Pierre Bérard

AND Bernard Helffer

Rola Kiwan

Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip

Keywords: Mathematics Subject Classification. 58C40, 49Q10 Spectral theory, Courant theorem, Laplacian, Nodal sets, Möbius strip. 1

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The question we are interested in was initially suggested by Étienne Ghys in March 2016, during a conference in Abu Dhabi.

We start with the standard strip, S ∞ := (0, π) × (-∞, +∞) , and we look at the standard Laplacian -∆ with Dirichlet condition at x = 0 and x = π, and we add the conditions, (1.1) u(x, y + πa) = u(π -x, y) , u ∈ H 2 loc (S ∞ ) , where a is a positive parameter. Equivalently, we look at the Laplacian on the flat Möbius strip M a , with Dirichlet boundary condition (see Section 2). According to Courant's nodal domain theorem [9, Chap. VI.6], an eigenfunction associated with the nth Dirichlet eigenvalue of M a has at most n nodal domains. The eigenvalue λ n is called Courant-sharp if there exists an associated eigenfunction with exactly n nodal domains. As usual, we list the eigenvalues in nondecreasing order, multiplicities accounted for, starting from the label 1.

Remark 1.1.

There are obvious restrictions for an eigenvalue to be Courant-sharp. Indeed, let k ≥ 2 be an integer. If the eigenvalue λ k satisfies λ k-1 < λ k = • • • = λ k+ -1 < λ k+ for some integer ≥ 2, then the eigenvalues λ k+1 , . . . , λ k+ -1 cannot be Courant-sharp. Our aim is to prove the following theorem. Equivalently, for the Laplacian with Dirichlet condition on the boundary of S ∞ and the periodicity condition (1.1), with a = 1, the only Courant-sharp eigenvalues are the first and second ones.

Here is a sketch of the proof. Since the Möbius strip is a surface with boundary (actually a quotient of a cylinder), we can apply [6, p. 524] to extend Pleijel's theorem [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF], and conclude that there exists an explicit number γ(2) < 1 such that lim sup

k-→+∞ ν(k) k ≤ γ(2) < 1 ,
where ν(k) denotes the maximal number of nodal domains of an eigenfunction associated with the eigenvalue λ k . This inequality implies that there are finitely many Courant-sharp eigenvalues only. Using Weyl's law with a controlled remainder term, and an adapted Faber-Krahn inequality, we give an upper bound and a condition to be satisfied by Courant-sharp eigenvalues. Together with Remark 1.1, these conditions limit the possible Courant-sharp eigenvalues of M 1 to the set {λ 1 , λ 2 , λ 6 , λ 7 }.

Courant's theorem implies that λ 1 and λ 2 are Courant-sharp. To conclude the proof, we determine the possible nodal patterns of the eigenfunctions associated with λ 6 and λ 7 .

Because of its symmetries, and higher eigenvalue multiplicities, the case of the "square" Möbius strip M 1 (a = 1) seems to be the most interesting. Other investigations corresponding to irrational or small a's could also be performed as in [15], leading to partial answers to the Courant-sharp question. In this paper, we shall only consider the case a = 1.

Looking for the Courant-sharp eigenvalues of the flat Möbius strip comes naturally in view of the known results for the square and for the flat tori. It is also natural to consider cylinders and Klein bottles. The following table displays some of the known results. (1) Other tori are treated in [START_REF] Helffer | Minimal partitions for anisotropic tori[END_REF], [START_REF] Bonnaillie-Noël | Spectral minimal partitions for a family of tori[END_REF].

(2) The Robin boundary condition is written ∂u ∂ν + hu = 0, where h is the Robin parameter, and ν the outward-pointing normal.

(3) The cylinders considered in the table are (0, π) × S 1 r , r ∈ 1 2 , 1 . (4) The flat Klein bottles considered in the table have fundamental domains (0, π) × (0, cπ), c ∈ {1, 2}, with sides {x = 0} and {x = π} identified, while sides {y = 0} and {y = cπ} are identified with reversed orientations. [START_REF] Bérard | Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders[END_REF] The case of spheres appears in [24,16]. (6) In the case of Neumann or Robin boundary condition, determining Courant-sharp eigenvalues is more difficult because one can only apply the Faber-Krahn inequality (see Section 7) to nodal domains which do not meet the boundary.

The paper is organized as follows. In Section 2, we describe the Möbius strip, and compute its spectrum using separation of variables. Sections 3-5 are devoted to the description of the nodal patterns for the first eigenspaces, E λ 2 , E λ 6 , and E λ 7 .

In Section 6, we give a Weyl law with a controlled remainder term, and consider isoperimetric and Faber-Krahn inequalities for the Möbius strip. In Section 7, we give an upper bound, together with a condition à la Faber-Krahn, to be satisfied by Courant-sharp eigenvalues. This section also contains the proof of Theorem 1.2.

In Section 8, we consider an Euler type formula for nodal patterns on the Möbius strip. In Section 9, we give examples of high energy eigenfunctions of M 1 with only two nodal domains.

Acknowledgement.

The second author would like to thank C. Léna for useful discussions. The authors would also like to thank the two anonymous referees for their constructive comments, and for pointing out several misprints.

2.

The Möbius strip M a 2.1. Presentation and geometry. Let S ∞ := (0, π) × (-∞, ∞), be the infinite strip with width π, equipped with the flat metric dx 2 + dy 2 of R 2 . Given a > 0, define the following isometries of S ∞ :

(2.1)

         σ a : (x, y) → (π -x, y + a π) , : (x, y) → (π -x, y) , τ t : (x, y) → (x, y + t), for t ∈ R . Define the groups (2.2)    G := σ k a | k ∈ Z , G 2 := σ k a | k ∈ 2 Z . The group G 2 is a subgroup of G, of index 2, generated by σ 2
a . The action of G on S ∞ is smooth, isometric, totally discontinuous, without fixed points. By [7, Section 2.4], we can consider the quotient manifolds with boundary

(2.3) C a := S ∞ /G 2 and M a := S ∞ /G ,
equipped with the flat metric induced from the metric of S ∞ .

The cylinder C a is the product manifold (0, π) × S 1 a , where S 1 a is the circle R/(2πaZ). One can view C a as the rectangle (0, π) × [0, 2πa] with the sides (0, π) × {0} and (0, π) × {2aπ} identified, (x, 0) ∼ (x, 2aπ). This rectangle is a fundamental domain for the action of G 2 on S ∞ .

The isometry σ a induces an isometry of C a , whose square is the identity. The Möbius strip M a can also be obtained as the quotient C a / {Id, σ a }. One can view M a as the rectangle R a := (0, π) × [0, aπ], with the sides (0, π) × {0} and (0, π) × {aπ} identified by (x, 0) ∼ (π -x, aπ). The rectangle R a is a fundamental domain of the action of G on S ∞ . In the sequel, we will mainly view M a as R a , together with the identification (x, 0) ∼ (π -x, aπ).

The isometry of S ∞ induces an isometry of C a and M a . The action t → τ t induces an isometric action of S 1 a on C a and M a . Physically, one can realize M a by making a paper model: a rectangular sheet of paper, [0, π] × [0, aπ], is twisted by a rotation of angle π about its symmetry axis x = π 2 , and the two horizontal sides {y = 0} and {y = aπ} are glued together. The fixed point set of in M a is a circle (the "soul") of length aπ, while the boundary ∂M a has length 2aπ.

The cylinder C a embeds isometrically in R 3 by

(2.4) (0, π) × S 1 a (u, v) → u, a cos( v a ), a sin( v a ) ∈ R 3 ,
and we can view the cylinder as a collection of segments (0, π) a attached to the circle {y 2 + z 2 = 1} ∩ {x = 0}, orthogonally to the plane {x = 0}.

Define the map F by

(2.5) (w, v) → w cos v a , (R + w sin v a ) cos 2v a , (R + w sin v a ) sin 2v a , where w = u -π 2 , (w, v) ∈ -π 2 , π 2 × [0, aπ], and R > 0. It is easy to see that F is a diffeomorphism from M a onto F (M a ) ⊂ R 3 provided that R > π 2 .
One can view the surface F (M a ) as the collection,

F (- π 2 , π 2 ), 2v a v ∈ [0, aπ] ,
of segments centered at the point 0, R cos 2v a , R sin 2v a , contained in the plane xOω a , where ω a = 0, cos 2v a , sin 2v a , and making in this plane an angle v a with the axis Ox. Remark 2.1. We will use the map F to visualize the topology of the nodal sets and nodal domains of the eigenfunctions of M 1 in three dimensions. Note that the map F is not an isometric embedding. As a matter of fact, it is not even conformal, but the vectors ∂ u F and ∂ v F are orthogonal. The surface M a , with the metric F * (dx 2 + dy 2 + dz 2 ) induced from the canonical metric in R 3 , has negative curvature. According to [13, § 15] and [10, Lecture 14], if the Möbius strip M a can be isometrically embedded into R 3 , then a > π 2 , and such embeddings actually exist for a > √ 3. where α p,q ∈ C, and the sum extends over the pairs (p, q) ∈ N • × Z such that λ(p, q, a) = λ(m, n, a). Since Φ • σ a (x, y) = Φ(π -x, y + πa) = (-1) p+q+1 α p,q sin(px) exp(iq y a ) , it follows that Φ • σ a = Φ if and only if the summation in (2.7) extends over the pairs (p, q) ∈ N • × Z such that λ(p, q, a) = λ(m, n, a), with the additional condition that p + q is odd. As a consequence, we have the following result.

Lemma 2.2. A complete family of real

   cos θ = √ A 2 + B 2 , sin θ = √ C 2 + D 2 ,
we write Φ as, (3.2) Φ β,θ (x, y) = cos θ sin(mx) sin(ny + α)

+ sin θ sin(nx) sin(my + β) .

Using the isometric S 1 action τ t : (x, y) → (x, y + t) on M 1 , we can assume that α = 0. We also observe that Φ β+π,θ (x, y) = (-1) n cos θ sin(mx) sin(n(y + π)) + (-1) m+1 sin θ sin(nx) sin(m(y + π) + β) , so that

(3.3) Φ β+π,θ (x, y) = (-1) n Φ β,θ (x, y + π) = (-1) n Φ β,θ (π -x, y) .
It follows that the nodal sets of Φ β+π,θ and Φ β,θ are symmetric with respect to [START_REF] Bérard | Dirichlet eigenfunctions of the square membrane: Courant's property, and A. Stern's and Å. Pleijel's analyses[END_REF][START_REF] Bérard | Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle[END_REF], one can reduce the parameter set further.

x = π 2 in R 1 or M 1 .

Proposition 3.1. To determine the nodal patterns of the Dirichlet eigenfunctions

Φ ∈ E [m,n] , for m, n ∈ N • with m + n is odd, it is sufficient to study the nodal properties of the family Φ β,θ , Φ β,θ (x, y) = cos θ sin(mx) sin(ny) + sin θ sin(nx) sin(my + β) , for β ∈ (0, π], and θ ∈ [0, π 2 ]. When [m, n] = [1, 2] or
(1) For [m, n] = [1, 2], it suffices to consider (β, θ) ∈ (0, π 2 ] × [0, π 2 ]. Indeed, (3.4)    Φ β+ π 2 ,θ = -Φ β,θ • τ -π 2 , Z(Φ β+ π 2 ,θ ) = τ π 2 Z(Φ β,θ ) . (2) For [m, n] = [2, 3], it suffices to consider (β, θ) ∈ (0, π 3 ] × [0, π 2 ]. Indeed, (3.5)    Φ β+ π 3 ,θ = -Φ β,θ • τ -π 3 , Z(Φ β+ π 3 ,θ ) = τ π 3 Z(Φ β,θ
) . For the analysis of nodal sets, we need the following definition.

Definition 3.2. A point (x, y) is a critical zero of a function Φ if Φ(x, y) = 0 and d (x,y) Φ = 0. A critical zero has order k ∈ N • if
the function Φ and its derivatives of order less than or equal to (k -1) vanish at (x, y), and a least one derivative of order k does not. A point (x, y) such that Φ(x, y) = 0 and d (x,y) Φ = 0 is called regular.

