ u t 2 .
Lemma 7.6. If I is a compact interval whose interior contains E 0 , then there exist t 0 > 0 and C such that if t < t 0 , then

N w I t , E t ≤ C • t • w I t . Proof. For each w ∈ H 1 β , we have a t (w t , w) = a t (u t , w) -a t (u t -w t , w)
so that the Cauchy-Schwarz inequality and the preceding lemma imply

|a t (w t , w) -a t (u t , w)| ≤ C • t u t a t (w) 1
2 . We also have using Cauchy-Schwarz and the preceding lemma.

u t , w -w I t , w ≤ C • t • u t w .
We now start again from (56). First in the bounding term, we have already seen that we could replace ãt (u t ) 1 2 by C u t . Thus from the triangle inequality, (56) and the two preceding estimates we obtain

a t (w I t , w) -E w I t , w ≤ C • t u t • a t (w) 1 2 + w ≤ C • t • u t • a t (w) 1 2 .
The claim follows using (57).

Lemma 7.6 has the following corollary that expresses, in the language of semiclassical analysis, that w I t is an order t quasimode.

Corollary 7.7. If I is a compact interval that contains E 0 there exists C and t 0 > 0 such that, for t < t 0 and each v ∈ dom(a t ), we have [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF]). Let (E t , u t ) be an eigenbranch of q t then there exists k ∈ Z such that

a t (w I t , v) -E t w I t , v ≤ C • t • w I t • v . Proof.
(58) lim t→0 E t = (k • π) 2 .
Proof. Suppose to the contrary that E 0 is not of the form (k

• π) 2 where k is an integer. Let n = inf{ ∈ Z | (π ) 2 > E 0 }. Choose a compact interval I ⊂ ](n -1) 2 π 2 , n 2 π 2 [ whose interior contains E 0 .
Let u t be a real-analytic eigenfunction branch of q t associated to E t . Let w t := P I at (u t ) be the projection of u t onto the modes of a t that have energy lying in I.

If ≥ n, then each eigenvalue of a t is at least (πn) 2 . (This follows from Lemma A.1 or simply the nonnegativity of ȧt .) Thus, since sup(I) < (πn) 2 , equation (41) implies that Π <n (w t ) = w t . Let C be as in Lemma 7.6, and apply Lemma 7.3 with ε = 1/(2C 2 ) to obtain κ so that ȧt (w

t ) ≥ κ 2t • w t 2
It follows that ȧt (w t )/ w t 2 is not integrable. This contradicts Theorem 4.2 of [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF].

The next proposition will be the starting point of the contradiction argument in the following sections. It says that a cusp form eigenbranch cannot limit to 0. Heuristically, the zeroth Fourier coefficient of a cusp form vanishes identically whereas an eigenvalue branche that limit to 0 must eventually live have nontrivial zeroth Fourier mode. However, because we have made a nontrivial change of variable, this fact requires an argument. Proposition 7.9. If E t is a real-analytic cusp form eigenvalue branch of q t , then the integer k appearing in (58) is positive.

Proof. Suppose to the contrary that lim t→0 E t = 0. Set I = [0, 1] and consider w t = w I t defined in (55). If > 0, the restriction of a to V is bounded below by π 2 > 1, thus we have Π 0 (w t ) = w t . On the other hand, the projection of u t onto >0 V equals u tu 0 t ⊗ 1. Let v 0 t denote the projection of u tw t onto V 0 . Since each V is a direct sum of eigenspaces of a, we have

a t (u t -w t ) = a t (v 0 t ) + a t (u -u 0 t ⊗ 1).
The quadratic form a is nonnegative and the restriction of a t to >0 V is bounded below by π 2 . Hence a(u tw t ) ≥ π 2 • u tu 0 t ⊗ 1 2 . By Lemma 7.5 we have

a t (u t -w t ) ≤ C • t 2 • u t 2 . Therefore, u t -u 0 t ⊗ 1 2 ≤ C • t 2 • u t 2 ,
(60) 1 0 η(t • x, y) • u t (x, b(t • x, y)) dx = 0 where b(x, y) = B( √ 1 -x 2 , y) and η(x, y) = (y/b(x, y)) • ∂ y B( √ 1 -x 2 , y). Inspection shows that there exists C such that sup {|η(t • x, y) -1|, (x, y) ∈ [0, 1] × [1, β]} ≤ C • t
for small t. Therefore, using the Cauchy-Schwarz inequality we find that (61)

1 0 (η(t • x, y) -1) • u t (x, b(t • x, y)) dx ≤ C • t • 1 0 |u t (x, b(t • x, y)) | 2 dx 1 2 .
Using the fundamental theorem of calculus and the Cauchy-Schwarz inequality we find that

|u t (x, b(t • x, y)) -u t (x, y)| 2 ≤ |b(t • x, y) -y| 2 b(t•x,y) y ∂u t ∂y (x, z) 2 dz.
Thus, since there exists C so that sup {|b(t • x, y) -y| , (x, y)

∈ [0, 1] × [1, β]} ≤ C • t
for small t, we have

1 0 |u t (x, b(t • x, y)) -u t (x, y)| 2 dx ≤ C 2 • t 2 1 0 β 1 ∂u t ∂y (x, y) 2 dy dx.
Since a t and q t are asymptotic at first order

t 2 S ∂u t ∂y (x, y) 2 dx dy ≤ a t (u t ) ≤ (E t + C • t) • u t 2
for sufficiently small t. By combining these estimates, we find that (62)

1 0 u t (x, b(t • x, y)) -u t (x, y) dx ≤ C • (E t + t) 1 2 • u t .
Since u 0 t (y) = 1 0 u(x, y) dx, we can combine (60), ( 61) and (62) to find that

u 0 t (y) ≤ C (E t + t) • u t .
Squaring and integrating over y ∈ [1, β], we find that (63)

β 1 u 0 (y) 2 dy ≤ 2C 2 • E 2 t + t 2 • u t 2 .
Without loss of generality, we have u t = 1. Therefore, if E t limits to zero, then (63) will contradict (59) for small t.

7.4. Bounds on the first variation of the eigenvalue. We can also use the nonconcentration of the spectral projection to give an O(t -1 ) lower bound on the first variation Ėt in the case that the projection onto the small modes is significant.

Proposition 7.10. Let I be a compact interval whose interior contains E 0 and I ⊂ ((k -1) 2 π 2 , (k + 1) 2 π 2 ). For each δ > 0, there exists κ > 0 and t 0 > 0 so that if t < t 0 and

Π <k (w t ) ≥ δ • u t , then (64) Ėt ≥ κ t .
Proof. Using the Cauchy-Schwarz inequality and the nonnegativity of ȧt we have

(65) ȧ(u t ) ≥ ȧ(w t ) -ȧ(w t ) 1 2 • ȧ(u t -w t ) 1 2 .
It follows from ( 25) that for all v ∈ Dom(a t )

(66) ȧt (v) ≤ 2t -1 • a t (v).
and hence ȧ(w t )

1 2 ≤ √ 2 • t -1 2 • a(w t ) 1 2 ≤ t -1 2 • (2 sup(I)) 1 2 • w t .
Moreover, by combining this with Lemma 7.5 we find ȧ(u tw t ) ≤ Ct u t 2 Thus, from (65) we obtain

ȧ(u t ) ≥ ȧ(w t ) -C • u t 2 .
Hence by applying Lemma 7.5 we have

Ėt • u t 2 = q(u t ) ≥ ȧ(u t ) -C • a(u t ) (67) ≥ ȧ(w t ) -C • u t 2 -C • q(u t ) ≥ ȧ(w t ) -C • u t 2
for t sufficiently small. As in the proof of Theorem 7.8, we have Π <k+1 (w t ) = w t = Π <k (w t ) + Π k (w t ). Since ȧ is non-negative and 'block-diagonal' we have ȧt (w t ) ≥ ȧt (Π <k (w t )).

Let C be as in Lemma 7.6, and apply Lemma 7.3 with ε = δ 2 /(2C 2 ) to obtain κ so that

ȧt (w t ) ≥ κ t Π <k (w t ) 2 - δ 2 2 w t 2 ≥ κ • δ 2 2t • w t 2
for t sufficiently small. Estimate (57) implies that w t 2 ≥ 1 2 u t 2 for t small, and therefore by combining the above inequalities, we prove the claim.

In contrast to Proposition 7.10, we have the following.

Lemma 7.11. There exists t 0 > 0 and C such that if t < t 0 , then Ėt

E t ≤ 2 t + 3C.
Proof. It follows from ( 25) that for all v ∈ Dom(a t )

(68) ȧt (v) ≤ 2t -1 • a t (v).
From (42), there exists C so that for sufficiently small t

a t (v) ≤ (1 + C • t) • q t (v), and qt (v) ≤ ȧt (v) + C • a t (v).
Thus, if u t is the real-analytic eigenfunction branch of q t associated to E t , then

Ėt • u t 2 = qt (u t ) ≤ ȧt (u t ) + C • a t (u t ) ≤ (2 • t -1 + C) • a t (u t ) ≤ (2 • t -1 + C) • (1 + C • t) • q t (u t ) = 2 • t -1 + C • (1 + C • t) • E t • u t 2 .
By choosing t 0 sufficiently small, we obtain the claim.

