Sylvain Golenia

Tristan Haugomat
email: tristan.haugomat@etudiant.univ-rennes1.fr

Gol Sylvain

Tristan Énia

Haugomat

ON THE A.C. SPECTRUM OF THE 1D DISCRETE DIRAC OPERATOR

Keywords: 2000 Mathematics Subject Classification. 47B36, 39A70, 46N50, 47A10, 81Q10 discrete Dirac, ac spectrum, 1-dimensional, Jacobi matrix

published or not. The documents may come

Introduction

We study properties of relativistic (massive or not) charged particles with spin-1/2. We follow the Dirac formalism, see [Di]. We shall focus on the 1-dimensional discrete version of the problem. In the introduction we stick to the case of Z and shall discuss the case of Z + := Z ∩ [0, ∞) in the core of the paper, see Section 5.1. The mass of the particle is given by m ≥ 0. For simplicity, we re-normalize the speed of light and the reduced Planck constant by 1. The discrete Dirac operator, acting on ℓ 2 (Z, C 2 ), is defined by

D m := m d d * -m ,
where d := Id -τ and τ is the right shift, defined by τ f (n) = f (n + 1), for all f ∈ ℓ 2 (Z, C). The operator D m is self-adjoint. Moreover, notice that

D 2 m = ∆ + m 2 0 0 ∆ + m 2 ,
where ∆f (n) := 2f (n) -f (n + 1) -f (n -1). This yields that σ(D 2 m ) = [m 2 , 4 + m 2 ]. To remove the square above D m , we define the symmetry S on ℓ 2 (Z, C) by Sf (n) := f (-n) and the unitary operator on ℓ 2 (Z, C 2 )

U := 0 iS -iS 0 . (1.1)
Clearly U = U * = U -1 . We have that U D m U = -D m . We infer that the spectrum of D m is purely absolutely continuous (ac) and that

σ(D m ) = σ ac (D m ) = [-m 2 + 4, -m] ∪ [m, m 2 + 4].
We shall now perturb the operator by an electrical potential V = (V 1 , V 2 ) t ∈ ℓ ∞ (Z, R 2 ). We set

H := D m + V 1 0 0 V 2 . (1.2)
Here, V i denotes also the operator of multiplication by the function V i . Clearly, the essential spectrum of H is the same as the one of D m if V tends to 0 at infinity. We turn to more refined questions. The singular continuous spectrum, quantum transport, and localization have been studied before [START_REF] Carvalho | Spectral and dynamical properties of sparse one-dimensional continuous Schrödinger and Dirac operators[END_REF][START_REF] Prado | Sparse 1D discrete Dirac operators I: Quantum transport[END_REF][START_REF] Carvalho | Sparse 1D discrete Dirac operators II: Spectral properties[END_REF][START_REF] Prado | Dynamical lower bounds for 1D Dirac operators[END_REF][START_REF] De Oliveira | Spectral and Localization Properties for the One Dimensional Bernoulli Discrete Dirac Operator[END_REF][START_REF] De Oliveira | Dynamical Delocalization for the 1D Bernoulli Discrete Dirac Operator[END_REF]. The question of the purely ac spectrum seems not to have been answered before. This is the purpose of our article.

We recall the following standard result for the Laplacian (the non-relativistic setting). For completeness we sketch the proof in Section 6.

Theorem 1.1. Take V ∈ ℓ ∞ (Z, R) and ν ∈ Z + \ {0} such that:

lim n→±∞ V (n) = 0, V | Z + -τ ν V | Z + ∈ ℓ 1 (Z + , R),
(1.3) then the spectrum of ∆ + V is purely absolutely continuous on (0, 4).

In the case of Z + and for ν = 1, the result has been essentially proved in [Wei] (in fact in the quoted reference, one focuses only on the continuous setting). The proof for the discrete setting can be found in [DoNe, Sim]. For ν > 1, it seems that it was first done in [Sto]. Note that for instance, (1.3) is satisfied by potentials like V (n) = (-1) n W (n), where W is decay to 0. We refer to [GoNe] and to [KaLa] for recent results in this direction.

An amusing and easy remark is the difference between Z + and Z. In the latter, it is enough to assume the decay hypothesis on the right part of the potential. This reflects the fact that the particle can always escape to the right even if the left part of the potential would have given some singular continuous spectrum in a half-line setting.