For eigenfunctions in dimension 2, the critical zeros are isolated, and their orders determine the structure of the nodal set locally. In the case of the Möbius strip M 1 , the eigenfunctions are defined globally on R 2 , and one defines a boundary critical zero as a critical zero of (the extended function) Φ, which lies on the boundary ∂M 1 . As a first step to Theorem 1.2, we prove the following result. According to Remark 1.1 and Table 2.1, the eigenvalue λ 2 = λ(1, 2) has multiplicity 4, so that λ 3 , . . . , λ 5 cannot be Courant-sharp. Since λ 6 = λ(3, 0) is simple, and since the associated eigenfunction sin(3x) has 2 nodal domains in M 1 , the eigenvalue λ 6 is not Courant-sharp. The eigenvalue λ 7 = λ(2, 3) has multiplicity 4. This implies that λ 8 , . . . , λ 10 cannot be Courant-sharp. To finish the proof of Proposition 3.3, it suffices to prove that λ 7 is not Courantsharp. This is the purpose of Lemma 5.1 in Section 5; the proof is by analyzing the possible nodal patterns of eigenfunctions in

E λ 7 = E [2,3] .
Although we already know that λ 2 is Courant-sharp, it is interesting to describe the possible nodal patterns of associated eigenfunctions. They indeed present some peculiar properties when compared to second Dirichlet eigenfunctions of a simplyconnected domain. In Section 4, we give the possible nodal patterns of eigenfunctions in E λ 2 = E [START_REF] Bérard | Remarks on the boundary set of spectral equipartitions[END_REF][START_REF] Bérard | Dirichlet eigenfunctions of the square membrane: Courant's property, and A. Stern's and Å. Pleijel's analyses[END_REF] , up to isometries given by the S 1 action, and to the symmetry with respect to x = π 2 . In Section 5, we give the possible nodal patterns of eigenfunctions in E λ(2,3) , up to isometries given by the S 1 action, and to the symmetry with respect to x = π 2 . As a consequence, we shall conclude that an eigenfunction in E λ 7 has at most 6 nodal domains, so that λ 7 is not Courant-sharp.

Analysis of the eigenspace E λ 2

In this section, we describe the nodal patterns of the eigenfunctions Φ

β,θ ∈ E [1,2] = E λ 2 , with β ∈ (0, π 2 ] and θ ∈ [0, π 2 ]
. According to Proposition 3.1, this is sufficient to determine all the possible nodal patterns, up to isometries. We consider the following cases and subcases.

(

) θ = 0 and θ = π 2 . (2) β = π 2 , three subcases: 0 < θ < π 4 , θ = π 4 and π 4 < θ < π 2 . (3) β ∈ (0, π 2 ), three subcases 0 < θ < θ β , θ = θ β and θ β < θ < π 2 1
(the function β → θ β is described below). The analysis of these cases can be done using the methods described in Section 5. For this reason, we shall not give full details here, but a mere description.

The figures below display the nodal lines in the fundamental domain R 1 ; recall that the Möbius strip M 1 is obtained by identifying the lines {y = 0} and {y = π} via (x, y) ∼ (π -x, y + π). The nodal lines appear in red, the Dirichlet boundary in blue. When the lines {y = 0} and {y = π} are not nodal, they appear as dashed black lines to indicate the Möbius identification. 2 )

The Möbius strip M 1 is not simply-connected (it retracts onto the soul circle). A consequence of the conjecture is the non-existence of interior critical zeros. A counter-example to the conjecture has been constructed by M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and N. Nadirashvili, with a domain Ω not simply connected. We refer to [19] and the recent paper [START_REF] Kennedy | A toy Neumann analogue of the nodal line conjecture[END_REF] for more details and references on the nodal line conjecture. , the nodal set consists in two simple curves which intersect at an interior critical zero of order 2; one curve is closed, the other hits the boundary at two critical zeros of order 2. As expected, there are two nodal domains in each cases. We refer to Figure 4.6 for a 3D-picture.

(a) 0 < θ < π 4 (b) θ = π 4 (c) π 4 < θ < π 2 Figure 4.2. Nodal sets Z(Φ π 2 ,θ ), for 0 < θ < π 2 4.3. Case β = 0 and 0 < θ < π 2 .
This case is similar to the preceding one. Indeed, recall that the nodal sets Z(Φ 0,θ ) and Z(Φ π 2 ,θ ) are isometric. We give the corresponding pictures for completeness. The reason is that the map F is not conformal, as pointed out in Remark 2.1.

Analysis of the eigenspace E λ 7

The purpose of this section is to finish the proof of Proposition 3.3 by proving the following lemma. Recall that This operation produces two intertwined strips, one Möbius strip (corresponding to the domains labelled "5" and "6"), and a doubly twisted (orientable) strip (corresponding to the domains labelled "1, 2, 3" and "4"). Proof. In this proof, we use the abbreviation1 Φ for Φ β,θ .

E λ 7 = E [2,3] , see
Assertion [START_REF] Bérard | Remarks on the boundary set of spectral equipartitions[END_REF]. Let ξ ∈ (0, π). Then, {x = ξ} ⊂ Z(Φ β,θ ) if and only if Φ(ξ, y) = cos θ sin(2ξ) sin(3y) + sin θ sin(3ξ) sin(2y + β) ≡ 0 , as a function of y.

If β = 0, choosing y = π 2 we find that sin(2ξ) = 0. This in turn implies that sin θ sin(3ξ) sin(2y + β) ≡ 0, and hence that sin(3ξ) = 0, a contradiction.

If β = 0, choosing y = 0, we find that sin(3ξ) = 0. This in turn implies that cos θ sin(2ξ) sin(3y) ≡ 0, and hence that sin(2ξ) = 0, a contradiction.

Assertion (2).

Take η ∈ [0, π], and assume that Φ(x, η) ≡ 0 as a function of x. Choosing x = π 2 , we find that sin(3η) = 0. This in turn implies that sin θ sin(3x) sin(2η + β) ≡ 0, which implies that sin(2η + β) = 0. The conditions sin(3η) = 0 and sin(2η + β) = 0 occur simultaneously if and only if (η, β) ∈ (0, 0); (π, 0); ( π 3 , π 3 ) .

Proposition 5.3. Assume that β ∈ [0, π 3 ], and that θ ∈ (0, π 2 ). Let (ξ, η) ∈ M 1 (or R 1 ) be a critical zero of Φ β,θ . The following properties hold.

(1) The point (ξ, η) satisfies,

(5.2) sin(3η) sin(2η + β) sin(ξ) = 0.

(

) If ξ ∈ (0, π), i.e., if (ξ, η) ∈ M 1 is an interior critical zero of Φ β,θ , then (5.3) (η, β) ∈ (0, 0); (π, 0); ( π 3 , π 3 ) , 2
and the point (ξ, η) has order 2. In particular, if β ∈ (0, π 3 ) and θ ∈ (0, π 2 ), the function Φ β,θ has no interior critical zero. If β ∈ 0, π (c) the boundary critical zero (ξ, η) can only have order at least 4 if sin(3η) sin(2η + β) = 0. In that case, the order is 4.

Proof. We use the abbreviation Φ for Φ β,θ in the proof.

Assertion [START_REF] Bérard | Remarks on the boundary set of spectral equipartitions[END_REF]. The point (ξ, η) is a critical zero of Φ if and only if it satisfies the system of equations Φ(x, y) = cos θ sin(2x) sin(3y) + sin θ sin(3x) sin(2y + β) = 0, (5.6a) ∂ x Φ(x, y) = 2 cos θ cos(2x) sin(3y) + 3 sin θ cos(3x) sin(2y + β) = 0, (5.6b) ∂ y Φ(x, y) = 3 cos θ sin(2x) cos(3y) + 2 sin θ sin(3x) cos(2y + β) = 0. (5.6c) Since θ ∈ (0, π 2 ), if (ξ, η) is a critical zero of Φ, the determinant of the linear system (5.6a) and (5.6b) must vanish at (ξ, η), sin(3η) sin(2η + β) (3 sin(2ξ) cos(3ξ) -2 cos(2ξ) sin(3ξ)) = 0, or equivalently sin(3η) sin(2η + β) sin 3 (ξ) (4 cos 2 (ξ) + 1) = 0 . Assertion [START_REF] Bérard | Dirichlet eigenfunctions of the square membrane: Courant's property, and A. Stern's and Å. Pleijel's analyses[END_REF]. If ξ ∈ (0, π), (5.2) implies that sin(3η) sin(2η + β) = 0. Since (ξ, η) is an interior critical zero, the system (5.6), and the assumption θ ∈ (0, π 2 ), imply that (5.7) sin(3η) sin(2η + β) = 0 ⇔ (η, β) ∈ (0, 0); (π, 0); ( π 3 , π 3 ) .

To conclude the proof of the assertion, we need the second derivatives of Φ.

∂ xx Φ(x, y) = -4 cos θ sin(2x) sin(3y) -9 sin θ sin(3x) sin(2y + β), (5.8a) ∂ xy Φ(x, y) = 6 cos θ cos(2x) cos(3y) + 6 sin θ cos(3x) cos(2y + β), (5.8b) ∂ yy Φ(x, y) = -9 cos θ sin(2x) sin(3y) -4 sin θ sin(3x) sin(2y + β).

(5.8c)

For (η, β) ∈ (0, 0); (π, 0); ( π 3 , π 3 ) , we have ∂ x 2 Φ β,θ (x, η) ≡ 0, ∂ y 2 Φ β,θ (x, η) ≡ 0 .
The critical zero (ξ, η) has order at least 3, if ∂ y Φ β,θ (ξ, η) = 0 and ∂ xy Φ β,θ (ξ, η) = 0. This system implies that 3 cos(3ξ) sin(2ξ) -2 cos(2ξ) sin(3ξ) cos(3η) cos(2η + β) = 0 .

The term in the parenthesis is equal to 2 sin 3 (ξ)(4 cos 2 (ξ) + 1), and hence does not vanish since ξ ∈ (0, π). The second factor does not vanish either since (η, β) ∈ (0, 0); (π, 0); ( π 3 , π 3 ) . It follows that the second derivative ∂ xy Φ β,θ does not vanish at (ξ, η). The proof of Assertion (2) is complete.

Assertion [START_REF] Bérard | Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle[END_REF]. Let ξ ∈ {0, π} and ε = cos(ξ). The only derivatives of Φ, of order less than or equal to 4, which are not identically identically zero on {x = ξ} are, (5.9f)

The point (ξ, η) is a boundary critical zero if and only if ∂ x Φ(ξ, η) = 0. This occurs either for (η, β) ∈ (0, 0); (π, 0); ( π 3 , π 3 ) , or for η ∈ 0, π 3 , 2π 3 , π , and θ given by cot θ = -εf (β, η). The critical point (ξ, η) has order at least 3 if and only if ∂ xy Φ(ξ, η) = 0, i.e., g(β, η) = 0. The boundary critical zero (ξ, η) has order at least 4 if and only if ∂ x 3 Φ(ξ, η) = 0 and ∂ xy 2 Φ(ξ, η) = 0 simultaneously. For this to occur, we must have sin(3η) sin(2η+β) = 0. The relations ∂ x 3 y Φ(ξ, η) = 0 and ∂ xy 3 Φ(ξ, η) = 0 can only occur simultaneously if cos(3η) cos(2η + β) = 0. According to the previous relation, this means that a boundary critical zero has order less than or equal to 4. Remark 5.4. We shall refine the analysis of Assertion (3) later on, in Subsection 5.4. Remark 5.5. The function f (β, y), defined in (5.4), and the function g(β, y), defined in (5.5), are crucial to understand the structure of the nodal sets. We study them in Subsection 5.3. Proposition 5.6. Define the set (5.10)

Z β := Z(Φ β,0 ) ∩ Z(Φ β, π
2 ). This is the set of common zeros of the eigenfunctions Φ β,θ , where β is fixed, and θ varies in (0, π 2 ). Define the function (5.11) P β (x, y) = sin(2x) sin(3y) sin(3x) sin(2y + β).

(1) For β ∈ 0, π In Figure 5.2(A), the nodal lines of Φ β,0 appear as dashed black lines; the nodal lines of Φ β, π 2 appear as dot/dashed purple lines; the lines {y = 0} and {y = π} are common to both functions and appear in red. The points of Z β \{y = 0} are marked "O" (in red). The "checkerboard" in Figure (B) displays the prohibitions: the nodal set Z(Φ β,θ ) cannot visit the domains in gray, and cannot cross the segments marked with "X". The "P" denote points through which the nodal set cannot pass. The points marked "R" denote regular points (in particular, the nodal set cannot hit the boundary at regular points). We will use similar checkerboards in Subsections 5.4 and 5.5, to describe the nodal sets, using Propositions 5.2, 5.3, 5.6 and Lemma 5.8 Notation 5.7. Given a bounded domain Ω, let δ(Ω) denote its smallest Dirichlet eigenvalue.

Let Φ ∈ E [START_REF] Bérard | Dirichlet eigenfunctions of the square membrane: Courant's property, and A. Stern's and Å. Pleijel's analyses[END_REF][START_REF] Bérard | Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle[END_REF] = E λ 7 be any eigenfunction, and let Ω be any nodal domain of Φ. Then, it is well-known that δ(Ω) = λ(2, 3) = 13. Clearly, any nodal domain of the function P β is an open set contained in some nodal domain of Φ β,0 or Φ β, π 2 . Lemma 5.8. For β ∈ [0, π 3 ] and θ ∈ (0, π 2 ), no nodal domain Ω P of the function P β can contain a closed nodal line of Φ β,θ , or a nodal line whose end points are located on Ω ∩ ∂M 1 .