Proof of the main theorem

Proposition 3.8 reduces the proof of Theorem 1.1 to the following.

Theorem 8.1. The family t → q t does not have a real-analytic cusp form eigenbranch.

The proof of Theorem 8.1 will be by contradiction. We will assume that there exists a real-analytic cusp form eigenvalue branch, E t . By the results of §7, we have (69) lim

t→0 E t = (π • k) 2 .
where k is positive. We aim to contradict the positivity of k.

8.1. Choosing β. The proof of Theorem 8.1 will rely on the estimates of solutions to ordinary differential equations made in Appendix B. To make the estimates less tedious, we will choose β to be sufficiently close to 1, where 'sufficiently close' will be determined by the integer k that appears in (69). However, the construction of the quadratic form q t depends on β. 9 Therefore, the integer k that appears in (69) depends a priori on β. In order to avoid circularity of reasoning, we will prove the following. Proposition 8.2. Let E t be a real-analytic eigenvalue branch associated to a realanalytic cusp form eigenfunction branch of t → q β t . For each β > 1, the family t → q β t has a real-analytic cusp form eigenfunction branch with associated eigenvalue branch E t . 10 9 We have suppressed this dependence from notation until now. 10 The respective eigenfunction branches will not be the same if β = β .

Proof. For each fixed t, since E t corresponds to a cusp form, it belongs to the spectrum of q β t for all β (see Lemma 2.7).By Theorem 3.3, there exists a realanalytic eigenvalue branch s → Ēt s of s → q β s such that Ēt t = E t . Since s → q β s has only countably many real-analytic eigenvalue branches, there exists some branch s → Ēt s such that the set of t with Ēt t = E t has an accumulation point. Thus, by real-analyticity, we have Ēt t = E t for all t . If for each t, the dimension of the eigenspace V t of q β t associated to E t is greater than one, then one can argue as in the proof of Proposition 3.7 to obtain a realanalytic cusp form eigenfunction branch of q β t associated to E t . Otherwise, by real-analyticity, for each t in the complement of discrete set A of t we have dim(V t ) = 1. Let t → u β t be a real-analytic eigenfunction branch of q β t associated to E t . Let t → u β t denote a real-analytic eigenfunction branch of q β t associated to E t . For each t, the pull-back of u t by the diffeomorphism ϕ β 0,t is a cusp form of E on T t . In turn, for each t, the pull-back of u t • ϕ β 0,t by (ϕ β 0,t ) -1 is a cusp form eigenfunction of q β t . Hence, if t / ∈ A, then the eigenfunction u β t is a cusp form for t / ∈ A. Thus, by Corollary 3.6, the branch t → u β t is a real-analytic cusp form eigenbranch of q β t .

As a consequence of Proposition 8.2, we may fix β to satisfy 11

(70) 1 < β < k k -1 .
It follows that for each < k and y ∈ [1, β] we have

(71) (π • ) 2 - E t y 2 < 0,
as soon as t is small enough.

In what follows, we will drop β from the notation for q β t .

8.2. Tracking. In this section we show that there exists a real-analytic eigenvalue branch λ t of a k t such that |λ * t -E t | is at most of order t. In §8.4, we will show to the contrary that |λ * t -E t | is at least of order t 2 3 . This will provide the desired contradiction.

Theorem 8.3 (Tracking). If E t is a cusp form eigenvalue branch with positive limit (kπ) 2 , then there exists t 0 > 0, C > 0, and a real-analytic eigenvalue branch λ * t of a k t so that for each t < t 0 ,

spec a k t [E -Ct, E + Ct] = {λ * t }.
Proof. It suffices to show that there exist positive constants t 0 and C so that the distance from E t to the spectrum of a k t is at most C • t for t < t 0 . Indeed, if this were so, then for each t we would have an eigenvalue λ t of a k t so that |E tλ t | ≤ Ct.

11 This choice of β is most probably not necessary but it will simplify the arguments in the appendix. In particular it implies that, on [1, β] and for < k, the Sturm-Liouville equations associated with a t have no turning point.

Suppose that there existed a sequence t n → 0 and eigenvalues λ tn = λ tn of a k tn so that |E tnλ tn | ≤ Ct n . Then we would have

1 t n • λ tn -λ tn ≤ 2C.
This would contradict the 'super-separation' of eigenvalues given in Theorem 10.4 in [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF]. Hence there exists t 1 > 0 such that if t < t 1 , then at most one eigenvalue λ t of a k t satisfies |E tλ t | ≤ Ct. Since the eigenvalues of the family a k t belong to real-analytic branches, the function t → λ t would be real-analytic.

We first proceed to prove that there exists a sequence t n going to 0 such that E t is at a distance of order t of the spectrum of a k t . Let u t denote a real-analytic eigenfunction branch associated to E t . By Corollary 7.7 and the fact that Π k is an orthogonal projection that commutes with a t there exists constants C qm and t 0 such that for each v and t < t 0 , then (72)

|a t (Π k (w t ), Π k (v)) -E t • Π k (w t ), Π k (v) | ≤ C qm • t • w t • Π k (v) .
Let B denote the set of t > 0 so that the distance from E t to spectrum of a k t is at least 2 • C qm • t, where C qm is the constant that appears in (72). By the remark above, we want to show that there exists T > 0 so that B∩ ]0, T [ is empty.

For each t ∈ B∩ ]0, s[, we apply a resolvent estimate to (72) and obtain w k t ≤ w t /2. Orthogonality then implies that for each t ∈ B∩ ]0, s[, there exists j t < k such that (73)

w t ≤ 2k • w jt t .
By applying Corollary 7.3 and using the fact that ȧ is both nonnegative and 'blockdiagonal' with respect to ⊕V , we find that

(74) ȧt (w t ) ≥ κ (2k) 2 • t • w t 2 for all t ∈ ]0, t 3 [ ∩B.
By Theorem 4.2 in [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF], the function ȧt (w t ) / w t 2 is integrable over ]0, t 3 [. Thus, by (64), the function 1/t is integrable over B∩ ]0, t 3 [. It follows that there exists a sequence (t n ) n≥1 , t n ∈ R + \ B with t n → 0. Let λ n be a real-analytic eigenbranch of a k tn with |E tnλ n (t n )| < 2C qm • t n . Suppose that there exists a sequence (t n ) n≥1 , t n ∈ B with t n → 0. We claim that there exists another constant C and another sequence (t

* n ) n≥1 , t * n ∈ B with t * n → 0 so that λn (t * n ) ≥ Ėt * n -C.
To see this, first note that by taking subsequences if necessary, we may assume that t n+1 < t n < t n for each n. Lemma 8.4. Let k > 0 and let t n be a sequence converging to zero. For each n ∈ Z + , let λ n be a real-analytic eigenbranch of the family

Let [t - n , t + n ] be the connected component of λ -1 n ([E t -2C qm t, E t + 2C qm t]) that
a k t . If lim n→∞ λ n (t n ) = k 2 π 2 , then lim n→∞ t n • λn (t n ) = 0.
Remark 8.5. Lemma A.1 as stated does not directly apply since the eigenvalue branch λ n depends on n. By keeping track of the constants in the proof of Lemma A.1, one can produce a version that directly implies Lemma 8.4. We prefer to give a direct proof here.

Proof. Let ψ n be a unit norm eigenfunction of a k tn with eigenvalue λ n (t n ). By the standard variational formula

λn (t n ) = ȧtn (ψ n (y)) = 2t n • ∞ 1 |ψ n (y)| 2 dy.
Since ψ n is an eigenfunction of a k tn with eigenvalue λ n (t n ), we have

(75) t 2 n • ∞ 1 |ψ n (y)| 2 dy = ∞ 1 λ n (t n ) y 2 -(πk) 2 |ψ n (y)| 2 dy.
It suffices to show that the right hand side of (75) tends to zero as n tends to infinity.