We now turn to the main result of the paper. For sake of simplicity we present the case of Z with electric perturbations. We refer to Section 3 for the main statements, where we deal with a mixture of magnetic and Witten-like perturbations and also with Z + . Theorem 1.2. Take V ∈ ℓ ∞ (Z, R 2 ) and ν ∈ Z + \ {0} with:

lim n→±∞ V (n) = 0, V | Z + -τ ν V | Z + ∈ ℓ 1 (Z + , R 2 ),
then the spectrum of H is purely absolutely continuous on -√ m 2 + 4, -m ∪ m, √ m 2 + 4 .

To study H we reduce the problem to a non-self-adjoint Laplacian-like operator which depends on the spectral parameter. This is due to a spin-up/down decomposition, see Proposition 4.2. This idea has been efficiently used in the continuous setting, e.g., [DES, BoGo, JeNe] and references therein, and seems to be new in the discrete setting. Then, we adapt the iterative process to the non-self-adjoint Laplacian-like operator and follow the presentation of [FHS]. We refer to [START_REF] Froese | A geometric approach to absolutely continuous spectrum for discrete Schrödinger operators, Review, Alp workshop[END_REF] for a recent survey about this technique.

Finally we present the organization of the paper. In section 2 we recall general facts about the free discrete Dirac operator and about hyperbolic geometry. Then in Section 3 we present the main results. Next in Section 4, we reduce the problem to a kind of Laplacian and adapt the transfer matrices technique. After that in Section 5, we prove the main results about absolutely continuous spectra. Finally we discuss briefly the case of the Laplacian. Notation: We denote by B(X) the space of bounded operators acting on a Banach space X. Let

Z k := Z ∩ [k, +∞[ for k ∈ Z and Z -:= Z \ Z + . For A, B ⊂ C, we set A ⋐ B if clA ⊂ intB,
where cl and int stand for closure and interior, respectively. Acknowledgments: We would like to thank Serguei Denisov, Stanilas Kupin, Christian Remling, and Hermann Schulz-Baldes for useful discussions. We would also like to thank Leonid Golinski for comments on the script. The research is partially supported by Franco-Ukrainian programm "Dnipro 2013-14".

General facts

2.1. The spectrum of the discrete Dirac operator. Let Z k := Z∩[k, +∞[ for k ∈ Z and G ∈ {Z k , Z}. We define d ∈ B ℓ 2 (G, C) by df (n) := f (n) -f (n + 1), for all f ∈ ℓ 2 (G, C) and n ∈ G. Clearly d is bounded. Its adjoint is given by d * f (n) = f (n) if G = Z k and n = k, f (n) -f (n -1)
otherwise, for all f ∈ ℓ 2 (G, C) and n ∈ G . Now for m ≥ 0 we define the discrete Dirac operator on ℓ 2 (G, C 2 ) by G) be the Laplacian on ℓ 2 (G, C) defined by

D (G) m := m d d * -m . It is easy to see that D (G) m is self-adjoint. Let ∆ (
∆ (G) f (n) := f (n) -f (n + 1) if G = Z k and n = k, 2f (n) -f (n -1) -f (n + 1)
otherwise, (2.1) for all f ∈ ℓ 2 (G, C). We study first the discrete Dirac operator on Z, we have

D (Z) m 2 = ∆ (Z) + m 2 0 0 ∆ (Z) + m 2 .
By Fourier transformation, we see that ∆ (Z) is non-negative and that its spectrum is [0,4]. Therefore, the spectrum of

D (Z) m 2 is [m 2 , 4 + m 2 ].
Relying on (1.1), we obtain:

Proposition 2.1. We have σ D (Z) m = σ ess D (Z) m = σ D (Z+) m = σ ess D (Z+) m = -m 2 + 4, -m ∪ m, m 2 + 4 . Proof. We have -D (Z) m -λ -1 = U D (Z) m -λ -1 U so ϕ -D (Z) m = U ϕ D (Z) m U , for all ϕ Borel measurable. Therefore, σ D (Z) m = - √ m 2 + 4, -m ∪ m, √ m 2 + 4 . By writing Z = Z -∪ Z + , we see easily that σ ess (D (Z) m ) = σ ess (D (Z+)
m ). To conclude, a direct computation shows that D (Z+) m has no eigenvalue.