Proof. Indeed, such a curve would bound a nodal domain ω of Φ β,θ , strictly contained in Ω P . On the one hand, by domain monotonicity of Dirichlet eigenvalues, we would have δ(ω) > δ(Ω P ) ≥ λ(2, 3) (the second inequality follows from the fact that the nodal domains of P β are contained in nodal domains of the functions Φ β,0 or Φ β, π

2 ). On the other hand, we have δ(ω) = λ(2, 3), a contradiction. f (β, y) andg(β, y). Recall the expression of the function f (β, y) defined in (5.4). For y ∈ (0, π 3 )

Analysis of the functions

∪ ( π 3 , 2π 3 ) ∪ ( 2π 3 , π), f (β, y) := 3 2
sin(2y + β) sin(3y) .

We have,

∂ y f (β, y) = 3 2 sin 2 (3y) g(β, y),
where g(β, y) = 2 sin(3y) cos(2y + β) -3 cos(3y) sin(2y + β).

was defined in (5.5). Expanding g(β, y), and using the identity 1 sin 2 (y) = 1 + cot 2 (y), we obtain g(β, y) = sin 5 (y) 2 cos β(1 + 5 cot 2 (y)) -sin β cot(y)(5 + 3 cot 4 (y)) .

It follows that

(5.12) g(β, y) = 0 and y ∈ (0, π) ⇔ cot β = 1 2 cot(y) (5 + 3 cot 4 (y)) 1 + 5 cot 2 (y) .

Lemma 5.9. The function,

(5.13) h(t) := 1 2 t(5 + 3t 4 ) 1 + 5t 2 ,
is an increasing bijection from R to R. As a consequence, the function

(5.14) y → arccot (h(cot y))
is an increasing bijection from (0, π) to (0, π). Furthermore,

(5.15) t -∞ -1 √ 3 0 1 √ 3 1 √ 3 ∞ h(t) -∞ -1 √ 3 0 1 √ 3 2 3 √ 3 ∞ y 0 π 6 π 4 π 3 π 2 2π 3 π arccot (h(cot y)) 0 π 6 arccot( 2 3 ) π 3 π 2 2π 3 π Proof. A straightforward computation gives h (t) = 5 2 (3t 2 -1) 2 (t 2 + 1) (1 + 5t 2 ) 2 .
The lemma follows.

As a consequence, given β ∈ (0, π 3 ), there exists a unique y β ∈ (0, π 3 ), such that β = arccot (h(cot y β )). It follows that, given β ∈ (0, π 3 ), the function y → f (β, y) varies as indicated (5.16).

(5.16)

y 0 y β π 3 2π 3 π ∂ y f (β, y) -0 + + + ∞ ∞ ∞ ∞ f (β, y) m β -∞ -∞
where (5.17 From Lemma 5.9, the definition of g(β, y) in (5.5), and the uniqueness of the zero of the function y → g(β, y), we deduce that (5.18)

       β = π 6 ⇒ y β = π 6 , β > π 6 ⇒ β > y β > π 6 , β < π 6 ⇒ β < y β < π 6
, and from the definition of g(β, y), see (5.5), (5.19)

   y π 3 -β = π 3 -y β , y π 6 = π 6 .
It turns out that y and β play a symmetric role. Instead of fixing β, we could fix some η ∈ (0, π 3 ). Then, there exists a unique β η = arccot(h(cot η)), and the function y → f (β η , y), has an infimum m βη at η.

The relation cot(β) = h (cot(y)) can also be rewritten as, (5.20) tan(β) = (tan(y)) , where

:= 2t 3 (5 + t 2 ) 3 + 5t 4 . We conclude that (5.22) β η ∼ 10 3 η 3 , as η → 0.

From (5.17), (5.14) and (5.18), we conclude that the function β → m β satisfies, (5.23)

   m π 3 -β = m β , m π 6 = 3 2 .
Writing m η = f (β η , η) and using (5.22), we obtain that (5.24) lim

η→0 m η = 1.
Finally from the definitions of m β = f (β, y β ) and of y β , and using (5.18), we conclude that (5.25)

       m β > 0, for β < π 6 , m β < 0, for β > π 6 , 1 ≤ m β ≤ 3 2 . Furthermore, m β = 1 if and only if β = 0 or β = π
2 , and m β = 3 2 if and only if β = π 6 . Define the value θ β by the relation

(5.26) cot(θ β ) = 3 2 sin(2y β + β) sin(3y β ) = 3 2 sin(2y β + arccot(h(cot y β ))) sin(3y β )) .
Equivalently, the value θ β is defined by the equations

(5.27) cot(β) = h (cot y β ) , 2 cos θ β sin(3y β ) -3 sin θ β sin(2y β + β) = 0 .
From the definition of f and (5.23), we conclude that (5.28)

       θ π 3 -β = θ β , θ π 6 = arccot(3/2) , lim β→0 θ β = lim β→ π 2 θ β = π 4 .
The graph of β → θ β is symmetric with respect to β = π 6 , the function decreases from π 4 to arccot(3/2) ≈ 0.588003 as β increases from 0 to π 6 , and then increases to The checkerboards associated with the function P β , P β (x, y) = sin(2x) sin(3y) sin(3x) sin(2y + β) , are given in Figure 5.5. According to Proposition 5.6, the nodal set Z(Φ) is contained in the set {P β ≤ 0} (the rectangles in white). The points marked "O" in the figure are the common zeros of the family of functions Φ β,θ , for a fixed β ∈ (0, π 3 ), and all θ ∈ (0, π 2 ). The function Φ does not vanish at the points marked "P", so that Z(Φ) does not pass through these points. The points marked "R" are regular points of the boundary, the partial derivative ∂ x Φ does not vanish at these points. (1) For y 0 ∈ 0, π 3 , 2π 3 , π , the function Ψ(x, y 0 ) only vanishes at the points x = π 3 and x = 2π 3 . The nodal set Z(Φ) cannot hit the corresponding horizontal lines elsewhere.

(2) For y 0 ∈ π 2 -β 2 , π -β 2 , the function Ψ(x, y 0 ) only vanishes at the point x = π 2 . The nodal set Z(Φ) cannot hit the corresponding horizontal lines elsewhere.

(3) By (5.6b) and (5.6c), at the common zeros marked "O", the tangent to the nodal line is oriented SouthEast-NorthWest, pointing to the white domains, except on the bottom line where the orientation is SouthWest-NorthEast. (4) According to Subsection 5.3 and (5.16), the function Ψ(0, y) has precisely one zero in each interval ( π 3 , π 2 -β 2 ) and ( 2π 3 , π -β 2 ). The nodal set Z(Φ) hits the boundary at a critical zero of order 2.

(5) According to Subsection 5.3 and (5.16), the function Ψ(π, y) has precisely one zero in each interval ( π 2 -β 2 , 2π 3 ) and (πβ 2 , π). The nodal set Z(Φ) hits the boundary at a critical zero of order 2. In the interval (0, π 3 ), the function Ψ(π, y) has 0, 1, or 2 zeros, depending on whether cot θ is less than, equal to, or larger than m β (defined in Equation (5.17)). In the first case, the nodal set does not hit the boundary. In the second case, it hits the boundary at a critical zero of order 3. In the third case, it hits the boundary at two critical zeros of order 2.

From these properties, we deduce the general aspect of the nodal set Z(Φ), with reference to the right hand picture in Figure 5.5.

Column 1:

The nodal set in the rectangles A and B is a simple curve with end points a boundary critical zero of order 2 and the common zero marked "O". Column 2: The nodal set in the rectangles C, D and E is a simple curve with end points the common zeros marked "O". Column 3: The nodal set in the rectangles F and G is a simple curve with end points the common zeros marked "O".

Column 4: In the rectangles J and K, the nodal set is a simple curve with end points the common zero marked "O" and a boundary critical zero of order 2. In the rectangle H, there are three different cases. 0 < θ < θ β : The nodal set is a simple curve joining the two common zeros marked "O". θ = θ β : The nodal set consists in two arcs entering the rectangle at the common zeros marked "O", and meeting at the boundary critical zero (π, y β ), of order 3.

θ β < θ < π 2 :
The nodal set consists in two disjoint arcs entering the rectangle at the common zeros marked "O", and hitting the boundary at two distinct critical zeros, each of order 2.

Remark 5.12. One can localize the position of the nodal curves by looking at their intersections with the lines {x = x 0 }, i.e., at the zeros of the function Ψ(x 0 , y), and by making use of (5.16).

One can summarize the preceding discussion as follows, see Figure 5.6. Fix some β ∈ (0, π 3 ). Let y β ∈ (0, π 3 ) and θ β ∈ (0, π 2 ), be defined by (5.12) and (5.26) respectively. The nodal sets of the family {Φ β,θ }, with β fixed and θ ∈ (0, π 2 ) present a bifurcation at θ = θ β , as illustrated by 2 . Their nodal sets have already been studied in [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF], see also [START_REF] Bérard | Dirichlet eigenfunctions of the square membrane: Courant's property, and A. Stern's and Å. Pleijel's analyses[END_REF] for more details. They are given in Figure 5.

λ 1 (Ω) A(Ω) ≥ πj 2 0,1 .
Proof. The proof is given in [25] (with a more detailed proof analyzing more carefully the regularity assumptions given in [6]). In the two papers additional assumptions are made which are only used in the proof of the isoperimetric inequalities. We have replaced them by Howards's result [START_REF] Howards | The isoperimetric problem on surfaces[END_REF]. Our proof gives also a variant of Léna's result on the torus (note that Léna gives an alternative proof). 6.3. Weyl formula with control of the remainder. Following the classical proof in Pleijel's foundational paper [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF], we need to find an explicit lower bound for the counting function, N (λ) = #{k | λ k < λ}, whose main term as λ → +∞ is given by the classical Weyl formula, (6.1)

N (λ) ∼ π 4 λ .
The coefficient π 4 in front of λ comes from the computation of 1 4π A(M 1 ) in 2D, where A(M 1 ) is the area of M 1 . In our case, we have A(M 1 ) = π 2 . This asymptotics is sufficient for showing that the number of Courant-sharp eigenvalues is finite. To actually determine the Courant-sharp eigenvalues, we need a lower bound for N (λ), valid for any λ ≥ 9. The case of the square was treated in [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF]. For the Möbius strip, we prove the following lower bound. Proof. Let D λ be the part of the closed disk of radius √ λ in the first quadrant. In view of Lemma 2.2 and (2.8), to each pair (m, n) ∈ D λ , such that m + n odd and m = 0, we associate a rectangle or a square, as follows:

•

If n > 0, we associate the rectangle R m,n = [m, m + 1] × [n, n + 2]. • If n = 0, (hence m odd positive), we associate the square R m,0 = [m, m + 1] × [1, 2].
The following lemma will be useful to continue the proof. Lemma 6.5.

Let I λ = {(m, n) ∈ D λ ; m + n odd and m = 0}. Then, (6.3) A(D λ ) ≤ (m,n)∈I λ A(R m,n ) + 2 √ λ - 1
Proof. To prove this lemma, it is enough to prove that,

D λ ⊂ [0, 1] × [0, √ λ] ∪ [0, √ λ] × [0, 1] ∪ (m,n)∈I λ R m,n .
To see that, let us prove that for any number (x, y) ∈ D λ , such that x > 1 and y > 1, there is a pair (m, n) ∈ I λ such that (x, y) ∈ R m,n . Consider the integer m such that m ≤ x ≤ m + 1. Two cases are possible here: ). This ends the proof of (6.3).

To finish the proof of Proposition 6.2, it is enough to consider the multiplicity of the eigenspace associated to each λ(m, n), one can indeed write,

N (λ) = (m,n)∈D λ ,m =0 A(R m,n ),
which is equivalent to (6.2). Before starting the proof of the theorem, we need the following two lemmas.

Upper bound for the Courant-sharp eigenvalues

Lemma 7.2. If λ is an eigenvalue of the Laplacian on M 1 with an associated eigenfunction u, and if the number of nodal domains ν(u) of u satisfies ν(u) ≥ 4, then we have:

(7.1) j 2 0,1 ν(u) π ≤ λ
Proof. If ν(u) ≥ 4, observing that the area of M = M 1 is π 2 , there exists one nodal domain D associated to u with area A(D) less than π 2 4 , hence less than π. Applying now (6.3) to D, we get:

(7.2) λ = λ 1 (D) ≥ πj 2 0,1 A(D) ≥ ν(u)j 2 0,1 π .
We now apply this lemma to a Courant-sharp eigenvalue λ k . By assumption, there exists an eigenfunction u k with k nodal domains. Applying Lemma 7.2 to u k , we get:

Proposition 7.3. If λ k is a Courant-sharp eigenvalue of the Laplacian on M 1 , with k ≥ 4, then, (7.3) λ k k ≥ (j 0,1 ) 2 π .
We can now give the proof of the theorem.