Let > 0. Since λ n (t n ) tends to (πk) 2 , there exists δ > 0 so that if |y -1| < δ, then λ n (t n ) • y -2 -(πk) 2 < /2 and thus (76) 1+δ 1 λ y 2 -(πk) 2 |ψ n (y)| 2 dy ≤ 2 • y 1 |ψ n (y)| 2 dy .
To estimate the remaining integral over [1 + δ, ∞), we will apply a standard convexity estimate from the theory of ordinary differential equations. 12 If n is sufficiently large, λ n (t n )/(πk) 2 < (1 + δ/4) 2 , and hence there exists η > 0 so that if y > z > 1 + δ/2, then

|ψ n (y)| 2 ≤ |ψ n (z)| 2 • exp - η t • (y -z) .
It follows that there exists t 0 so that if t < t 0 , then

∞ 1+δ |ψ(y)| 2 dy ≤ 4(πk) 2 • ∞ 1 |ψ(y)| 2 dy. Since λ n (t n )/y 2 -(πk) 2 ≤ (πk) 2 (77)
for sufficiently large n, we may combine (77) with (76) to show that (75) is less than the given for sufficiently large n. 8.3. Crossings. In this subsection, we show that w k s is smaller than u s for s near a crossing time, a value of the parameter t such that E t belongs to the spectrum of a 0 t . Then we show that there exists a sequence of crossing times t n and intervals of width O(t 8 3 ) about the crossing times on which w k t is smaller than u t .

The proof of the first result depends on the analysis contained in Appendix B.

Proposition 8.6. Given ρ < 1, there exists η > 0 and t 0 > 0 such that if t < t 0 and

(78) dist E t , spec a 0 t ≤ η • t 5 3 , then w k t ≤ ρ • u t .
Proof. Let ψ 0 t be an eigenfunction of a 0 t with eigenvalue

λ 0 t satisfying |E t -λ 0 t | < η • t 5 3 . We have (E t -λ 0 t ) • u t , ψ 0 t = (a t -q t )(u t , ψ 0 t ) = t • b t (u t , ψ 0 t ) + O(t 2 ) • u t • ψ 0 t . and hence (79) E t -λ 0 t • u t , ψ 0 t ≥ t • b t (u t , ψ 0 t ) -O(t 2 ) • u t • ψ 0 t .
In Appendix B, we prove that there exists κ > 0 so that

b t (u t , ψ 0 t ) ≥ κ • t 2 3 • w k t -t δ • u t • ψ 0 t .
Hence by applying the Cauchy-Schwarz inequality to the left hand side of (79), we find that

E t -λ 0 t u t • ψ 0 t ≥ κ • t 5 3 • w k t -t δ • u t -O(t 2 ) u t • ψ 0 t . Let η = ρ • κ/2, and use (78) to find that ρ 2 • u t ≥ w k t -O(t δ ) • u t -O(t 1 3 ) • u t .
The claim follows by choosing t 0 sufficiently small. Proposition 8.7. For all η > 0, there exists δ, s 0 > 0 such that if s < s 0 , E s ∈ spec(a 0 s ), and t ∈ s, s

+ δ • s 8 3 , then dist E t , spec(a 0 t ) ≤ η • s 5 3 .
Proof. Let λ 0 t be the eigenvalue branch of a 0 t such that E s = λ 0 s . By Lemma 6.5, we have λ 0 t = c • t 2 for some c > 0, and hence λ0 t = 2 • t -1 • λ 0 t . Using the fact that a t and q t are asymptotic and the fact that ȧ is non negative, there exists a constant C such that Ėt ≥ -CE t for all sufficiently small t. Thus, for even smaller t we obtain

d dt ln λ 0 t E t ≤ 3 • t -1
Since E s = λ s , integration over [s, t] and exponentiation gives (80)

λ 0 t E t ≤ t s 3 . If t ≤ s + δ • s 8 3 , then t s 3 ≤ 1 + δ • s 5 3 3 ≤ 1 + 4 • δ • s 5 3
where the last inequality holds for s ≤ s 1 = (2δ) -3 5 . By combining this with (80), one finds that for t ∈ s, s + η • s 8 3 , we have (81)

λ 0 t -E t ≤ E t • 4 • δ • s 5 3
Using Lemma 7.11, and λ ≥ 0, we have that

(82) d dt ln E t λ 0 t ≤ 3 • t -1 .
An argument similar to the one above gives that

E t -λ 0 t ≤ λ 0 t • 4 • δ • s 5 3 . for t ∈ s, s + δ • s 8 3 .
Since by assumption lim t→0 E t = (πk) 2 , there exists s 2 so that if

t < s 2 + η • s 8 3 2 , then E t ≤ 2 • (πk) 2 . Therefore, by (81), we have that λ 0 t | s ≤ t ≤ s + η • s 8 3
is bounded above by 3(πk) 2 for s < s 0 = min{s 1 , s 2 }. In sum, if s < s 0 and

t ∈ s, s + η • s 8 3 , then λ 0 t -E t ≤ 3(πk) 2 • δ • s 5 3 .
The claim follows by chosing δ carefully.

We wish to estimate from below the size of the set of t for which (78) holds true. This is accomplished by the following proposition.

Proposition 8.8. Let δ > 0. There is a sequence t n of crossing times such that

(83) lim n→∞ n • t n = k • ln(β).
and if n = m are large enough, then the intervals t n , t n + δ • t Proof. By Lemma A.1, there exists, ν * > 0 so that λ * t = (πk

) 2 + ν * • t 2 3 + O t 4 3
. It also follows from Proposition 8.3 that there exists M so that

(84) (πk) 2 + ν * • t 2 3 -M • t < E t < (πk) 2 + ν * • t 2 3 + M • t
for sufficiently small t. By Lemma 6.5, the eigenvalues of a 0 t have the form c n • t 2 where c n = (1/4 + r 2 n ) and r n is the increasing sequence of positive solutions to the equation 2r = tan(r ln(β)). Standard asymptotic analysis shows that (85)

r n = nπ + π 2 ln(β) + o(1). Fix 0 < ν - 0 < ν * < ν + 0 . For each ν ∈ [ν - 0 , ν + 0 ] and each n ∈ Z, there exists a unique t ν n ∈ R + so that (86) c n • (t ν n ) 2 = (πk) 2 + ν • (t ν n ) 2 3 .
We drop the dependence in ν from the notation for a moment. If we set

x n = c -1 6 n and y n = c 1 6 n • t 1 3 n then (86) becomes y 6 n = (π • k) 2 + ν • x 2 n • y 2 n .
By the analytic implicit function theorem, for x near 0, there exist a unique analytic function Y (x) so that By inspecting of the first few coefficients in the Taylor expansion of Y 3 , we find that

Y (x) 6 = (π • k) 2 + ν • x 2 • Y (x) 2 . t n t + n t E t c n • t 2 (⇡k) 2 + ⌫ • t 2 3 (⇡k) 2 + ⌫ + • t 2 3 (⇡k) 2
Y (x) 3 = π • k + ν 2 • (πk) 1 3 • x 2 + O(x 3 ).
Thus, since lim n→∞ x n = 0, and

t n = c -1 2 n • Y 3 (c -1 2 n ) we find that (87) t ν n = (πk) • c -1 2 n + τ • c -5 6 n + O(c -1 n ).
where τ = ν • (πk)

1 3 /2. Choose > 0 so that if ν ± = ν * ± , then ν - 0 ≤ ν -< ν * < ν + ≤ ν + 0 . Define t ± n = t ±ν n .
By applying the intermediate value theorem to λ t -E t , there exists t n ∈ ]t - n , t + n [ so that E tn = λ tn . See Figure 3. Since c n is increasing to infinity, the sequence t n is decreasing to zero.

Moreover, since

ν ± = ν * ± ε (88) t n = (πk) • c -1 2 n + τ * • c -5 6 n + o(c -5 6 n ).
where

τ * = ν * • (πk) 1 3 /2. From (85) we have c -1 n = σ 2 n 2 • 1 + O 1 n
where σ = ln(β)/π. By substituting this into (87) and (88) we find that

t ± n = (πk) • n -1 + τ • σ 5 3 • n -5 3 + O ± (n -2 )
and

t n = (πk) • n -1 + τ * • σ 5 3 • n -5 3 + o(n -5 3 ).
The first claim follows. Moreover, since ν ± = ν * ± ε, we have

t + n -t n ∼ ε • (πk) 1 3 2 • n 5 3 t n -t - n ∼ ε • (πk) 1 3 2 • n -5 3 t - n -t + n+1 ∼ ε • (πk) 1 3 • n -5 3 t 8 3 n = O(n -8 3 ).
It follows that, for all sufficiently large n, we have [t n , t n + δ • t

8 3 n ] ⊂ [t - n , t + n ]. Since the intervals {[t - n , t + n ]
} are disjoint, the claim is proven. 8.4. Relative variation and the contradiction. In this section we derive the desired contradiction. In particular, we prove the following. Theorem 8.9. If E t is a cusp form eigenvalue branch with a positive limit, then there exists t 0 > 0 and c > 0 so that if t < t 0 , then

E t -λ * t > c • t 2 3 .
The proof will consist of two types of lower estimates. The first depends on the fact that near each crossing the 'relative variation' Ėtλ * t is at least of order O(t -1 ). The second shows that away from the crossings the relative variation is not too negative. Define K(t, ρ) = s ∈ ]0, t] w k s ≤ ρ • u s . If ρ < 1, then it follows from Proposition 7.10 that there exists κ > 0 so that for s ∈ K(t, ρ) we have (89) Ės ≥ κ • s -1 .