2.2.

A few words about the Poincaré half-plane. We shall use extensively some properties of the Poincaré half-plane. It is defined by:

H := {x + iy | x ∈ R, y > 0}, endowed with the metric ds 2 = dx 2 + dy 2 y 2 ,
Recall that the geodesic distance is given by:

d H (z 1 , z 2 ) = cosh -1 1 + 1 2 |z 1 -z 2 | 2 ℑz 1 • ℑz 2 ≤ |z 1 -z 2 | ℑ(z 1 ) ℑ(z 2 ) . (2.2)
We turn to the study of (hyperbolic-)contractions.

Lemma 2.2. Given a, b ∈ cl(H) and c > 0, we set:

ϕ a,b,c (z) := -a -(b + cz) -1 -1 . (2.3) It is a contraction of (H, d H ).
Moreover, if a, b ∈ H, then ϕ a,b,c is a strict contraction of (H, d H ). More precisely, we have

d H (ϕ a,b,c (z 1 ), ϕ a,b,c (z 2 )) ≤ 1 1 + ℑ(a) • ℑ(b) d H (z 1 , z 2 ) , (2.4) for all z 1 , z 2 ∈ H. Moreover we have d H (ϕ a,b,c (z 1 ), ϕ a,b,c (z 2 )) ≤ (1 + ℑ(b)|a|) 2 (ℑ(b) • ℑ(a)) 2 ,
for all z 1 , z 2 ∈ H.

Proof. First, since z → cz is a hyperbolic isometry, it is enough to consider ϕ a,b := ϕ a,b,1 . Then, using that z → z + w is a contraction when w ∈ cl(H) and that z → -1/z is an isometry, we obtain that ϕ a,b is a contraction. We turn to the second statement. A direct computation yields that if h ∈ H then

-(h + H) -1 = B |.| i 2ℑ(h) , 1 2ℑ(h) ⊂ H. (2.5) Given C > 0, as in the proof of [FHS][Proposition 2.1], if z 1 , z 2 ∈ H with min (|z 1 |, |z 2 |) ≤ C note that d H (z 1 + a, z 2 + a) ≤ C C + ℑ(a) d H (z 1 , z 2 ). Since z → -z -1 is an isometry of H and z → -(b + z) -1 is a contraction of H, we use (2.5) for h = b to have that d H (ϕ a,b (z 1 ), ϕ a,b (z 2 )) = d H a -(b + z 1 ) -1 , a -(b + z 2 ) -1 ≤ (ℑ(b)) -1 (ℑ(b)) -1 + ℑ(a) d H -(b + z 1 ) -1 , -(b + z 2 ) -1 ≤ 1 1 + ℑ(a) • ℑ(b) d H (z 1 , z 2 ) ,
for all z 1 , z 2 ∈ H. So we obtain (2.4). By (2.5) for h = a we know that -

(a + H) -1 is an Euclidean ball of diameter ℑ(a) -1 , hence |ϕ a,b (z 1 ) -ϕ a,b (z 2 )| ≤ (ℑ(a)) for all z 1 , z 2 ∈ H. Given C > 0, if z ∈ H with |z| ≤ C we have ℑ - 1 a + z ≥ ℑ(a) (C + |a|) 2 . So if z ∈ H, by (2.5) for h = b, | -(b + z) -1 | ≤ (ℑ(b)) -1 , so ℑ (ϕ a,b (z)) ≥ ℑ(a) (ℑ(b)) -1 + |a| 2 .
So we obtain that

d H (ϕ a,b (z 1 ), ϕ a,b (z 2 )) = cosh -1 1 + 1 2 |ϕ a,b (z 1 ) -ϕ a,b (z 2 )| 2 ℑ(ϕ a,b (z 1 ))ℑ(ϕ a,b (z 2 )) ≤ |ϕ a,b (z 1 ) -ϕ a,b (z 2 )| ℑ(ϕ a,b (z 1 )) ℑ(ϕ a,b (z 2 )) ≤ (ℑ(b)) -1 + |a| 2 (ℑ(a)) 2 = (1 + ℑ(b)|a|) 2 (ℑ(b) • ℑ(a)) 2 ,
for all z 1 , z 2 ∈ H.