Proof of Theorem 7.1. Assume that λ k is Courant-sharp. By Remark 1.1, N (λ k ) ≤ k -1. Using (6.2) and ( 7.3), we conclude that P ( √ λ k ) ≥ 0, where (7.4) 7.1 lists the eigenvalues λ k less than or equal to 65 which could be Courant-sharp, taking Remark 1.1 into account, together with the corresponding ratio λ k π (j 0,1 ) 2 (which only makes sense for k ≥ 4). The only eigenvalue which satisfies the Faber-Krahn condition k ≤ λ k π (j 0,1 ) 2 is λ 7 . According to Lemma 5.1, the eigenvalue λ 7 is not Courant-sharp. Taking Proposition 3.3 into account, the proof of Theorem 1.2 is complete.

P (x) := ( π j 2 0,1 - π 4 )x 2 + 2x -2 . Eigenvalue (m, n) λ k λ k π (j0,1) 2 λ 1 (1, 0) 1 - λ 2 (1, 2), ( 2

An Euler-type formula for the Möbius band

The Möbius band is a non-orientable surface with boundary, with Euler characteristic 0, and one boundary component. Otherwise stated, it is a real projective plane with one disk removed. A natural question is whether the non-orientability character can be detected in the nodal patterns. The purpose of this section is to give a positive answer. where ω(D) = 0 if all the nodal domains are orientable, and ω(D) = 1 if one nodal domain in non-orientable. In the latter case, it turns out that there is exactly one non-orientable nodal domain.

Proof. We refer to [4] for the proof of Property 8.2 in the more general framework of partitions.

Remark 8.3. The Euler-type formula (8.1) actually holds for more general partitions, [START_REF] Bérard | Remarks on the boundary set of spectral equipartitions[END_REF]20].

One can easily check that Property 8.2 is true for the nodal domains which appear in Section 5, with ω(D) = 0 for all nodal patterns, except for the nodal pattern in Figure 5.6 (C) for which ω(D) = 1, note that the nodal domain labeled "2" is homeomorphic to a Möbius strip.

Formula (8.2) can also be verified on the nodal patterns displayed in the following figures.

Theorem 1 . 2 .

 12 When a = 1, the only Courant-sharp eigenvalues of the Dirichlet Laplacian on the Möbius strip M 1 are the first and second eigenvalues.

Figure 2 . 1 . 2 . 2 .

 2122 Figure 2.1. The Möbius strip embedded in R 3 , with boundary and soul

Proposition 3 . 3 .

 33 For the Möbius strip M 1 , the Dirichlet eigenvalues λ 1 and λ 2 are Courant-sharp. The eigenvalues λ 3 , . . . , λ 10 are not Courant-sharp. Proof. The first assertion is a direct consequence of Courant's nodal domain theorem [9, Chap. VI.6].

4. 1 . 2 .

 12 Case θ = 0 or θ = π In this case, the eigenfunctions are decomposed and the nodal sets explicit, see Figure 4.1. When θ = 0, Figure (A), there are two disjoint nodal lines (in M 1 ), hitting the boundary at critical zeros of order 2, and no interior critical zero. When θ = π 2 , Figure (B) & (C), there are two nodal lines which intersect at an interior critical zero of order 2, and hit the boundary at critical zeros of order 2. As expected, in each case, there are two nodal domains.

2 (c) Φ π 5 , π 2 Figure 4 . 1 .

 25241 Figure 4.1. Nodal sets Z(Φ β,0 ), Z(Φ 0, π 2 ), and Z(Φ π 5 , π

  Figures 4.1(B) and (C) display second Dirichlet eigenfunctions with an interior critical zero, and a nodal set which contains a closed curve, the arc x = π 2 . We refer to Figure 4.5 for a 3D-picture. The existence of interior critical zeros is a consequence of the multiplicity of λ 2 which is 4 for M 1 . For domains Ω ⊂ R 2 , Pólya's nodal line conjecture states that a second Dirichlet eigenfunction cannot have a closed nodal line. The conjecture has been proved for convex domains by A. Melas (smooth convex domains) and G. Alessandrini (general convex domains).

4. 2 . 2 .

 22 Case β = π 2 and 0 < θ < π In this case, a bifurcation occurs at θ = π 4 . The different patterns are illustrated in Figure 4.2. For θ = π 4 , Figure (B), the line y = π 2 is contained in the nodal set Z(Φ π 2 , π 4 ). It hits the boundary part {x = 0} at a critical zero of order 2, and the boundary part {x = π} at a critical zero of order 4. There is another nodal component hitting the boundary at this point, a closed curve. When 0 < θ < π 4 , Figure (A), the nodal set consists in two disjoint simple curves with end points critical zeros of order 2 on the boundary. No interior critical zeros for both patterns. When π 4 < θ < π 2 , Figure (C)

2 Figure 4 . 3 .

 243 Figure 4.3. Nodal sets Z(Φ 0,θ ), for 0 < θ < π 2

Figure 4

 4 Figure 4.4 (C) presents another peculiarity. The nodal domain labelled "1" is not simply connected, and actually homeomorphic to a Möbius strip, and it is not orientable. We refer to Figure 4.8 for 3D-pictures.

2 Figure 4

 24 Figure 4.4. Nodal set Z(Φ β,θ ), for 0 < β < π 2 and 0 < θ < π 2

Figure 4 . 5 . 2 )Figure 4 . 7 .Figure 4 . 8 . 2 ) 4 . 1 .

 4524748241 Figure 4.5. Second eigenfunction with an interior critical zero and a closed nodal line (β = π 5 , θ = π 2 )

5 . 1 . 2 .

 512 Recall that the Möbius strip M 1 is obtained by identifying the lines {y = 0} and {y = π} via (x, y) ∼ (π -x, y + π). The figures below represent the nodal sets in the fundamental domain R 1 , or on the topological 3D-representation given by the map F defined in (2.5), see Remark 2.1. The nodal lines appear in red in the fundamental domain R 1 , the Dirichlet boundary in blue. When {y = 0} ∼ {y = π} is not a nodal line, the dashed black lines indicate the Möbius identification. Special values θ = 0 and θ = π The eigenfunctions Φ β,0 (actually independent of β), and Φ β, π 2 play a special role. Their nodal sets are explicit; they are the unions of horizontal and vertical segments, see Figure5.1. There are both interior and boundary critical zeros, all of order 2. There are 6 nodal domains in each case.

Figure

  Figure (A) illustrates the following property: cutting the Möbius strip along the soul circle yields a strip with half the width, and twice the length of the original strip. This strip is doubly twisted and hence orientable. Figure(B)illustrates what happens when one cuts the strip in the y-direction, at exactly one-third the width. This operation produces two intertwined strips, one Möbius strip (corresponding to the domains labelled "5" and "6"), and a doubly twisted (orientable) strip (corresponding to the domains labelled "1, 2, 3" and "4").

2 Figure 5 . 1 ., π 2 ) 5 . 2 .Proposition 5 . 2 .( 1 )

 251252521 Figure 5.1. Nodal sets Z(Φ β,0 ) and Z(Φ π 5 , π 2 )

3

  and θ ∈ (0, π 2 ), any interior critical zero of Φ β,θ lies on the line {y = β}.(3) If ξ ∈ {0, π}, i.e., if (ξ, η) is a boundary critical zero of Φ β,θ , then (ξ, η) has order k ∈ {2, 3, 4}. More precisely, (a) the point (ξ, η) is a boundary critical zero if ξ ∈ {0, π}, and    either (η, β) ∈ (0, 0); (π, 0); ( π 3 , π 3 ) , or η ∈ 0, π 3 , 2π 3 , π and cot θ = -cos(ξ) f (β, η), where the function f (β, y) is defined by, the boundary critical zero (ξ, η) has order at least 3 if and only if g(β, η) = 0 as well, where the function g(β, y) is defined on [0, π] 2 by (5.5) g(β, y) = 2 cos(2y + β) sin(3y) -3 cos(3y) sin(2y + β) ;

∂

  x Φ(ξ, y) = 2 cos θ sin(3y) + 3ε sin θ sin(2y + β), (5.9a) ∂ xy Φ(ξ, y) = 6 cos θ cos(3y) + 6ε sin θ cos(2y + β), (5.9b) ∂ x 3 Φ(ξ, y) = -8 cos θ sin(3y) -27ε sin θ sin(2y + β), (5.9c) ∂ xy 2 Φ(ξ, y) = -18 cos θ sin(3y) -12ε sin θ sin(2y + β), (5.9d) ∂ x 3 y Φ(ξ, y) = -24 cos θ cos(3y) -54ε sin θ cos(2y + β), (5.9e) ∂ xy 3 Φ(ξ, y) = -54 cos θ cos(3y) -24ε sin θ cos(2y + β).

3

  and θ ∈ (0, π 2 ), the set Z β consists in the line {y = β}, and a finite number of regular points for Φ β,θ on M 1 \{y = β}.(2) For β ∈ (0, π 3 ) and θ ∈ (0, π 2 ), the set Z β consists in finitely many regular points for Φ β,θ . (3) For β ∈ (0, π 3 ), the nodal set Z(Φ β,θ ) does not meet the set {P β > 0}, and only meets the nodal sets Z(Φ β,0 ) and Z(Φ β, π 2 ) at the points in Z β . (4) For β ∈ 0, π 3 , the nodal set Z(Φ β,θ ) does not meet the set {P β > 0}, and only meets the nodal sets Z(Φ β,0 ) and Z(Φ β, π 2 ) at the points in Z β , or along the common line {y = β}. This proposition, whose proof is clear, is best understood by looking at Figure 5.2 (see also Figures 5.5 and 5.9).

Figure 5 . 2 .

 52 Figure 5.2. The checkerboards for β = 0 and θ ∈ (0, π 2 )

  β + β) sin(3y β ).

Figure 5 . 3 .

 53 Figure 5.3. Graph of the function f (β, •), for β = π8

π 4 when β continues increasing to π 3 . 5 . 10 .

 3510 Remark As pointed out above, when looking at the set {g(β, y) = 0}, β and y play symmetric roles. Given β, there is a unique zero y β , and conversely, given y, there is a unique zero β y . In order to visualize the functions m β and θ β , it is simpler to look at β as a function of y, and to visualize m βy and θ βy as functions of y. Figure5.4 displays the graphs of the functions β y = arccot (h(cot y)), m βy and θ βy .

Figure 5 . 4 .

 54 Figure 5.4. Graphs as functions of y

( a )Figure 5 . 5 .

 a55 Figure 5.5. Localization of the nodal set Z(Φ β,θ )

  Figure 5.6 (in this figure, β = π 4 ). When θ = θ β , Figure (B), the nodal set consists in three simple regular curves which do not intersect in M 1 . Two of these curves hit the boundary at the critical zero (π, y β ), of order three. One of them hits the boundary part {x = 0}; the other hits the boundary part {x = π}×( π 3 , 2π 3 ). The two end points of the third curve belong to {x = 0} and {x = π} respectively. Except for the afore mentioned boundary critical zero of order 3, the boundary critical zeros have order 2. There is no interior critical zero. When 0 < θ < θ β , Figure (A), the nodal set consists in three non-intersecting simple regular curves. Two of them have one end point on each part of the boundary. The third has two distinct end points on {x = π}. There are no interior critical zero. The boundary critical zeros all have order 2. When θ β < θ < π 2 , Figure (C), the nodal set consists in two non-intersecting simple regular curves. They both have one end point on {x = 0}, the other on {x = π}.These end points are critical zeros of order 2. There are no interior critical zeros. In all three cases, the number of nodal domains (on M 1 ) is 3.

2 Figure 5 2 5. 5 .

 2525 Figure 5.6. Z(Φ β,θ ), with 0 < β < π 3 fixed, and 0 < θ < π 2 5.5. Special values β = 0 and β = π 3 , with θ ∈ (0, π 2). The nodal sets Z(Φ π 3 ,θ ) are nicer to visualize. They are displayed in Figure5.7. The functions Φ 0,θ are Dirichlet eigenfunctions of the square (0, π)2 . Their nodal sets have already been studied in[START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF], see also[START_REF] Bérard | Dirichlet eigenfunctions of the square membrane: Courant's property, and A. Stern's and Å. Pleijel's analyses[END_REF] for more details. They are given in Figure 5.8. Recall that Z(Φ π 3 ,θ ) = τ π 3 Z(Φ 0,θ ), see (3.5) and Figure 5.10.

Proposition 6 . 3 .

 63 8. Recall that Z(Φ π 3 ,θ ) = τ π 3 Z(Φ 0,θ ), see (3.5) and Figure 5.10. 6.2. Faber-Krahn inequality. If Ω is an open connected set of M 1 with a piecewise C 1 boundary, and such that A(Ω) ≤ π, then:

Proposition 6. 4 .