Hence, since λ * t = O(t -1 3 ), there exists t * > 0 so that if t < t * and ρ < 1, then

(90) Ės -λ * s ≥ κ 2 • s -1 .
for each s ∈ K(t, ρ). We will integrate this estimate near the crossings to obtain the following.

Lemma 8.10. For each ρ < 1, there exists t 0 > 0 and γ(ρ) > 0 so that for each t < t 0 , we have

K(t,ρ) Ės -λs ds ≥ γ(ρ) • t 2 3 .
Proof. By (90) the integrand is positive on K(t, ρ), it suffices to show the same estimate holds for a subset G of K(t, ρ).

To define this subset, we first combine Proposition 8.6, Proposition 8.7, and Proposition 8.8 to find δ > 0, N ≥ 2, and a monotone sequence {t n } so that for each n I n,δ = t n , t n + δ • t where τ = k • ln(β). The subset G will be defined as a union of I n over sufficiently large n.

We have I n,δ s -1 ds = ln(1 + δ t 5 3 n ) and hence there exists N * ≥ N so that if n ≥ N * we have (92)

I n,δ s -1 ds ≥ δ 2 • t 5 3 n .
Thus, from (91) we find that if N ≥ N * , then

(93) 2 τ 5 3 n≥N +2 t 5 3 n ≥ n≥N +2 n -5 3 ≥ ∞ N +2 x -5 3 dx = (N +2) -2 3 ≥ t N 4τ 2 3
.

Since the intervals I n,δ are disjoint, by combining (90), (92), and (93), we find that (94)

G N Ės -λ * s ds ≥ γ • t 2 3 N , where γ = κ • δ • τ • 2 -16 3 and G N : = n≥N +2
I n,δ .

Let t 0 = t N * . If t < t 0 , then t ∈ [t N +1 , t N ] for some N ≥ N * . We have

t N +2 + t 8 3 N +2 ≤ t N +1 ≤ t and hence G N ⊂ K(t, ρ) and t 2 3 N ≥ t 2 3
. Therefore, (94) implies the claim.

To bound the relative variation on the complement of K(t, ρ), we will use the following.

Proposition 8.11. There exists C and t 0 > 0 such that, if t < t 0 , then

(95) Ėt ≥ w k t 2 u t 2 • λ * t -C • t -1 9 .
Proof.

, v r t ) = 2t -1 • a k t (ψ * t , v r t ) -(πk) 2 ∞ 1 ψ * t (y) • v r t (y) dy = -2(πk) 2 • t -1 ∞ 1 ψ * t (y) • v r t (y) dy. (98) Since ψ * t , v r t = 0, we have ∞ 1 ψ * t • v r t dy = ∞ 1 ψ * t • v r t • (1 -y -2 ) dy
The large y asymptotics of ψ r t and v * t can be analysed using the same methods used in Appendix B for v k t . In particular, the estimate proven in Lemma B.7 holds with r estimated in Lemma B.5. In particular, for each α < 2 3 , there exists a constant so that for sufficiently small t

∞ 1+2t α |ψ * t | 2 dy ≤ C • t 2-2α • w t 2 ,
and

∞ 1+2t α |v r t | 2 dy ≤ C • t 2-2α • w t 2 .
If y ≤ 1 + 2t α , then (1y -2 ) ≤ 8t α for sufficiently small t. Therefore, by splitting the domain of integration into [1, 1 + 2t α ] and [1 + 2t α , ∞[ and using the Cauchy-Schwarz inequality, we find that (99)

∞ 1 ψ * t • v r t dy ≤ 9 • t α • ψ * t • v r t + C • t 2-2α • w t 2
for sufficiently small t.

We claim that v r t = O(t

3 ) • w t . Indeed, by applying Lemma 7.6 with v ∈ V k , we have

|a k t (v k t , v) -E • v k t , v | ≤ C • t • w t • v for some constant C. Thus, since the eigenvalue λ * satisfies |E t -λ * | < C • t we find that (100) |a k t (v r t , v) -E • v r t , v | ≤ 2C • t • w t • v . By definition, v r
t is a projection onto eigenspaces of a k t whose associated eigenvalues are distinct from λ * . By Lemma A.1, there exists δ > 0 so that such eigenvalues differ from λ * by at least δ • t 2 3 . Because of (100), we can thus apply a resolvent estimate (e.g. Lemma 2.1 [START_REF] Hillairet | Spectral simplicity and asymptotic separation of variables[END_REF]) to find that

(101) v r t ≤ 2C δ • t 1 3 • w t .
By substituting (101) into (99) and setting α = 5/9, we find a constant C so that

∞ 1 ψ * t • v r t dy ≤ C • t 8 9 • w t 2 .
By combining this estimate with (98), (97), and (96), we obtain a constant C so that Ėt

• u t 2 ≥ λ * • w * t 2 -2C • t -1 9 • w t 2 .
By orthogonality

w * t 2 = w k t 2 -w r t 2
, and hence by (101) and Lemma A.1, we have a constant C so that

λ * • w * t 2 = λ * • w k t 2 -C • t 1 3 • w t 2 .
The desired result follows.

Corollary 8.12. There exists C such that for each ρ ∈ ]0, 1[, there exists t 0 > 0 such that if 0 < t < t 0 , then

[0,t]\K(t,ρ)

Ės -λ * s ds ≥ C • ρ 2 -1 • t 2 3 .
By making the change of variable y = t , we find that

∞ 0 v t • w dx + µ 0 x • g t 2 3 • x • v t • w dx = t -2 3 • (λ -µ) ∞ 0 f (t 2 3 • x) • v t • w dx.
where

f (z) = 1 (z + 1) 2 and g(z) = z + 2 (z + 1) 2
This leads us to set s = t 1 3 and define for each v ∈ C ∞ 0 ([0, ∞[) the quadratic forms

A s (w) = ∞ 0 (w ) 2 dx + µ ∞ 0 x • g(s 2 • x) • w 2 dx and N s (w) = ∞ 0 f (s 2 • x) • w 2 dx. Define H := L 2 ([0, ∞), 1
x 2 +1 dx) and D := H 1 ([0, ∞]) (i.e. the set of functions u ∈ L 2 ([0, ∞)) such that the distributional derivative also is in L 2 ).

Then for each s > 0, the form N s is a bounded quadratic form on H and A s is a closed quadratic form on H with domain D s .

Since w → w maps bijectively

C ∞ 0 ([1, ∞]) onto C ∞ 0 ([0, ∞]
), the function v is an eigenfunction of A s with respect to N s with eigenvalue ν = s -2 • (λµ).

It follows from the perturbation theory of generalized eigenvalue problems (see §VII.6 in [Kato]) that the eigenvalues of A s with respect to N s can be organized into real-analytic eigenvalue branches for s > 0. 13 Since the generalized eigenvalue problem A s (u, v) = ν • N s (u, s) corresponds to a Sturm-Liouville problem with Neumann condition at x = 0, the eigenspaces are 1-dimensional. Hence, we may enumerate the real-analytic eigenvalue branches ν i s so that for each i ≥ 0 and s > 0 we have (104) ν i s < ν i+1 s . Lemma A.2. For each i, there exists s 0 > 0 and C so that if s < s 0 then (105) νi s ≤ C • s. In particular, there exists a so that for small s > 0 (106) ν i s = a + O(s 2 ). Moreover, -a/(2µ) 2 3 is a zero of the derivative of the Airy function A -.

Proof. First, we show that each ν s i is bounded. To this end, define

B(v) = ∞ 0 (v (x)) 2 + 2µ • x • v(x) 2 dx.
13 At s = 0 the domains of As and Ns change, and hence analytic perturbation theory can not be applied.

Since g is bounded above by 2, we have A s (v) ≤ B(v) for each s > 0 and v ∈ C ∞ 0 ([0, ∞[). Note that for each s ≤ 1, we have N s (v) ≥ N 1 (x) and hence (107)

A s (v) N s (v) ≤ B(v) N 1 (v) .
Integration by parts shows that the eigenfunctions of B with respect to N 1 are solutions the Sturm-Liouville problem

-v (x) + 2µ • x • v(x) = v(x) (1 + x) 2 .
Standard convexity estimates on solutions to ordinary differential equations imply that each eigenfunction belongs to the domain D s of A s for each s > 0. In particular, the sum of the first i eigenspaces of B with respect to N 1 belongs to D s .

Therefore, using (107), the minimax principle, and (104) we find that ν i s is bounded by the i th eigenvalue of B with respect to N 1 .