We shall also need the following technical lemmata.

Lemma 2.3. Suppose that a, b ∈ cl(H), c > 0, and z ∈ H. We have:

d H (ϕ a,b,c (z), i) ≤ (|b| + c|z|) 2 cℑz + 1 (|b| + c|z|) |a| + (cℑ (z)) -1 cℑ (z) .
Proof. First we have:

|ϕ a,b,c (z)| = 1 a -(b + cz) -1 ≤ 1 ℑ a -(b + cz) -1 ≤ 1 ℑ -(b + cz) -1 = |b + cz| 2 ℑ (b + cz) ≤ (|b| + c|z|) 2 cℑ (z) .
Then we obtain:

ℑ (ϕ a,b,c (z)) = ℑ - 1 a -(b + cz) -1 = ℑ a -(b + cz) -1 a -(b + cz) -1 2 ≥ ℑ -(b + cz) -1 |a| + |b + cz| -1 2 ≥ ℑ (b + cz) |b + cz| 2 |a| + ℑ (b + cz) -1 -2 ≥ cℑ (z) (|b| + c|z|) 2 |a| + (cℑ (z)) -1 2 .
Finally with (2.2) we infer:

d H (ϕ a,b,c (z), i) ≤ |ϕ a,b,c (z) -i| ℑ (ϕ a,b,c (z)) ≤ |ϕ a,b,c (z)| + 1 ℑ (ϕ a,b,c (z)) ,
which yields the result.

Lemma 2.4. For all n ∈ Z 1 there exist

A n , B n , C n , D n ∈ R[X 1 , Y 1 , Z 1 , . . . , X n , Y n , Z n ], such that C n (ω) + D n (ω)ζ = 0 and ϕ x1,y1,z1 • • • • • ϕ xn,yn,zn (ζ) = - A n (ω) + B n (ω)ζ C n (ω) + D n (ω)ζ , for all ω := (x 1 , y 1 , z 1 , . . . , x n , y n , z n ) ∈ cl(H) 2 × R * + n and ζ ∈ H.
We point out that the continuity of A n , B n , C n , and D n with respect to the coefficients will be crucial in Proposition 4.8. The proof is straightforward. We give it for completeness.

Proof. We prove the result by induction. Let n = 1, we have andD 1 (x, y, z)

ϕ x,y,z (ζ) = -x -(y + zζ) -1 -1 = - y + zζ xy -1 + xzζ = - A 1 (x, y, z) + B 1 (x, y, z)ζ C 1 (x, y, z) + D 1 (x, y, z)ζ with A 1 (x, y, z) := y, B 1 (x, y, z) := z, C 1 (x, y, z) := xy -1,
:= xz. Moreover C 1 (x, y, z) + D 1 (x, y, z)ζ = xy -1 + xzζ = (y + zζ) x - 1 y + zζ = 0 for all (x, y, z) ∈ cl(H) 2 × R * + and ζ ∈ H because this is the product of two elements of H. Now suppose that we have proved the existence of A n , B n , C n , D n and prove the existence of A n+1 , B n+1 , C n+1 , and D n+1 . Let ω := (x 1 , y 1 , z 1 , . . . , x n+1 , y n+1 , z n+1 ) ∈ cl(H) 2 × R * + n+1 and ζ ∈ H. Let ω := (x 1 , y 1 , z 1 , . . . , x n , y n , z n ), we have ϕ x1,y1,z1 • • • • • ϕ xn+1,yn+1,zn+1 (ζ) = ϕ x1,y1,z1 • • • • • ϕ xn,yn,zn (ϕ xn+1,yn+1,zn+1 (ζ)) = - A n (ω) -B n (ω) A 1 (x n+1 , y n+1 , z n+1 ) + B 1 (x n+1 , y n+1 , z n+1 )ζ C 1 (x n+1 , y n+1 , z n+1 ) + D 1 (x n+1 , y n+1 , z n+1 )ζ C n (ω) -D n (ω) A 1 (x n+1 , y n+1 , z n+1 ) + B 1 (x n+1 , y n+1 , z n+1 )ζ C 1 (x n+1 , y n+1 , z n+1 ) + D 1 (x n+1 , y n+1 , z n+1 )ζ = - A n+1 (ω) + B n+1 (ω)ζ C n+1 (ω) + D n+1 (ω)ζ , with A n+1 (ω) := A n (ω)C 1 (x n+1 , y n+1 , z n+1 ) -B n (ω)A 1 (x n+1 , y n+1 , z n+1 ), B n+1 (ω) := A n (ω)D 1 (x n+1 , y n+1 , z n+1 ) -B n (ω)B 1 (x n+1 , y n+1 , z n+1 ), C n+1 (ω) := C n (ω)C 1 (x n+1 , y n+1 , z n+1 ) -D n (ω)A 1 (x n+1 , y n+1 , z n+1 ), D n+1 (ω) := C n (ω)D 1 (x n+1 , y n+1 , z n+1 ) -D n (ω)B 1 (x n+1 , y n+1 , z n+1 ).
Finally since ϕ xn+1,yn+1,zn+1 is a contraction by Lemma 2.2, we have that ϕ xn+1,yn+1,zn+1 (ζ) ∈ H and