 4 The counting function of the Dirichlet eigenvalues of the Möbius strip M 1 satisfies,

Figure 6 . 1 .

 61 Figure 6.1. The square R m,0 (red) and the rectangles R m,n (color depending on n)

Theorem 7 . 1 .

 71 This section is inspired by the paper of C. Léna on the torus [23]. For k / ∈ {1, 2, 7}, the eigenvalues λ k of M 1 are not Courant-sharp.

Property 8 . 2 .

 82 Let Ω be a bounded connected open set in R 2 , with piecewise C 1,+ boundary. The following theorem appears in[20]. Proposition 8.1. Let D = {D i } k i=1 be a nodal k-partition of Ω, where the D i 's are the k nodal domains of some Dirichlet eigenfunction Φ in Ω, and ∂D = Z(Φ). Let b 0 denote the number of connected components of ∂Ω, and b 1 the number of connected components of ∂D ∪ ∂Ω. Let C i (∂D) denote the set of interior critical zeros of Φ, and C b (∂D) the set of boundary critical zeros. Given x ∈ C i (∂D), let ν(x) denote the number of nodal semi-arcs at x; given y ∈ C b (∂D), let ρ(y) denote the number of nodal semi-arcs hitting the boundary at y. Then, (8.1)k = 1 + b 1 -b 0 + 1 2 x∈C i (∂D) (ν(x) -2) + 1 2 y∈C b (∂D) ρ(y) .In the case of the Möbius strip, we expect the following Euler-type formula to hold. It takes non-orientability into account. Let D be nodal k-partition of the Möbius strip M 1 . With the previous notation, the following relation holds, (8.2) k = ω(D) + b 1 -b 0 + 1 2 x∈C i (∂D) (ν(x) -2) + 1 2 y∈C b (∂D) ρ(y) ,

Figure 8 .

 8 1 displays nodal patterns containing one Möbius strip, and one or two non-simply connected, orientable nodal domains. Figure 8.2 displays the nodal patterns of the function cos θ sin(x) cos(my) + sin θ sin(mx) cos(y) , with m = 6, for the values θ = 0.18 π and θ = 0.4 π. In both cases, one of the nodal domains is non-orientable, homeomorphic to a Möbius strip (left), or to a Möbius with holes (right).

Figure 8 . 1 .Figure 8 . 2 .

 8182 Figure 8.1. Nodal sets of sin(3x) and sin(5x)

Figure 9 . 1 .

 91 Figure 9.1. Example à la Stern[START_REF] Bérard | Remarks on the boundary set of spectral equipartitions[END_REF] 4]

Figure 9 . 2 .

 92 Figure 9.2. Example à la Stern [1, 6]

Table 1 .

 1

	Case	Boundry Cond.	Courant-sharp eigenvalues	References
	Square	Dirichlet	{λ 1 , λ 2 , λ 4 }	[26, 2]
	Square	Neumann	{λ 1 , λ 2 , λ 4 , λ 5 , λ 9 }	[17]
	Square	Robin	{λ 1 , λ 2 , λ 4 }	[11] (h large, Remarks 1.3)
	Square	Robin	{λ 1 , λ 2 , λ 4 , λ 9 }	[12] (h > 0 small, Remarks 1.3)
	Torus	-	{λ 1 , λ 2 }	[3, 23] (resp. for the flat equilateral or square tori)
	Triangle	Dirichlet	{λ 1 , λ 2 , λ 4 }	[3] (equilateral triangle)
	Disk	Dirichlet/Neumann	{λ 1 , λ 2 , λ 4 }	[15, 18]
	Möbius strip	Dirichlet	{λ 1 , λ 2 }	The present paper
	Cylinder	Dirichlet	{λ 1 , λ 2 }	[5] (See Remarks 1.3)
	Klein bottle	-	{λ 1 , λ 2 }	[5] (See Remarks 1.3)

1 Remarks 1.3.

Dirichlet spectrum of the Möbius strip, case a

  = 1. In the case a = 1, the Dirichlet eigenvalues of M 1 have higher multiplicities. Let λ(m, n) denote λ(m, n, 1).

	Dirichlet eigenfunctions of the Möbius strip a ), sin(mx) sin(n y M a , equipped with the flat metric, is (2.8) sin(mx), m ∈ N • odd, and sin(mx) cos(n y a ), m, n ∈ N • , m + n odd, with associated eigenvalues λ(m, n, a) = m 2 + n 2 a 2 . Definition 2.3. Let Φ be a Dirichlet eigenfunction of -∆ on M a . The nodal set of Φ is defined as (2.9) Z(Φ) = {x ∈ M a | Φ(x) = 0} , denoted, λ 1 < λ 2 ≤ λ 3 ≤ • • • , and listed in nondecreasing order, with multiplicities, starting from the label 1. The first Dirichlet eigenvalues of M 1 are given in the following table, with eigen-functions given by (2.8). Eigenvalue (m, n) λ(m, n) Multiplicity λ 1 (1, 0) 1 1 λ 2 , λ 3 , λ 4 , λ 5 (1, 2), (2, 1) 5 4 λ 6 (3, 0) 9 1 λ 7 , λ 8 , λ 9 , λ 10 (2, 3), (3, 2) 13 4 λ 11 , λ 12 , λ 13 , λ 14 (1, 4), (4, 1) 17 4 λ 15 , λ 16 , λ 17 , λ 18 , λ 19 (3, 4), (4, 3), (5, 0) 25 5 λ 20 , λ 21 , λ 22 , λ 23 (2, 5), (5, 2) 29 4 λ 24 , λ 25 , λ 26 , λ 27 (1, 6), (6, 1) 37 4 λ 28 , λ 29 , λ 30 , λ 31 (4, 5), (5, 4) 41 4 λ 32 , λ 33 , λ 34 , λ 35 (3, 6), (6, 3) 45 4 λ 36 (7, 0) 49 1 λ 37 , λ 38 , λ 39 , λ 40 (2, 7), (7, 2) 53 4 λ 41 , λ 42 , λ 43 , λ 44 (5, 6), (6, 5) 61 4 i.e., as the closure in M 2.3. Notation 2.4. In the sequel, the Dirichlet eigenvalues of the Möbius strip M 1 are λ 45 , . . . , λ 52 (1, 8), (8, 1), (4, 7), (7, 4) 65 8

a ∪ ∂M a of the set of (interior) zeros of Φ. The nodal domains of Φ are the connected components of M a \Z(Φ).

Table 2 .1. First Dirichlet eigenvalues of M 1 3. Analysis of the first eigenspaces

 2

	When m, n ∈ N • (with m+n odd), the eigenvalue λ(m, n) = m 2 +n 2 has multiplicity
	at least 4. In this case, the corresponding eigenspace, denoted E λ(m,n) , contains the
	subspace E [m,n] consisting of eigenfunctions of the form
	(3.1)	Φ(x, y) = sin(mx) (A cos(ny) + B sin(ny)) + sin(nx) (C cos(my) + D sin(my)) .
	Note that E λ 2 = E [1,2] and E λ 7 = E [2,3] , while E λ 15 = E [3,4] ⊕ E [5,0] , see Table 2.1. Since
	we are interested in nodal sets, we may assume, without loss of generality, that Φ is
	normalized by A 2 + B 2 + C 2 + D 2 = 1. The function in the first parenthesis on the right hand side of (3.1) can be written as √ A 2 + B 2 sin(ny + α), and the function in the second parenthesis as √ C 2 + D 2 sin(my + β), for some α, β ∈ (-π, π]. Defining
	θ ∈ [0, π 2 ] by,

Table 2 . 1 .

 21 Lemma 5.1. An eigenfunction Φ associated with the Dirichlet eigenvalue λ 7 of the Möbius strip M 1 has at most six nodal domains. As a consequence, the eigenvalue λ 7 is not Courant-sharp. = 0 or θ = π 2 , Φ β,θ has 6 nodal domains, see Subsection 5.1; (2) when β ∈ (0, π 3 ) and θ ∈ (0, π 2 ), Φ β,θ has 3 nodal domains, see Subsection 5.4; (3) when β ∈ 0, π 3 and θ ∈ (0, π 2 ), Φ β,θ has 4 nodal domains, see Subsection 5.5. Subsections 5.2 and 5.3 contain technical preliminaries.

	Taking Proposition 3.1 into account, we consider the family of eigenfunctions,
	(5.1)	Φ β,θ (x, y) = cos θ sin(2x) sin(3y) + sin θ sin(3x) sin(2y + β) ,
	for β ∈ [0, π 3 ] and θ ∈ [0, π 2 ], and examine three cases. More precisely, we will prove that,
	(1) when θ

Table 7 .

 7

1. Courant-sharp vs Faber-Krahn

Since the quadratic function P is negative for x ≥ 8, Courant-sharp Dirichlet eigenvalues of M 1 must be smaller that 64. The last step is to check the eigenvalues smaller than 64. Table

  [START_REF] Bérard | Inégalités isopérimétriques et applications[END_REF] Å. Pleijel. Remarks on Courant's nodal theorem. Comm. Pure. Appl. Math. 9 (1956), 543-550. 2, 24, 28 PB: Université Grenoble Alpes and CNRS, Institut Fourier, CS 40700, 38058 Grenoble cedex 9, France. Email address: pierrehberard@gmail.com BH: Laboratoire Jean Leray, Université de Nantes and CNRS, F44322 Nantes Cedex, France and LMO (Université Paris-Sud). Email address: Bernard.Helffer@univ-nantes.fr RK: American University in Dubai, P.O.Box 28282, Dubai, United Arab Emirates. Email address: rkiwan@aud.edu

We shall do so in the subsequent proofs, whenever there is no ambiguity.

We summarize the analysis of the nodal sets for β = π 3 . Write (5.30) Φ π 3 ,θ (x, y) = cos θ sin(2x) sin(3y) + sin θ sin(3x) sin(2y + π 3 ) = -2 sin(x) sin(yπ

3 ) Γ θ (x, y -π 3 ) , where (5.31) Γ θ (x, y -π 3 ) = cos θ cos(x)(4 cos 2 (yπ 3 ) -1) + sin θ(4 cos 2 (x) -1) cos(yπ 3 ) . 3 )) sin(3x) sin(2(yπ 3 )), and apply Proposition 5.6: the nodal set Φ π 3 ,θ can only visit the white domains. It passes through the points marked "O", does not pass through the points marked "P". The points marked "R" are regular points at the boundary. The corresponding checkerboard appears in Figure 5.9. Analyzing the functions Γ θ (x, 0) and Γ θ (π, y -π 3 ), one finds that there are three cases:

(1) when 0 < θ < π 4 , the nodal line entering "U" at ( 2π 3 , 2π 3 ) does not hit y = π 3 , and hits the boundary in {π} × ( π 3 , 2π 3 ) and there is a similar nodal line entering "V" at ( 2π 3 , 0); (2) when π 4 < θ < π 2 , the nodal line entering "U" at ( 2π 3 , 2π 3 ) crosses y = π 3 , does not hit the boundary {π} × (0, 2π

3 ), and exits V at ( 2π 3 , 0);

(3) when θ = π 4 , the nodal set in U ∪ V consists in two line segments, one from ( 2π 3 , 2π 3 ) to (π, π 3 ), and a symmetric one from ( 2π 3 , 0) to (π, π 3 ). In the first two cases, the critical zeros have order 2; in the third case, the critical zero (π, π

3 ) has order 4, as indicated in Proposition 5.3, Assertion (3). In the three cases, the eigenfunction Φ has four nodal domains.

Finally, Figure 5.10 illustrates the relations (3.5). More precisely, one can write (5.33) Proof. We note indeed that an open set in M 1 can be considered as an open set of a flat Klein bottle. We have just to delete one line x = const on the Klein Bottle.


J-M Petit
email: petit@obs-besancon.fr.

J J Kavelaars

B J Gladman

J L Margot

P D Nicholson

R L Jones

J Wm Parker

M L N Ashby

A Campo Bagatin

P Benavidez

J Coffey

P Rousselot

O Mousis

P A Taylor

The extreme Kuiper Belt binary 2001 QW 322

The study of binary Kuiper Belt objects help probe the dynamic conditions present during planet formation in the Solar System. We report on the mutualorbit determination of 2001 QW 322 , a Kuiper Belt binary with a very large separation whose properties challenge binary-formation and evolution theories. Six years of tracking indicate that the binary's mutual orbit period is ≈25-30 years, that the orbit pole is retrograde and inclined 50-62 • from the ecliptic plane, and, most surprisingly, that the mutual orbital eccentricity is < 0.4. The semimajor axis of 105,000-135,000 km is 10 times that of other near-equal mass binaries. Because this weakly bound binary is prone to orbital disruption by interlopers, its lifetime in its present state is likely less than

Gy.