In the remainder of the argument we drop the superscript i and focus on an individual real-analytic eigenfunction branch u s with eigenvalue ν s . For s > 0, we have νs = Ȧs (u s )

N s (u s ) -ν s • Ṅs (u s )
N s (u s ) where • indicates differentiation with respect to s. A computation gives that for each w Ȧs

(w) = 2s • µ ∞ 0 x 2 • g (s 2 • x) • w(x) 2 dx, and
Ṅs (w) = 2s ∞ 0 x • f (s 2 • x) • w(x) 2 dx.
Let u s be a real-analytic eigenfunction branch of A s with respect to N s associated to the real-analytic eigenvalue branch ν s . Integration by parts gives

(108) -u s (x) + µ • x • g s 2 • x • u s (x) = ν s • f (s 2 • x) • u s (x).
Let M be the upper bound on ν s proven above. If s ≤ 1 and x > x 0 := max{1, M/µ}, then

µ • x • g(s 2 • x) -ν s • f (s 2 • x) ≥ µ 2 and hence u s (x) ≥ µ 2 • u s for s ≤ 1. It follows that (u 2 s ) (x) ≥ µ • u 2 s (x) for x ≥ x 0 . Thus, since N (u s ) is finite, we find that for x 0 ≤ x ≤ y (109) u s (y) 2 u s (x) 2 ≤ exp - √ µ • y exp - √ µ • x .
Integrating from x 0 to 2x 0 , we find a constant C (that depends on x 0 ) such that, for y > 2x 0 we have

y 2 • u(y) 2 ≤ C • y 2 exp(- √ µy) • 2x 0 x 0 u(x) 2 1 + x 2 dx.
From this we find constants C such that that

Ȧs (u s ) ≤ C • s • N s (u s ).
Proof. The quadratic form b t is controlled by a t meaning that there exists a constant C such that for u, v ∈ dom(a t )

|b t (u, v)| ≤ C • a t (u) 1 2 • a t (v) 1 2 .
Thus, proposition 7.5 and the fact that a t (ψ

0 ) = O( ψ 0 ) imply that b t (u t -w t , ψ 0 ⊗ 1) = O(t) • u t • ψ 0 ,
and hence it suffices to bound b t (w t , ψ 0 ) from below.

Observe also that proposition 7.5 also implies that u t ∼ w t in the limit t → 0 so that we can freely replace u t by w t and vice-versa in each (multiplicative) estimate.

By the discussion §6.2 and §7.2, for each t we can uniquely write

w t (x, y) = ≤k λ∈spec(a t )∩It ψ λ (y) • e (x)
where each ψ λ (y) is an eigenfunction of a t with eigenvalue λ ∈ I t . Set From (26) we have

b t w t , ψ 0 ⊗ 1 = β 1 1 0 ∇ t w t • 0 x • p(y) x • p(y) 0 • ∇ t ψ 0 (y) * dxdy.
where ∇ t f = [∂ x f, t∂ y f ] and p(y) is defined in (24). Since ∂ x ψ(y) ≡ 0, and e (x) = 2 -1 2 cos( πx), we find that

b t w t , ψ 0 ⊗ 1 = -2 -1 2 π • 1 0 x • sin( πx)dx • β 1 p(y) • v t (y) • t • (ψ 0 ) (y) dy
If = 0, then sin( πx) ≡ 0, and so b t (w t , ψ 0 ⊗ 1) = 0. For 0 < < k, apply Lemma B.3 below to find that

(114) b t (w t , ψ 0 ⊗ 1) = O (t) • v t • ψ 0 .
Since w t and w t are orthogonal if = , we have

k-1 =1 v t 2 = 2 -1 2 k-1 =1 w t 2 ≤ w t where (a + , a -) ∈ C 2 , h ,r µ,± (y) = ± t -2 • W -1 y 1 r(z) • v µ,∓ (z) dz, and
W = v µ,+ • v µ,--v µ,-• v µ,+ is the Wronskian.
In particular, for each and each t, there exists (a t,+ , a t,-) ∈ C 2 so that the function v t of (112) satisfies

v Et = a t,+ + h ,r t Et,+ • v Et,+ + a t,-+ h ,r t Et,-• v Et,-
The eigenfunction ψ 0 of a 0 t satisfies (124) with r = 0, and hence there exists

(c + , c -) ∈ C 2 so that ψ 0 = c + • v 0 λ 0 ,+ + c -• v 0 λ 0 ,-. The integral in (116) is equal to β 1 g • ± a t,± + h ,r t Et,± • v Et± • ± c ± • t • v 0 λ 0 ,± dy.
By expanding the product of sums, one obtains a sum of 2 3 integrals. By substituting the expressions (122) and (121), integration by parts, and applying standard estimates, we find that each integral is O(t).

For example, consider the terms of the form (126)

a t,± • c ± β 1 g • f 0 λ 0 f Et 1 4 exp i t y 1 ± f Et 1 2 ∓ f 0 λ0 1 2 • (1 + ε * ) dy.
Since > 0, an elementary computation shows that there exists δ > 0 so that if z ∈ [1, β] and t is sufficiently small, then

(127) δ ≤ f Et (z) 1 2 ± f 0 λ 0 (z) 1 2
Thus, we may integrate by parts to find a constant C so that the integral in (126) is at most

C • t • g C 1 .
It follows that all the terms of this form are bounded above by

(128) C • t • g C 1 • ψ 0 • ± |a t,± | .
We also have terms of the form

(129) c ± β 1 g • h ,r t Et,± • f 0 λ 0 f Et 1 4 exp i t y 1 ± f µ 1 2 ∓ f 0 µ 1 2 • (1 + ε * ) dy.
We integrate by parts as above, but this time we need to also bound h ± = h ,r t Et,± and its derivative. From (121) and (122) we find that there exists t 1 so that if t < t 1 , then |t • W| ≥ 1. From ( 121) and (119) we find that for each (130) sup

y∈[1,β] v Et,∓ (y) ≤ 2 √ δ 1
for all sufficiently small t. Hence, using the Cauchy-Schwarz inequality and Lemma B.5 we have, for all y ∈ [1, β],

|h ± (y)| ≤ 1 t • y 1 r(y) • v Et,∓ (y) dy ≤ 1 t • y 1 r(y) 2 dy 1 2 • y 1 v Et,∓ (y) 2 dy 1 2 ≤ 1 t • C • t • u t • β -1 • 2 • δ -1 2 1
for all sufficiently small t. For the derivative of h ,r µ,± , we have

h ± (y) ≤ 3 t • |r(y)| • 2δ -1 2 .
Applying Cauchy-Schwarz and Lemma B.5 gives

β 1 h ± (y) dy ≤ 6 • δ -1 2 t β -1 • β 1 |r(y)| 2 dy 1 2 (131) ≤ 2 • δ -1 2 t β -1 • C • t • u t .
Finally, we apply integration by parts to (129). The resulting terms that do not contain h ± have uniformly bounded C 0 norm. The term that contains h ± can be bounded using (131). It follows that all the term of this form are bounded by

(132) C • t • u t • ψ 0 .
The final step consists in bounding |a ± | by v t to control the terms of eq. (128). Using (123) and (125) we have

m |a + | 2 + |a -| 2 1 2 ≤ v t + (β -1) • 2 √ δ 1 sup [1,β] {|h + (y)| + |h -(y)|}
By orthogonality we have v t ≤ u t and, using the bound on |h ± (y)| we finally obtain

|a + | 2 + |a -| 2 1 2 ≤ C • u t .
This finishes the proof.

B.2. The proof of Lemma B.4. As in the previous subsection, the function v k t is a solution to the inhomogeneous equation ( 118) with µ = E t and r defined by (117). However, for = k, the function

f k t (y) = E t y 2 -k 2 π 2
is negative for large y. In fact, since E t decreases to (πk) 2 , the function f k t changes sign nearer and nearer to y = 1. Since the solution v k t belongs to L 2 (R, y -2 dxdy), we expect it to decay exponentially as soon as y moves away from 1. For y near 1, we will approximate v t using Airy functions. In this subsection we will make these approximations precise and use them to give a proof of Lemma B.4 B.2.1. Normalization of ψ 0 . By Lemma 6.5, ψ 0 is a constant multiple of ψ defined in (36). Because both sides of the estimate in Lemma B.4 are homogeneous functions of degree 1 in ψ 0 , it suffices to assume that ψ 0 = ψ.

Let |f | 0 denote the supremum norm of f over [1, β].

Lemma B.6. There exists t 0 > 0 such that if t < t 0 and λ 0 ∈ I, then

(133) 1 2 ≤ |ψ| 0 ≤ 2 β, (134) ln
(β) 2 ≤ ψ ≤ ln(β) and (135) |t • ψ | 0 ≤ sup(I).
Proof. We have

(136) ψ(y) = ω + (r, y) + (2r) -1 • ω -(r, y) where ω + (r, y) = y 1 2 • cos(r • ln(y)), ω -(r, y) = y 1 2 • sin(r • ln(y)), and λ 0 = t 2 • (1/4 + r 2 ). In particular, for sufficiently small t (137) inf(I) 2t ≤ r ≤ sup(I) t .
Thus, since |ω ± | 0 ≤ √ β and |ω + | 0 ≥ 1, the triangle inequality applied to (136) implies that (133) holds for sufficiently small t.