A combination of survey strategies and adaptive optics technologies has led to a surge in the discovery rate of binary minor planets. Since 2001, newly-discovered binaries in the main asteroid and Kuiper belts have been announced at the rate of about seven per year [START_REF] Merline | Asteroids III[END_REF][START_REF] Noll | The Solar System beyond Neptune[END_REF]. There are now more than 100 binaries known, nearly half of which are Kuiper Belt Objects (KBOs). Measurements of the frequency of binary objects and their sizes and orbital configurations constrain their formation and evolution mechanisms, theories of planetesimal accretion and disruption, and the collisional history of the Kuiper belt.

Discovering and studying the mutual orbits of binary systems is currently the only way to directly determine KBO masses. Assuming that the optical properties of the KBO binaries are representative of the whole KBO population, one can link mass to apparent magnitude and hence estimate the total mass of the Kuiper Belt without requiring assumptions on albedo and density. Recent HST observations [START_REF] Margot | [END_REF] indicate that KBOs display a wide range of albedos (8-40%, assuming unit density), which complicate estimating the total mass of the Kuiper belt using a luminosity function. Combined with thermal infrared observations, phase-function photometry or star occultation observations, direct determination of KBO masses leads to the determination of their density and bulk composition.

Here we report the mutual-orbit determination of the large-separation Kuiper Belt binary, 2001 QW 322 (4). This KBO was discovered in data acquired 24 August, 2001 at the Canada-France-Hawaii Telescope by the Canada-France Ecliptic Plane Survey team. The two components had identical magnitudes of m R ≃ 24.0 within the measurement uncertainties, implying essentially equal sizes. Only one other equal-component binary was known at the time, asteroid (90) Antiope, with a magnitude difference of ∼0.1 mag (5). However 2001 QW 322 was obviously exceptional because the measured separation of ∼4 ′′ at its distance of 43 AU corresponds to a sky-projected physical separation of 125,000 km (about one third of the distance from Earth to the Moon), far larger than any other small body binary.

The large separation implied a mutual-orbit period of at least several years. Six years of tracking using 4-to-8 m class telescopes (Fig. 1) resolved that 2001 QW 322 , an object in the main classical Kuiper belt [START_REF] Gladman | The Solar System beyond Neptune[END_REF], has a low-eccentricity mutual orbit with a separation of 105,000-135,000 km, greater than any other known minor-planet binary (7). The separation is so large that this nearly equal-mass binary should be incredibly fragile to dynamical disruption, and its continued existence in the middle of the main Kuiper Belt puts strong constraints on the history of the belt [START_REF] Petit | [END_REF].

Within the roughly 0.1-magnitude observational uncertainties at discovery, the two components had identical brightnesses, a finding confirmed in 2002 (9). Assuming identical albedos, object size is proportional to square root of flux, so it is unclear which component is the largest (and thus the primary). We therefore obtained higher-precision photometric observations from 8-m class telescopes (Very Large Telescope (VLT), Gemini-North and Gemini-South) between 2002 and 2007 including broadband colors in R-band, I-band, and V -band (Tables S1, S2 andS3). From these data, it appears that the relative magnitudes of the two components are essentially the same in all measured colors; the color similarity implies a surface similarity that strengthen the assumption of equal albedos. The plentiful R-band data give a mean relative magnitude m b -m a = -0.03 ± 0.02. For what follows, we therefore take the two components to have the same physical size. Using the magnitude r G0303 = 23.7 for both components, measured at a geocentric distance of ∆=43.4 AU, we derive a radius of r=54 km for an assumed r-band albedo p=0.16. The color of 2001 QW 322 is on the blue extreme of the color distribution of the cold classical Kuiper belt (10) or the Kuiper Belt core (11). At the 1 σ confidence level, both components could have a light curve of amplitude ∼0.4 magnitude, but no rotation period could be derived.

We measured (Table S4) Right-Ascension (RA) and Declination (DEC) of the center of mass (assuming equal mass components) of the binary, along with position angle and separation between components (Fig. 1). Fits to the astrometric measurements up to and including 2006 were equally consistent with a large-eccentricity mutual orbit either pole aligned or anti-aligned with the line of sight, as well as a low-eccentricity orbit viewed nearly edge-on. Additional measurements were acquired in October 2006 (from Gemini-South), and fall 2007 (September from Gemini-South, October from (VLT, November from Multi Mirror Telescope (MMT) and VLT), which ruled out nearly all e > 0.3 solutions. This is surprising as it is difficult to imagine how to bind two small bodies that never come closer to each other than 85,000 km. The mutual orbit of 2001 QW 322 is retrograde, with a pole's ecliptic latitude between -50 • and -62 • , and is viewed somewhat edge-on (inclined at 55-70 • to the line of sight).

There remain three main groups of acceptable orbits: one group with e ≈ 0.2, a ≈ 114, 000 km, P ≈ 27 yr, ρ ≈ 0.94 g cm -3 , another group with nearly-circular orbits e ∼ < 0.05, a ≈ 128, 000 km, P ≈ 29 yr, ρ ≈ 1.11 g cm -3 , and a smaller group with e ∼ < 0.4, a ≈ 105, 000 km, P ≈ 18 yr, ρ ≈ 1.6 g cm -3 (Fig. 2). We mildly prefer the non-circular orbits which have a goodness-of-fit 10% better than for the nearly-circular group. The stability region for binary orbits can be expressed in terms of the ratio a/R H , where R H is the Hill radius (7), with prograde orbits becoming instable for a/R H greater than 0.3-0.4 (12), while the limit of stability of retrograde orbits is 0.5-0.6. This may explain why the first such system was found to be retrograde. It also indicate that 2001 QW 322 is very fragile to disruption as mentioned by [START_REF] Petit | [END_REF]. For our acceptable orbits, a/R H varies mostly between 0.27 and 0.32, with the first group of orbits centered at 0.29, and the second group centered at 0.31.

According to Kepler's third law, the total mass of the binary is 0.9-2.4 × 10 18 kg; most of the estimated values are between 1.1 × 10 18 and 1.5 × 10 18 kg. With the photometricallyderived nominal size of r=54 km for each component (assumed albedo of 0.16), the density of 2001 QW 322 (Fig. 2, bottom-left) is likely 0.8-1.2 g cm -3 . This is a little higher than compa-rably sized outer Solar System bodies (Figure 5 of 13; 0.6-0.8 g cm -3 ). Our nominal albedo of 0.16 is about double that estimated from optical and thermal infrared photometry for similarsize KBOs (14,15), but about a factor of 2 below that of (58534) Logos/Zoe (p = 0.37 ± 0.04;

2), which is of comparable size. Estimated density from eqs. S2 and S3 is proportional to the assumed albedo to the power 3/2. Halving our albedo would increase our radius estimates by √ 2 and decrease the estimated density by a factor of 2 3/2 = 2.8, below the range of published densities (13) for such small bodies.

The nominal densities shown in Fig. 2 are at the boundary between the density of a lowporosity pure-water ice body, and that of a mixture of water ice and silicate rocks (13). A thermal detection, mutual eclipse, or stellar occultation by the binary (all unlikely) would be needed to further constrain the size, albedo, density, and hence the bulk composition of 2001 QW 322 .

Given the very large separation (Fig. 3), such a binary is difficult to create and maintain.

Of all the proposed KBO binary formation scenarios (16)(17)(18)(19), only the collision of two bodies close to a third one (16) can simply explain the primordial formation of such a system (7).

A study of the long-term stability of the large-separation KB binaries [START_REF] Petit | [END_REF] concluded that the major destabilizing factor is unbinding due to direct collisions of impactors on the secondary.

Applying their method to the newly determined orbital and physical parameters for 2001 QW 322 and our nominal albedo, we find that the lifetime of this binary is 0. The origin corresponds to the center of mass, assuming equal mass for both components. The southern "a" component has been moving westward, while the northern "b" component has been moving to the east.

Supporting material

Discovery and tracking

2001 QW 322 was discovered on Aug. 24, 2001, while conducting a search for satellites of Uranus (S1). The measured separation was 125,000 km, and the two components had identical magnitudes of m R ≃ 24.0. Making common (in 2001) assumptions of 4% albedos and densities ∼1 g cm -3 , radii of order 100 km were derived. Assuming the separation was equal to the semimajor axis, a period estimate of ≈12 years resulted (S2). A mistaken use of diameter instead of radius for the period calculation caused the initial period estimate to be a factor of √ 8 too low; it should have been ∼ 11 years instead of four years for the 100-km radius case. This was the second similar-sized component binary known at the time. Since then, binaries with similar-sized component have been discovered in most dynamical classes: the near-Earth asteroid Hermes (S3), several small main-belt asteroids (e.g. S4), the Trojan Patroclus (S5), and several KBO binaries (S6).

We began a tracking program to determine both the heliocentric orbit of the system as well as the mutual orbit of the binary. By 2003 sufficient observations were acquired to establish the heliocentric orbit (see below) and yielded a relatively 'normal' trans-neptunian object in the so-called 'main' classical belt (S7), with semimajor axis a ⊙ =44 AU, eccentricity e=0.024, inclination i=4.8 • , and a distance of R=44 AU from the Sun at time of discovery. However, in the two years after discovery the on-sky separation had only changed by about an arcsecond, implying an orbital period of at least a decade. At this point, and for several more years, it was unclear if the mutual orbit had a large eccentricity (with an orbital plane in the plane of the sky) or a nearly-circular orbit viewed close to edge-on.

Both possibilities remained open for the following years and estimates for the orbital period increased to well past a decade. By 2006 it became clear that the closest projected separation would occur in 2007, and we acquired a series of high-quality observations on 8-meter class telescopes. These observations have resulted in the virtual elimination of the high-eccentricity solutions, and we thus here report that 2001 QW 322 has a low-eccentricity orbit with a recordbreaking separation for a small-body binary, having an orbital period measured in decades.

Photometric observations

Apparent magnitudes are related to physical size and albedo by Russell's formula (S8):

m X = M ⊙,X -2.5 log 10 p X r R 2 1 ∆ 2 AU (S1)
where M ⊙ is the Sun's brightness at 1 AU in the appropriate filter X, R is the heliocentric distance (in the same units as the object radius r), and ∆ AU is the geocentric distance in AU. One thus relates the object's apparent magnitude, its distance to the Sun and the Earth, its radius r and its geometric albedo p. From this we derive the dimensionless ratio r/R expressing the body's size as a function of measured and/or assumed quantities:

r R = 110 × ∆ AU 44 × 10 M ⊙,X -m X 5 × 0.16 p X . ( S2
)
Note that the estimated radius r depends principally on measured quantities (m X , ∆ AU , R and M ⊙,X ), and only weakly on the assumed value of the albedo p X (inverse square root). In the above formula, we scale distances from the Earth to 44 AU since this is the typical distance to the main classical Kuiper Belt. We elected to use p X =0.16 as our nominal scaling value since it is in the range of the most recent estimates, and a factor of four from the lower and upper extreme values for albedo of small bodies (hence the derived radius is within a factor of two of its real value). For the CFHT R filter and the AB system used in our initial observations, M ⊙ =-26.94 (S9-S11). Our observations occured at opposition, at a geocentric distance of ∆=43.3 AU (and heliocentric distance of 44.3 AU). Hence each component of magnitude m R =24 gives an estimate of r=50 km for the nominal albedo p R =0.16.

Because our initial flux measurments showed that both components of 2001 QW 322 have essentially the same apparent magnitude (implying identical sizes under the assumption of similar albedo), we obtained photometric observations in the R-band, I-band and V -band filters from 8-m class telescopes (VLT and Gemini North and South telescopes) between 2002 and 2007. Tables S1, S2 and S3 give the relative magnitude of the two components (component "a" minus component "b"), together with the apparent magnitude of each component. Table S1 lists Column two of Table S1 implies a possible variation of the relative magnitudes of up to 0.6 magnitude at a 2.2 σ confidence level. This level of confidence is derived from the uncertainties reported in the same column, which are those reported by the PHOT or ALLSTAR tasks in IRAF 1 . However, a close look at the variation of the relative magnitude in Gemini-N i G0302 filter in 2005, all taken within 20 minutes, indicates that the uncertainty reported here (photon counts statistical fluctuations) is probably underestimated, making the previous variations more likely a result at the 1.5 σ confidence level only. But we cannot quantify this more precisely. Similarly, the apparent magnitude data seem to imply a possible variability of the apparent magnitude of up to 0.45 magnitude for component "a", and up to 0.35 magnitude for component "b". For the fall 2007 VLT data, part of the variation could be due to phase effects. For the other VLT data spanning 1 to 4 days, the effect, if real, is more likely to be due to a rotation/shape related light curve. But again, this result is likely only at just above 1 σ confidence level. A more definitive answer would require to get very accurate photometry (0.05 magnitude uncertainty or better) in at least 3 colors over two or more sequential nights. This can only be achieved using an 8-m class telescope or larger because the target is moving, preventing too long exposures.