The same estimate applies for ω -. Hence the triangle inequality and (137) imply that (134) holds for sufficiently small t.

The bound on t • ψ is proven in a similar fashion using the fact that (139)

ψ (y) = -r + 1 2r • y -1 • ω -(r, y)
together with (137).

B.2.2. Localization near y = 1. The following proposition provides a quantitative description of the concentration of solutions to t The function L -1 t (r) is a solution to (118) on [1 + t α , ∞), and hence w := v -L -1 t (r) is a solution to the homogeneous equation (120). It follows from (142) that (144) w 2 (y) ≥ t α-2 • w 2 (y).

2 • v + f k t • v = r near y = 1. Proposition B.7. Let k ∈ Z + . For each α ∈ ]0, 2 3 [, there exists t 0 > 0 and C, such that if v is a solution to t 2 • v + f k t • v = r and t < t 0 , (140)
∞ 1+2t α |v(y)| 2 dy ≤ C • t -2α • ∞ 1 |r| 2 + exp -t 3α-2 2 • ∞ 1+t α v 2 (
if y ∈ [1 + t α , ∞[. In particular, w 2 is convex, moreover w 2 is non-negative and in L 2 ([1 + t α , ∞), y -2 dy) since v ∈ L 2 ([1 + t α , ∞), y -2 dy) and L -1 t r ∈ L 2 ([1 + t α , ∞), dy) ⊂ L 2 ([1 + t α , ∞), y -2 dy). This implies that lim y→∞ w 2 (y) = 0. Indeed, since w 2 is convex, (w 2 ) has a limit m in R ∪ {+∞}. If this limit is positive then it implies that w 2 (y) ≥ m 2 y for large y and this contradicts the fact that w 2 (y)y -2 is integrable. In particular, (w 2 ) is bounded so that by integrating (144) we find that w 2 ∈ L 1 ([1 + t α , ∞), dy). The argument also shows that (w 2 ) is non positive for large y so that w 2 has a limit when y → ∞. Since w 2 is integrable, this limit is 0.

For each y ∈ [1 + t α , ∞) the function e y , that is defined by e y (z) = w 2 (y) • exp -t α-2 2

• (zy) satisfies e y (z) = t α-2 • e y (z) with e y (y) = w 2 (y) and lim z→∞ e y (z) = 0. Therefore, by comparison with (144), and using the maximum principle, we find that if z ≥ y, then w 2 (z) ≤ e z (y). Applying this to z = y + t α , we find that for each y ≥ 1 + t α w 2 (y + t α ) ≤ exp -t 3α-2 2 w 2 (y).

By integration we obtain Proof. Use that y -2 ≤ 1, absorb the integral with v on the right-hand side in the left-hand-side and use Lemma B.5 to control the integral with r.

Corollary B.9. There exist C and t 0 > 0 so that if t < t 0 (146)

β 1+2t α g(y) • v k t (y) • (tψ) (y) dy ≤ C • t 1-α • u t • ψ .
Proof. Use the boundedness of g, the Cauchy-Schwarz inequality, Proposition B.7, Lemma B.5, and Lemma B.6.

The preceding corollary holds for each α ∈]0, 2 3 [. However we will want this contribution to be o(t 2 3 ) so that we will need to take α ∈]0, 1 3 [. B.2.3. The Airy approximation. For small t, the function f k t has a simple zero near y = 1. Thus, to approximate solutions of t 2 • v + f k Et • v = r near y = 1 we will use solutions to Airy's differential equation w (x)x • w = 0 where x = y -1.

We first describe the link to Airy's equation. If we define W (x) := v k t (x + 1), then we have (147) t 2 • W (x) + (kπ) 2 -E (x + 1) 2 • W (x) = r(x).

where r(x) = r(x + 1). Let ρ be the smooth function that satisfies 1 (x + 1) 2 = 1 -2x + x 2 • ρ(x).

By substituting the latter expression into (147) and by dividing by 2E t , we find that

- t 2 2E t • W + x • W + 1 2 • (kπ) 2 E t -1 • W = r 2E t - x 2 2 • ρ • W.
Setting

s = t √ 2E t (148) z s = 1 2 • 1 - (kπ) 2 E t , (149)
we have

(150) -s 2 • W (x) + (x -z s ) • W (x) = R t (x) where R t (x) = (2E t ) -1 • r(x) -2 -1 • x 2 • g(x)
• W (x). In the next few subsections, we will analyse the solutions to (150). But first, we provide an estimate of the L 2 ([0, 3t α ]) norm of R t .

Lemma B.10. For each α < 1 3 , there exists C > 0 and t 0 > 0 such that for each t < t 0 (151)

3t α 0 |R t (x)| 2 dx ≤ C • t 2(2α+ 1 3 ) • u t 2 .
Proof. We have E ≥ (kπ) 2 ≥ 1, and hence by Lemma B.5, r E

2 ≤ r t 2 ≤ C • t 2 • u t 2 .
Let ψ * = ψ * t be the tracking eigenvalue branch associated to E = E t . The eigenvalue λ * t corresponding to ψ * is in a O(t) neighbourhood of E t . Moreover, when t tends to 0, Proposition (A.1) implies that λ t is at a distance of order t 2 3 of the rest of the spectrum. Using (103), (117), Lemma B.5 and a resolvent estimate, we have

v k t -ψ * t = O(t 1
3 ) u t , and hence (152)

3t α 0 x 2 • g(x + 1) • (v t -ψ * t )(x + 1) 2 dx ≤ C • t 4α • t 2 3 u t 2 .
The tracking eigenfunction ψ * satisfies (118) with r = 0. Hence by Proposition B.7, if α ≤ α < 2/3, then 3t α 2t α

x 2 • g(x + 1) • ψ * (x + 1)

2 dx ≤ C • t 4α • exp -t 3 α-2 2 ψ * 2 .
Observe that, by orthogonality, ψ * ≤ u t , hence

3t α 0 x 2 • g(x + 1) • ψ * (x + 1) 2 dx ≤ C • t 4 α + t 2α exp -t 3 α-2 2 • ψ * 2 .
Since 2(2α + 1 3 ) < 2 (because α < 1 3 ), we may thus take α = 1 2 and the biggest term is then of order t 2(2α+ 1 3 ) . The claim follows. One checks that (157)

W ± (x) = A ± s -2 3 (x -z s )
defines a basis of solutions to (154). It follows from well-known identities that the Wronskian of A + A --A + A -:= {A + , A -} is 2. Hence the Wronskian of {W + , W -} is 2s -2 3 . Therefore, by the method of variation of constants, for each x > 0, the function

(161) |W x (x)| ≤ 2 • s -4 3 • s 1 3 • s 1 2 • R • x -3 4 and (162) |W x (x)| ≤ 2 • s -4 3 • s -1 3 • s 1 6 • R • x -1 4 .
Proof. Using the Cauchy-Schwarz inequality, we have for x ∈ [0, x]

x x R(z) • W -(z) dz ≤ x 0 R(z) 2 dz 1 2 • ∞ x W -(z) 2 dz 1 2
Lemma B.15. For all α ∈ ] 7 33 , 1 3 [, there exists δ > 0 such that (178)

a + W + [0,2t α ] = O(t ∞ ) • u t
where O(t ∞ ) is a function that is of order t n for each n, and

(179) a -W -[0,2t α ] ≥ 1 2 • w k t -C • t δ u t .
Proof. Using the behavior of the norm of A ± we find that

a + W + [0,2t α ] = O(t ∞ ) • a + W + [2t α ,3t α ] and a -W -[2t α ,3t α ] ≤ C • a -W -[0,2t α ]
. We thus have

a + W + [0,2t α ] = O(t ∞ ) • a + W + [2t α ,3t α ] ≤ O(t ∞ ) • W [2t α ,3t α ] + a -W -[2t α ,3t α ] + W p,t [2t α ,3t α ] ≤ O(t ∞ ) • u t + W -[0,2t α ] + t δ u t ≤ O(t ∞ ) • u t + W [0,2t α ] + a + W + [0,2t α ] + W p,t [0,2t α ] ≤ O(t ∞ ) a + W + [0,2t α ] + u t .
Estimate (178) then follows by absorbing the norm of a + W + into the left hand side.

To prove estimate (179), we first observe that by using the triangle inequality we find that

W [0,2t α ] ≤ W p,t [0,2t α ] + a -W -[0,2t α ] + a + W + [0,2t α ] .
The first term on the righta hand side is O(t δ ) u t and the last one is O(t ∞ ) u t so that we obtain a -W -[0,2t α ] ≥ W [0,2t α ] -O(t δ ) u t .

The claim then follows by observing that Corollary B.8 implies that

W [0,2t α ] ≥ 1 2 w k t -O(t 1-α ) u t .
Lemma B.16. We have

2t α 0 g • W -• t • ψ dx = (πk) • t • g(1) • A --s -2 3 z s + O t 4 3
for t small.

Proof. From (139), we have

ψ (x) = -(x + 1) -1 2 • r + 1 2r
• sin(r • ln(x + 1)).

Thus, the integral we want to estimate can be written as

  Figure 2. The triangle T c,w in the upper half plane. Proposition 4.1. Each nonconstant Neumann eigenfunction on T c,w is a cusp form and hence an eigenfunction of the modified quadratic form E β . Proof. The eigenvalue of a nonconstant Neumann eigenfunction on T is at least 1/4 [Jdg07]. 5 Thus, the claim follows from Corollary 2.2 and Lemma 2.5.

n

  and t m , t m + δ • t

Figure 3 .

 3 Figure 3. The crossing t .

n

  belongs to K(1/2, ρ), the intervals I n and I n+1 are disjoint, and for each n ≥ N(91) τ 2n ≤ t n ≤ 2τ n

2 3 •

 3 x + 1, letting v(x) = v t 2 3 • x + 1 and w(x) = w t 2 3 • x + 1 ,and dividing by t 4 3

  (112) v t (y) = λ∈spec(a t )∩It ψ λ (y).andw t (x, y) = v t (y) • e (x).By linearity (113) b t (w t , ψ 0 ⊗ 1) = ≤k b t w t , ψ 0 ⊗ 1 .

B.2. 4 .

 4 The inhomogeneous, semi-classical Airy equation. According to (150), an estimate of v will result from estimating the solutions to(153) s 2 • W + (xz s ) • W = R for s -2 3• z s in a fixed compact set. We first construct solutions to the associated homogeneous equation (154)s 2 • W + (xz s ) • W = 0using Airy functions. In particular, it is well-known that there exists a basis {A + , A -} of solutions to -A (x) + x • A(x)

R 0 R

 0 • W -+ W -(x) x • W + is a solution to (153).Lemma B.11. For each compact set K ⊂ R, there exists C such that, for eachx > 0, if s -2 3 • z s ∈ K and x ∈ [0, x], then (159) |W x (x)| ≤ C • s -|W x (x)| ≤ C • s -4 3 • s -1 3 • R where R denotes the L 2 norm of R over [0, x]. Moreover, there exists M so that if x ∈ M • s 2 3 , x ,then

  x)A -(s -2 3 (xz s )) sin(r • ln(x + 1)) dx,where we have set a 0 (x) := -(x + 1) -1 2 g. Denote by I(t) the integralI(t) = r 2t α 0 a 0 (x)A -(s -2 3 (xz s )) exp(ir • ln(x + 1) dx.

  contains t n . The graph of λ n must cross the lines E t ± 2C qm t, and hence λn (t ± n ) ≥ Ėtn ± 2C qm . This establishes the existence of the desired sequence t *

	n . By Lemma 4.3 in [HlrJdg11], the function t → | Ėt -ȧt (w t )| is bounded. Thus, since t * n ∈ B, we can apply (64) to find a constant κ > 0 so that Ėt * κ n ≥ t * n
	for all sufficiently large n. Hence, λn (t * n ) ≥ κ/t * n -C, and thus lim inf t→∞ t * n • λn (t * n ) ≥ κ.
	But this contradicts Lemma 8.4 below.

  y) y -2 dy .Thus, since α < 2/3, there exists t 1 > 0 such that if t ≤ t 1 then for all y ≥ 1 + t α ≥ (kπ) 2 • t α .For each smooth function ϕ with support in ]1 + t α , ∞[, defineL t (ϕ) = -t 2 • ϕ + (kπ) 2 -E t y 2 • ϕ.Extend L t to a self-adjoint operator on L 2 ([1 + t α , ∞[, dy). It follows from (142) that the spectrum of L t lies in [(kπ) 2 •t α , ∞[. Hence L t is invertible and the operator norm of L -1 t is bounded above by t -α k 2 π 2 . Therefore

	Proof. By Proposition 8.3 and Lemma A.1, there exists C so that if t is sufficiently
	small, then
	(141) Hence, for y ≥ 1 + t α , one finds that E t ≤ (kπ) 2 + C • t (kπ) 2 -E t y 2 ≥ (kπ) 2 • 2 • t α + t 2α 2 3 , 1 + 2t α + t 2α -C • t	2 3 .
	we have
	(142) y 2 (143) E t (kπ) 2 -∞ 1+t α L -1 t (r) 2 dy ≤	t -2α (kπ) 4	∞ 1+t α	|r(y)| 2 dy.

  We now use (143) and the triangle inequality to obtainv [1+2t α ,+∞) ≤ w [1+2t α ,+∞) + L -1 t (r) [1+2t α ,+∞) ≤ exp -t [1+t α ,+∞),y -2 + Ct -α r [1;+∞) [1+t α ,+∞),y -2 + L -1 t (r) [1+t α ,+∞),y -2 + Ct -α r [1;+∞)The claim follows.Corollary B.8. For each α ∈]0, 2 3 [, there exists C and t 0 such that, for each t < t 0 ,

	This implies
	∞ 1+2t α	w 2 (y) dy ≤	3α-2 2 1 -exp -t exp -t 3α-2 2	•	1+2t α 1+t α	w 2 (y) dy
	It follows that for t small enough we have
	∞ 1+2t α	w 2 (y) dy ≤ exp -t	3α-2 2	1+t α 1+2t α	w 2 (y) dy
	(145)	≤	1 2	exp -t	3α-2 2	1+t α 1+2t α	w 2 (y) y -2 dy
		1 2	exp -t	3α-2 2	1+t α ∞	w 2 (y) y -2 dy
	3α-2 2 /2 w ≤ exp -t 3α-2 2 /2 v ∞ 1+2t α v k t (y)
								3α-2 2	•	1+t α ∞	w 2 (y) dy

∞ 1+2t α w 2 (y) dy ≤ exp -t 2 dy y 2 ≤ C • t 2-2α u t 2 .

See, for example, Theorem VI.2.1[Kato].

See page 206 of[LaxPhl] under the heading 'A related quadratic form'.

However, because q 0 is not closed on the domain H 1 β (S), the family qt is not analytic at t = 0 in the sense of[Kato].

has been partly supported by the ANR programs METHCHAOS and NOSEVOL.

Proof. By definition, if s ∈ [0, t] \ K(t, ρ), then w k t 2 / u t 2 ≥ ρ 2 and hence from Lemma 8.11, we find that Ėtλ * t ≥ ρ 2 -1 • λ * t -C. By using Lemma A.1, we find C and t 0 so that t < t 0

The claim follows from integration.

Finally, we use Lemma 8.10 and Corollary 8.12 to prove Theorem 8.9. This will complete the proof of the main theorem.

Proof of Theorem 8.9. Apply Lemma 8.10 with ρ = 1/2. Then apply Corollary 8.12 with ρ = ρ 0 ≥ 1/2 such that

Since s → Ėsλ * s is positive on K(t, ρ 0 ) ⊃ K(t, 1 2 ), we find that

Since lim t→0 E tλ * t = 0, we have the desired conclusion.

Appendix A. Eigenvalue branches of a t

In this appendix, we compute the asymptotics of each real-analytic eigenvalue branch of a t for each ∈ Z + . Proposition A.1. Let ∈ Z + and let t → λ t be a real-analytic eigenvalue branch of a t for t > 0. Then (103)

.

where a = 2(π ) 2 2 3 • (-ζ) and ζ is a zero of the derivative of the Airy function A -defined in (155). Moreover,

To prove Proposition (A.1), we will first transform the eigenvalue problem into an eigenvalue problem that is easier to analyse. If v is an eigenfunction of a t with respect to • with eigenvalue λ, then for each w ∈ C ∞ 0 ([0, ∞[) and t > 0, we have

where we have set µ = 2 π 2 . Hence

A similar argument shows that

Therefore, (105) holds, and via integration we find a so that (106) holds true. Continuity of solutions to ordinary differential equations with respect to coefficients applies to (108) with fixed initial conditions u s (0) = 0 and u s (0) = 1. In particular, we have a solution u 0 to

. Estimate (109) applies to u 0 , and hence it follows from (155) that v is a multiple of A -. The function u satisfies the Neumann condition u (0) = 0 and hence v (-(2µ) -2 3 • a) = 0 as desired.

Proof of Proposition A.1. If v t is a real-analytic eigenfunction branch of of a t associated to the eigenvalue branch λ t , then v s 3 is a real-analytic eigenfunction branch of A s with eigenvalue branch ν s = s -2 (λ s 3µ). Lemma A.2 implies that

3 ).