Putting all the R-band relative magnitudes together, one derives a mean relative magnitude of m b -m a = -0.03 ± 0.02 (Fig. S1). So for all intents and purposes, we will assume equal magnitudes for both components.

Comparing column 2 of Tables S1, S2 and S3, the relative magnitudes of the two components are essentially the same in all measured colors. Within the photon count statistical errors, the colors of the two components are identical. Several KBO binaries observed with HST appear to have similar colors (S12). This result supports the idea that both components have similar albedos. Since they also appear to have the same magnitude, they are most likely of the same physical size. Using the r G0303 filter data from Gemini North, we can make a better estimate of the size of 2001 QW 322 as a function of albedo. For this, we take for the magnitude of both components r G0303 = 23.7, measured at a geocentric distance of ∆=43.4 AU. In this filter, the AB magnitude of the Sun is -26.93 (S9), yielding a size of r=54 km for the nominal albedo p R =0. 16.

Comparing our VLT data with the V-R versus R-I color-color plot in Fig. 2 from (S13), we see that 2001 QW 322 is on the blue extreme of the cold classical belt. This statement is also supported by our Gemini-N data, when compared to the Sloan g', r' and i' colors of Kuiper belt Core objects given by (S14) in their Table 1 (Fig. S2).

The mutual orbit

Our orbit determination software uses the separation and position angle of the secondary with respect to the primary to solve for the orbital parameters in the two-body problem. The mutual orbital plane orientation is assumed constant, but heliocentric motions and light-time corrections are taken into account. Astrometric positions and their errors are specified at the mid-time of the exposure sequence. Starting from thousands of initial conditions, the software adjusts for seven parameters (six orbital elements plus the mass of the system) in the nonlinear ordinary least squares problem with a Levenberg-Marquardt technique. The covariance matrix and post-fit residuals are computed and inspected. Binary orbits have been computed with this algorithm for near-Earth asteroids (S15, S16), main-belt asteroids (S17, 18), Kuiper belt objects (S6), dwarf planets, and binary stars.

For 2001 QW 322 , we used 92 measurements at 46 epochs, resulting in 85 degrees of freedom. The best-fit reduced χ 2 value was 0.526. Orbital fits with reduced χ 2 values exceeding the overall best-fit value by more than the statistically significant increase of 0.096 were eliminated. The valid solutions seem to cluster into three regions of the chi-square space, with the best-fit results near (e ≈ 0.2, a ≈ 114, 000 km, P ≈ 27 yr) and (e ≈ 0.4, a ≈ 105, 000 km, P ≈ 18 yr), and somewhat worse fits near (e ∼ < 0.05, a ≈ 128, 000 km, P ≈ 29 yr). Thanks to the high Signal-to-Noise ratio of most of our observations, and to the six years of arc, we are able to see the paralax of the object and resolve the degeneracy of which component is in the foreground and which is in the background. Hence we are able to resolve the initial degeneracy between prograde and retrograde orbits.

The Hill radius denotes the distance from a body at which the tidal forces due to the Sun and the gravitational force due to the body, both acting on a test particle, are in equilibrium. This roughly represent a region of stability for periodic orbits like the mutual orbit of binary objects. It is given by

R H = a ⊙ M b 3M ⊙ (S3)
where M b is the total mass of the binary, a ⊙ is its semi-major axis around the Sun and M ⊙ the mass of the Sun. As mentioned above, the heliocentric semimajor axis is a ⊙ =44 AU. Hence, the ratio a/R H varies mostly between 0.27 and 0.32, with the first group of orbits centered at 0.29, and the second group centered at 0.31. It is interesting to note that the only other "binary" systems that have such a large value of the ratio a/R H are irregular satellites of the giant planets.

From the dynamics of the mutual orbit, one can also derive an estimate of the size of the binary. Assuming both components have the same radius and density ρ, Newton's form of Kepler's third law gives the dimensionless ratio a r

3 = 2Gρ 3π P 2 (S4)
where a is the binary's mutual semimajor axis, G is the gravitational constant, and P is the period of the orbit. From the orbital fitting we obtain pairs of P and a, so taking ρ ≃1 g cm -3 ,

r ≃ 4.3 × 10 -3 a P 2/3 . ( S5
)
where P is given in years and a in the same units as r. The e ≈ 0.2 orbits yield a radius r ≃ 54 km, while e ≈ 0.05 orbits yield r ≃ 58 km. Note that the effect of the assumed density goes like ρ -1/3 , so a change of density by a factor of two results in only a 26% variation of the estimated radius. The radius uncertainty thus is dominated by the unknown density rather than one's conclusion about the mutual semimajor axis and orbital period.

It is reassuring that using two different kinds of measurements (astrometry and photometry) and common assumptions (ρ ≃1 g cm -3 and p ≃0.16) we get similar results for the inferred size of each component of 2001 QW 322 .

Combining the mass derived from Kepler's third law and the photometricaly derived nominal size r=54 km for each component, we compute the density of 2001 QW 322 . For the nominal albedo, the density is in the range 0.8-1.2 g cm -3 , which is at the limit between a mostly water-ice body and a mixture of water-ice and rocks. This density is smaller than that measured for the largest KBOs: Pluto and Charon (2.03±0.06 and 1.65±0.06 g cm -3 respectively; S19), Eris (2.26±0.25 g cm -3 ; S20) and ( 136108) 2003 EL 61 (3.0±0.4 g cm -3 ; S21), but it is denser than other KBO binaries for which a density have been published: 0.3-0.8 g cm -3 for (47171) 1999 TC 36 (S22) and 0.5-1.0 g cm -3 for (26308) 1998 SM 165 (S23).

Estimating the hydrostatic pressure at the center of each component by

P 0 = 2 3 πGρ 2 r 2 (S6)
(e.g. Stansberry et al. , 2006), we find P 0 ≃ 0.4 MPa. This pressure is insufficiant to remove much void space out of cold, particulate water-ice (S24), allowing a rather large porosity of both components and potentially a sizeable fraction of rocky material to be incorporated in them.

Formation scenarios and constraints from the orbit

All formation scenarios proposed to date involve three or more bodies. Formation scenarios by collision (reaccretion of the fragments of a crater or of the fully disrupted body, or rotational fission induced by an impact), involving only two precusor bodies, tend to create smaller secondary-to-primary mas ratios and smaller separation binaries. Rotational fission (involving only one body) followed by tidal evolution produce binaries with angular momentum one order of magnitude smaller than the KBO binaries with wide separations between components (S25, S26). This angular momentum deficit is also the rule for direct-collision formation scenarios (S25, S27-S31).

Four main scenarios have been proposed to explain the formation of Kuiper Belt binaries (S32-S35), each with their own characteristic outcomes. Only the collision of two bodies in the Hill sphere of a third one (cL 2 L, S32) claims to be able to directly form very larger separation binaries. (S33) claim that their mechanism of interaction of two large bodies with a swarm of small planetesimals (L 2 l) always dominate over cL 2 L. This is true for the overall efficiency of binary formation, as L 2 l is very efficient at small separations. But it is unable to directly generate any binary with such a large separation. Similarly, mechanisms proposed by (S34) and (S35) fall short of producing binaries like 2001 QW 322 .

The exchange reaction proposed by (S34) can produce very large separations, but only with very large mutual-orbit eccentricities. Creating 2001 QW 322 by this mechanism would require further evolution of the original, large eccentricity binary. Tidal dissipation is usually a good candidate to circularize orbits, although this would reduce the separation in the same process. The circularization timescale (S36), in the most favorable case of synchronous rotation, is given by

τ e = - e ė ≃ 2ae 2 ȧ = 2Q s 21k 2s m s m p a r s 5 1 n s , (S7)
where m s and m p are the masses of the secondary and primary respectively, r s is the radius of the secondary, n s = 2π/P is the mutual orbit frequency, Q s is the secondary component dissipation factor, and k 2s its Love number. Values of Q s ∼ 100 are usually assumed for small icy bodies, reaching 10 for the most dissipative bodies (S37). k 2s values range from 0.1 for dissipative bodies to 0.0001 for very elastic bodies (S37). For Q s /k 2s ∼ 100 (a lower limit) τ e ∼ 10 18 yr, although (S6) is derived in the favourable case of the components being already despun to synchronous rotation, even that is unlikely. We compute a despinning time scale (S37) of ∼ > 2 × 10 16 yr, so tides clearly have had no appreciable effect on this system. (S38) studied the long term stability of moderate to large separation Kuiper Belt binaries and concluded that the most efficient mechanism to disrupt them is the direct collision of an interlopper onto the secondary and kicking it out of its orbit. Using the same method for the newly determined orbital parameters of 2001 QW 322 , and still assuming our nominal albedo, we estimate a disruption lifetime for 2001 QW 322 of 0.3-1 Gy. If the formation of 2001 QW 322 is primordial, likely due to the cL 2 L mechanism, then it is one of a few survivors of a population at least 100 times larger. 2001 QW 322 could also be a transitory object, evolving due to perturbation by a swarm of small KBO interloppers. Moderate separation KB binaries (separation of 1-2 ′′ ) can have their separation gradually changed by small perturbations, either tightening them, or unbinding them over the age of the Solar System. However, asserting the likelihood of this hypothesis would require a better knowledge of the parent population of moderate separation KB binaries. g ′ -r ′ color versus r ′ -i ′ color for the Gulbis et al. (2006) Kuiper belt core objects (solid circles) and for the two components of 2001 QW 322 (diamonds). Both components have an r ′ -i ′ color that is in the middle of the range for the Core Kuiper belt objects, but a g ′ -r ′ color on the blue extreme of it.

(

  CFHT) is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.Observations at Palomar Observatory are carried out under a collaborative agreement between Cornell University and the California Institute of Technology.Observations made with ESO Telescopes at the La Silla or Paranal Observatories under programme IDs 069.C-0460, 071.C-0497, 072.C-0542, 074.C-0379, 075.C-0251 and 380.C-0791.The Gemini Observatory is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministrio da Cincia e Tecnologia (Brazil) and SECYT (Argentina).Observations obtained at WIYN Observatory, a joint facility of the University of Wisconsin-Madison, Indiana University, Yale University, and the National Optical Astronomy Observatories.Observations obtained at the William Herschel Telescope, at Roque de los Muchachos Observatory (La Palma, Canary Islands, Spain), operated by the Instituto de Astrofisica de Canarias.Observations reported here were obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona.

Fig. 1 .

 1 Fig. 1. (Top): image of the Kuiper Belt binary 2001 QW 322 from VLT, on September 3rd, 2002. The separation between the two components at the time was 3.4 ′′ . (Middle): image from Gemini-South on September 17th, 2007; the separation was 1.8 ′′ . On both panels, the components are circled. On all panels, North is up and East is left. The "a" component is the most southern one. (Bottom): relative position of the two components of 2001 QW 322 .

Fig. 2 .

 2 Fig.2. Range of plausible orbital parameters and densities for 2001 QW 322 . These have been obtained by non-linear minimisation of a χ 2 statistics, starting from several thousand initial conditions, retaining only those solutions with a reduced χ 2 close enough to the overall bestfit, as explained in (7). The fitted parameters are the binary's orbital period P , its semimajor axis a, its eccentricity e, its pole orientation and its argument of pericenter. Eccentricity (A), density (B) and semimajor axis (C) are given as a function of the period, and the pole's ecliptic latitude versus pole's ecliptic longitude (D). The density was derived assuming a radius for each component of 54 km. (D) shows the precision with which the mutual-orbit pole is now known on the sky.

Fig. 3 .

 3 Fig.3. Secondary-to-primary mass ratio versus average separation (in units of the primary's radius). Represented by the dashed box are the known binary asteroids, all on the left side of the plot (the largest separation barely exceeds 100). Also shown are the most-extreme outerplanet irregular satellites and several other binary KBOs. The mass ratio q is estimated from the published difference in magnitude between the components, assuming an equal albedo and density for both components. The error bar on the separation for 2001 QW 322 accounts only for the uncertainty in the estimated semimajor axis a, but does not account for the (unlikely) possibility that the radius could be off by up to a factor of two. Nevertheless, with equal mass component and a separation that is more than 2000 times the radius of each component, 2001 QW 322 clearly stands out in the top-right corner of this diagram as the widest-orbit, near-equal mass binary of the Solar System.

  photometric measurements for R-band filters: the R BESS filter on VLT/FORS1 in 2002, R SP ECIAL filter on VLT/FORS2 for all other VLT images, r G0303 filter on Gemini-N in 2005 and r filter on Gemini-S in 2006 and 2007. Table S2 lists photometric measurements for V -band filters: the V BESS filter on VLT/FORS2 in 2003 and 2007, and g G0301 filter on Gemini-N in 2005. Table S3 lists photometric measurements for I-band filters: the I BESS filter on VLT/FORS2 in 2003 and 2007, and i G0302 filter on Gemini-N in 2005.

Fig. S1 .

 S1 Fig. S1. Magnitude of Northern component minus magnitude of Southern component for all observations in R-band (VLT R BESS , VLT R SP ECIAL and Gemini r filters). Errorbars are 1-σ uncertainties. The dotted line correspond to the weighted average of all those measurements.

  Fig. S2. g ′ -r ′ color versus r ′ -i ′ color for theGulbis et al. (2006) Kuiper belt core objects (solid circles) and for the two components of 2001 QW 322 (diamonds). Both components have an r ′ -i ′ color that is in the middle of the range for the Core Kuiper belt objects, but a g ′ -r ′ color on the blue extreme of it.

Table S1 .

 S1 List of photometric measurements acquired by our team on 2001 QW 322 in R band filters (VLT R BESS , VLT R SP ECIAL and Gemini r). The second column (m b -m a ) gives the magnitude difference between the 2 components (Northern component minus Southern component) as directly measured from the images, before applying any zeropoint calibration; the uncertainty reported corresponds to the statistical error in the measured flux. The uncertainty is the 1-σ photon count statistical fluctuation, as reported by the PHOT or ALLSTAR tasks in IRAF. The magnitude reported separately for each component are either absolute photometry directly tied to the Landolt photometric system, or relative to the first measurement in a series (see notes). .01013 -0.10 ± 0.06 23.85 ± 0.05 23.75 ± 0.04 R SP ECIAL VLT 8m 2004 10 11.99947 -0.04 ± 0.07 23.81 ± 0.06 23.76 ± 0.06 R SP ECIAL VLT 8m b 2004 10 12.14095 -0.13 ± 0.08 24.06 ± 0.08 23.86 ± 0.07 R SP ECIAL VLT 8m b 2004 10 13.13509 0.31 ± 0.11 23.61 ± 0.08 24.01 ± 0.11 R SP ECIAL VLT 8m b 2004 10 15.16448 0.14 ± 0.11 23.90 ± 0.10 24.05 ± 0.12 R SP ECIAL VLT 8m b 2005 07 08.44718 -0.11 ± 0.07 23.66 ± 0.05 23.55 ± 0.04 Photometry on 2002 09 03.00102 is absolute, tied to the Landolt photometric standard system; other measurements are relative to this one. Same as ( a ), but with reference photometry on 2007 10 10.00426.

	0 2002 09 03.00102 0.25 ± 0.09 23.80 ± 0.05 24.04 ± 0.07 . 4 0 . 5 0 . 6 b UT Date m a m b m b -m a 2002 09 03.99922 -0.33 ± 0.17 24.18 ± 0.14 23.85 ± 0.11 2003 10 28r G0303 Filter R BESS R BESS	Telescope VLT 8m a VLT 8m a Gemini-N 8m
	0 2005 07 08.45190 0.01 ± 0.09 23.72 ± 0.06 23.73 ± 0.06 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 g ' ☞ r ' 0 . 0 0 . 1 0 . 2 0 . 3 r ' ✑ i ' a 2005 07 08.49226 -0.12 ± 0.08 23.75 ± 0.06 23.63 ± 0.06 2005 07 08.49698 -0.13 ± 0.10 23.82 ± 0.07 23.69 ± 0.06 2005 08 02.12595 -0.10 ± 0.10 23.92 ± 0.07 23.82 ± 0.07 R SP ECIAL 0 . 9 1 . 0 r G0303 r G0303 r G0303 r 2006 10 28.04920 0.10 ± 0.10 r 2006 10 28.05355 -0.19 ± 0.09 r 2007 09 17.01976 -0.03 ± 0.11 r 2007 09 17.02459 -0.04 ± 0.06 2007 10 10.00426 0.04 ± 0.13 23.96 ± 0.10 24.00 ± 0.10 R SP ECIAL 2007 10 10.02030 -0.11 ± 0.10 23.81 ± 0.08 23.70 ± 0.07 R SP ECIAL 2007 10 10.03712 0.13 ± 0.10 23.92 ± 0.07 24.05 ± 0.08 R SP ECIAL 2007 11 12.03120 -0.02 ± 0.13 23.81 ± 0.09 23.79 ± 0.09 R SP ECIAL	1 Gemini-N 8m . 1 Gemini-N 8m Gemini-N 8m VLT 8m Gemini-S 8m Gemini-S 8m Gemini-S 8m Gemini-S 8m VLT 8m c VLT 8m c VLT 8m c VLT 8m c
	a

b Same as ( a ), but with reference photometry on 2004

10 11.99947. c

Table S2 .

 S2 Same as TableS1but for images acquired in V band filters (VLT V BESS and Gemini g). .00556 -0.13 ± 0.09 24.44 ± 0.06 24.31 ± 0.06 V BESS VLT 8m 2005 07 08.45680 0.03 ± 0.17 24.21 ± 0.11 24.24 ± 0.13 g G0301 Gemini-N 8m 2005 07 08.46152 0.27 ± 0.15 23.94 ± 0.10 24.21 ± 0.12 g G0301 Gemini-N 8m 2005 07 08.48264 -0.29 ± 0.16 24.33 ± 0.13 24.04 ± 0.10 g G0301 Gemini-N 8m 2005 07 08.48737 -0.33 ± 0.17 24.16 ± 0.14 23.83 ± 0.11 g G0301 Gemini-N 8m 2007 10 10.01232 0.00 ± 0.08 24.27 ± 0.06 24.27 ± 0.07 V BESS VLT 8m

	UT Date 2003 10 28	m b -m a	m a	m b	Filter	Telescope

Table S3 .

 S3 Same as Table S1 but for images acquired in I band filters (VLT I BESS and Gemini i). .01355 -0.05 ± 0.09 23.30 ± 0.07 23.25 ± 0.06 I BESS VLT 8m 2005 07 08.46604 -0.33 ± 0.13 23.63 ± 0.10 23.30 ± 0.08 i G0302 Gemini-N 8m 2005 07 08.47006 -0.30 ± 0.10 23.49 ± 0.07 23.19 ± 0.06 i G0302 Gemini-N 8m 2005 07 08.47409 -0.14 ± 0.09 23.42 ± 0.07 23.28 ± 0.06 i G0302 Gemini-N 8m 2005 07 08.47812 -0.09 ± 0.09 23.40 ± 0.07 23.31 ± 0.06 i G0302 Gemini-N 8m 2007 10 10.02872 0.07 ± 0.07 23.30 ± 0.05 23.37 ± 0.05 I BESS VLT 8m

	UT Date 2003 10 28	m b -m a	m a	m b	Filter	Telescope

Table S4 .

 S4 Separation (in arcseconds) and position angle (PA; origin is North, positive values to the East) for all astrometric observations of 2001 QW 322 . The "UT Date" column corresponds to the time at center of observation, with no light travel time correction applied. The position (RA and DEC columns) is given for the center of mass of the binary, assuming equal masses for the components. The "δ Sep" column give the uncertainty estimate on the separation (in arcsecond). The uncertainty on the position angle (not given here) is simply δ Sep divided by Sep, in radian. The last column indicates the telescope used to acquire these data.

	UT Date	RA	DEC	Sep (") P.A. ( • ) δ Sep (")	Telescope
	2001 07 27.42296 20 34 48.37 -18 49 50.4	3.61	-70.0	0.20	CFHT 3.6m a
	2001 08 24.24085 20 32 43.41 -18 57 51.4	3.89	-68.7	0.17	CFHT 3.6m
	2001 08 24.26181 20 32 43.33 -18 57 51.7	3.87	-72.9	0.18	CFHT 3.6m
	2001 08 24.28289 20 32 43.24 -18 57 52.0	3.96	-69.9	0.12	CFHT 3.6m
	2001 08 25.25689 20 32 39.25 -18 58 07.3	3.74	-67.8	0.23	CFHT 3.6m
	2001 08 25.27753 20 32 39.16 -18 58 07.7	3.83	-71.4	0.09	CFHT 3.6m
	2001 09 18.16406 20 31 17.28 -19 03 23.0	4.09	-72.0	0.35	Hale 5m b
	2001 09 18.17234 20 31 17.26 -19 03 23.1	4.23	-67.5	0.35	Hale 5m
	2001 09 18.17648 20 31 17.25 -19 03 23.1	3.95	-67.1	0.35	Hale 5m
	2001 09 19.18914 20 31 14.60 -19 03 33.5	4.16	-73.2	0.25	Hale 5m
	2001 09 19.26248 20 31 14.40 -19 03 34.2	4.38	-79.3	0.53	Hale 5m
	2002 06 07.52857 20 43 12.03 -18 23 55.4	3.62	-65.7	0.12	CFHT 3.6m
	2002 06 07.53979 20 43 12.00 -18 23 55.5	3.54	-65.0	0.12	CFHT 3.6m
	2002 09 03.00102 20 37 16.13 -18 47 51.8	3.47	-65.0	0.05	VLT 8m c
	2002 09 03.99922 20 37 12.44 -18 48 06.3	3.27	-63.2	0.24	VLT 8m
	2003 07 29.05982 20 45 02.95 -18 24 18.4	3.26	-60.2	0.33	ESO 2.2m c
	2003 07 31.29834 20 44 52.60 -18 25 00.9	3.21	-57.3	0.10	ESO 2.2m
	2003 10 28.00556 20 40 46.07 -18 41 47.5	2.93	-55.4	0.06	VLT 8m
	2003 10 28.01013 20 40 46.08 -18 41 47.5	2.92	-55.2	0.04	VLT 8m
	2003 10 28.01355 20 40 46.08 -18 41 47.5	2.93	-54.9	0.06	VLT 8m
	2004 09 16.13690 20 46 45.72 -18 24 48.1	2.62	-43.9	0.35	Hale 5m
	2004 09 16.16437 20 46 45.64 -18 24 48.5	2.55	-48.2	0.35	Hale 5m
	2004 10 11.99947 20 45 52.88 -18 28 26.4	2.35	-40.9	0.04	VLT 8m
	2004 10 12.14095 20 45 52.74 -18 28 27.0	2.37	-42.0	0.04	VLT 8m
	2004 10 13.13509 20 45 51.84 -18 28 31.0	2.22	-43.0	0.10	VLT 8m
	2004 10 15.16448 20 45 50.27 -18 28 37.9	2.30	-42.2	0.11	VLT 8m
	2005 07 08.44718 20 56 46.34 -17 49 58.1	2.00	-25.6	0.04	Gemini-N 8m d
	2005 07 08.47006 20 56 46.25 -17 49 58.6	2.06	-27.6	0.07	Gemini-N 8m
	2005 08 02.12595 20 54 57.80 -17 57 53.8	1.92	-27.9	0.10	VLT 8m
	2005 09 26.17135 20 51 24.96 -18 12 53.2	2.00	-24.0	0.12	WIYN 3.5m e
	2005 09 26.23657 20 51 24.79 -18 12 53.9	1.96	-27.5	0.12	WIYN 3.5m
	2005 10 04.11739 20 51 08.31 -18 14 02.6	1.89	-23.1	0.15	Hale 5m
	Continued on Next Page . . .

Table S4 -

 S4 Continued 2005 10 04.15318 20 51 08.25 -18 14 02.9

		1.87	-27.7	0.14	Hale 5m
	2006 10 21.89960 20 55 52.77 -18 01 16.1	1.60	4.1	0.24	WHT 4.2m f
	2006 10 28.04920 20 55 52.45 -18 01 17.9	1.64	9.1	0.06	Gemini-S 8m d
	2006 10 28.05355 20 55 52.46 -18 01 17.9	1.66	5.5	0.05	Gemini-S 8m
	2007 09 16.19381 21 02 04.68 -17 41 44.0	1.84	32.4	0.18	Hale 5m
	2007 09 16.20003 21 02 04.66 -17 41 44.0	1.86	26.7	0.15	Hale 5m
	2007 09 16.23572 21 02 04.54 -17 41 44.6	1.83	32.8	0.14	Hale 5m
	2007 09 17.01976 21 02 01.99 -17 41 55.5	1.79	32.8	0.07	Gemini-S 8m
	2007 09 17.02459 21 02 01.98 -17 41 55.6	1.88	32.5	0.04	Gemini-S 8m
	2007 10 10.01232 21 01 06.39 -17 45 53.6	1.80	37.1	0.07	VLT 8m
	2007 10 10.02030 21 01 06.38 -17 45 53.6	1.82	34.2	0.05	VLT 8m
	2007 10 10.03712 21 01 06.35 -17 45 53.8	1.80	34.2	0.05	VLT 8m
	2007 11 03.08849 21 00 55.56 -17 46 38.0	1.83	35.4	0.09	MMT 6.5m g
	2007 11 03.09269 21 00 55.56 -17 46 38.0	1.92	40.4	0.11	MMT 6.5m
	2007 11 12.03120 21 01 05.14 -17 45 57.0	1.85	37.6	0.09	VLT 8m

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