By differentiating λ s 3 = µ + s 2 • ν s , we find that

By Lemma A.2, both νs and ν s are bounded. Therefore, λs 3 = O(s -1 ) and hence λt = O(t -1 3 ).

Appendix B. The off-diagonal estimates

Let (E t , u t ) be a real-analytic eigenbranch of q t such that lim t→0 E t = E 0 = (π • k) 2 for some positive integer k. For a fixed constant C > 0, let

As in §7.2, let w t denote the orthogonal projection of u t onto the sum of the eigenspaces of a t whose eigenvalues lie in I.

The purpose of this appendix is to prove the following fact that is crucially used in the proof of Proposition 8.7.

Proposition B.1. Let η > 0. There exists κ > 0, δ > 0, and t 0 > 0 such that, if t < t 0 and if ψ 0 is an eigenfunction of a 0 t with eigenvalue λ 0 satisfying

Remark B.2. The condition on λ 0 is only used to ensure that, when t tends to 0, λ 0 tends to k 2 π 2 .

Thus, by summing (114) over ∈ {0, . . . , k -1}, we obtain

For = k, we have

x sin(kπx) dx = (-1) k = 0 Thus, from Lemma B.4 and Lemma 5.1, there exists κ > 0 so that

The latter estimate, combined with (113), (115), and the triangle inquality, yield the claim.

The remainder of this appendix is devoted to proving the preceding lemmas.

Lemma B.5. There exists t 0 > 0 and C so that if t < t 0 , then for each

Proof. Multiply both sides of (117) by a smooth function with compact support φ and integrate over y ∈ [1, ∞[, then integrate by parts to obtain

φ ⊗ e so that by applying Lemma 7.7 to the test function φ ⊗ e , there exists t 0 > 0 and C such that for t < t 0 , we have

Recalling that the L 2 -norm on the right hand side has the weight y -2 , this implies that

The claim follows since y 2 ≥ 1 on the interval over which we integrate.

The strategy of the proof of Lemma B.3 is as follows. By (117), the function v t is a solution to the inhomogeneous equation ( 118)

where µ = E t and (119)

The function ψ 0 t is a solution to the homogeneous equation ( 120)

where µ = λ 0 . Our choice of β in (70) implies that f µ is bounded below by a constant δ 1 > 0 for all small t, < k, and µ ∈ I. Hence we can use WKB type estimates to find a basis v ± of solutions to the homogeneous equation (120). We will then use 'variation of parameters' to express each solution to to (118) in terms of this basis, and we use Lemma B.5 to provide control of the inhomogeneous term r. Finally, we will estimate the integral in (116) using a Riemann-Lebesgue type estimate.

Proof of Lemma B.3. For < k and µ ∈ I, we have f µ ≥ δ 1 > 0, and hence we can apply Theorem 6.2.1 in [Olver], to obtain a basis (v µ,+ , v µ,-) of solutions to the homogeneous equation (120) that satisfy

where, for µ ∈ I and < k, the smooth functions ε and ε have C 0 norm that is uniformly O(t).

Observe that v µ,± have L 2 ([1, β]) norms that are uniformly bounded above and away from 0. Moreover, since v µ,+ and v µ,-are highly oscillatory for small t, an integration by parts argument shows that the

for all sufficiently small t. Here • denotes the L 2 ([1, β]) norm. By the method of 'variation of constants', each solution to

Thus, from (158) and the triangle inequality, we have

Estimate (159) then follows from Lemma B.12 below.

To prove (160) we apply a similar argument to

Lemma B.12. There exists C so that if x ≥ 0 and s

Moreover, there exists a constant M so that if x > M • s 2 3 , then

and

(167)

Proof. The proof is a straightforward consequence of the continuity and known asymptotics of A ± and A ± . From (155) and integration by parts we find that (168)

, as u tends to ∞. Thus there exists u * so that if u ≥ u * , then

and, using (156), (170)

The expressions on the left hand sides of ( 169) and ( 170) are continuous in u, and hence are bounded by a constant C for u ∈ Ǩ ∪ [0, ∞[ where u ∈ Ǩ ⇔ -u ∈ K. By (157) and the change of variable r = s -2 3 • (yz s ), we have

A -(r) 2 dr.

where u s (x) = s -2 3 • (xz s ). Since for each x > 0 and u s (x) ≥sup K estimate (164) follows.

Moreover, from ( 169) and (170) we have

and

(173)

where W t = v(x + 1), g(x) = g(x + 1), and ψ(x) = ψ(x + 1). By assumption, the C 1 norm of g and hence g is uniformly bounded. The function W t satisfies the inhomogeneous equation ( 153) with s = t/ √ E t , and the inhomogeneity R t satisfies (151). To estimate W t and hence (174) we write

where W p,t = W 3t α is the particular solution to (153) defined by (158) for x = 3t α . The function W h,t is a solution to the associated homogeneous equation.

Lemma B.13. For each α ∈ ] 13 42 , 1 3 [ there exists δ > 0 such that

for all t sufficiently small.

Proof. Integration by parts gives (175)

Using Lemmas B.10 and B.11, one finds that there exists C such that, for x ∈ [0, 3t α ]

Thus, using (B.6), we conclude that the first term on the right side of (175) is O(t 2α-2 3 ) • u t • ψ . To bound the second term on the right side of (175), we take 2 3 > α > α and estimate the integrals over [0, t α ] and [t α , 3t α ] separately. Observe that since α < 2 3 then t α > M s 2 3 . Using Lemmas B.10 and B.11, we find C so that for all sufficiently small t

and

By combining these estimates and using (135), we find that

The claim will follow provided we can choose α < 1 3 and α > α so that each power of t appearing on the righthand side is greater than 2/3. This gives us a solution set which is the open triangle in R 2 bounded by the lines α < 1/3, 2α + α = 1, and 3αα/4 = 5/6. The two latter lines intersect for α = 13 42 . The claim follows.

The same kind of argument allows us to estimate the norm of W p,t Lemma B.14. For all α ∈ ] 7 33 , 1 3 [, there exists δ > 0 and C > 0 such that (176)

Proof. As above we consider α > 1 3 and take some α > α. Using Lemmas B.10 and B.11, one finds that

The claim will follow provided we can find α < α and α < 1 3 such that 5α 2 -3 α 4 -1 6 > 0 and 2α -α 2 -2 3 > 0. The solution set is here a quadrilateral whose projection on the α-axis is the interval ] 7 33 , 1 3 [. Finally, we consider the integral corresponding to the homogeneous part W h,t of W t :

(177)

There exist constants a + , a -, depending on t, such that

where W + and W -are as defined in (157) with the parameter s and z s defined in (148).

We first prove a lemma that roughly says that in the decomposition W = W p,t + a + W + + a -W -the L 2 norm is mainly supported by a -W -.

Integration by parts shows that

where we have set a 1 (x) = a 0 (x)(x + 1). Since α < 1 3 < 2 3 and s is of order t, and since A -is rapidly decreasing, the boundary term at 2t α is O(t ∞ ). Observe that we have a global 1 r prefactor in front of the integral term. Thus, when the ∂ x is applied to a 1 , we gain 1/r, that is, something of order t. When ∂ x hits the Airy function, we lose a s -2 3 so that the global prefactor is of order s -2 3 r which is O(t 1 3 ). Summarizing, integrating by parts gains at least a prefactor t 1 3 . By repeated integration by parts we thus observe that we can write, for each N

where the a k, are some constants and the remainder term R N can be written

for some smooth functions a k, . If we fix some order t M then, using that A -and all its derivatives are rapidly decreasing we can find N such that the remainder R N is O(t N ). This tells us that I(t) admits a complete asymptotic expansion of the form

From the first integration by parts we see that a 00 = i g(0).

and the second term is then of order t 1 3 . The claim follows by taking the imaginary part.

We will use the following to verify that the leading order term does not vanish.

Lemma B.17. We have

where -ζ is a zero of the derivative of A -.

Proof. From (148) we have Corollary B.18. There exists κ > 0 and t 0 > 0 so that if t < t 0 , then

for t sufficiently small. Proof. Let ζ be the zero of A -that comes from Lemma B.17. Since A -is a nontrivial solution to a second order differential equation, A -can not vanish at a zero of the derivative A -. Hence, for sufficiently for small t, we have |A -(-s

By arguing as in the proof of Lemmas B.17 and B.12 and using s ∼ t, we find c 1 > 0 so that

where k 1 = ∞ sup(K) |A -(u)| 2 and t is sufficiently small. In particular, (180)

Hence the claim follows from Lemma B.16.

The estimate in the latter corollary is homogeneous so that we can multiply W - by a -.

Using Lemma B.15 we then have Putting all the different pieces together yields the estimate.