C n+1 (ω) + D n+1 (ω)ζ = C n (ω) + D n (ω)ϕ xn+1,yn+1,zn+1 (ζ) × (C 1 (x n+1 , y n+1 , z n+1 ) + D 1 (x n+1 , y n+1 , z n+1 )ζ) = 0,
by induction. This finishes the proof.

Main results

In this paper, we study a Jacobi-like version of

D (G) m + V , given V = (V 1 , V 2 ) t ∈ ℓ ∞ (G, R 2 ) and W = (W 1 , W 2 ) t ∈ ℓ ∞ (G, C 2 ), where G ∈ {Z + , Z}. Let H (G) m,V,W := m + V 1 d d * -m + V 2 , (3.1) where d := d + W 1 + W 2 τ with τ f (n) := f (n + 1) and d = Id -τ .
Its study is motivated by the fact that it is an intermediate model before the study of discrete Dirac operator in weighted spaces, which would be the treated in another place. The perturbation in W i is a magnetic/Witten-like perturbation. When the perturbation is purely magnetic, one could simply gauge away the W . However, in some situations, even if one could remove the magnetic perturbation, it is sometimes important to be able to lead the analysis till its very end in order to be able to choose the best [sequence of] gauge [s]. Having W i real is motivated by the fact that this corresponds to a dicrete relativistic analog of the Witten Laplacian.

We start with the main result on Z + . It will be proved in Section 5.1.

Theorem 3.1. Take V ∈ ℓ ∞ (Z + , R 2 ) and W ∈ ℓ ∞ (Z + , C 2 ) with: lim n→∞ V (n) = lim n→∞ W (n) = 0, (3.2) then σ ess (H (Z+) m,V,W ) = - √ m 2 + 4, -m ∪ m, √ m 2 + 4 . Assuming also that there exist ν 1 , ν 2 ∈ Z + \ {0} such that W 1 (n) = -1 and W 2 (n) = 1, for all n ∈ Z + , V -τ ν1 V ∈ ℓ 1 (Z + , R 2 ) and W -τ ν2 W ∈ ℓ 1 (Z + , C 2 ), (3.3) then the spectrum of H (Z+) m,V,W is purely absolutely continuous on - √ m 2 + 4, -m ∪ m, √ m 2 + 4 .
We discuss briefly the necessity of the first line of (3.3).

Remark 3.2. Assume that there is

n 0 such that W 1 (n 0 + 1) = -1 and W 2 (n 0 + 1) = 1, then the operator is a direct sum of H (Zn 0 +1)
m,V,W and of a finite matrix. Therefore, it is easy to construct embedded eigenvalues for this operator and this is an obstruction for the result of the theorem. For instance suppose that

W 1 (k) = W 2 (k) = 0 for all k = n 0 and V 1 = V 2 = 0.
Recalling that in a finite dimensional Hilbert space X the spectra of M N and of N M are equal, for M, N ∈ B(X), we have that the point spectrum of

H (Z+) m,0,W 2 is exaclty the one of m 2 Id {1,2,...n0} +          1 -1 0 . . . 0 0 -1 2 -1 . . . 0 0 0 -1 2 . . . 0 0 0 0 . . . . . . . . . . . . 0 0 0 . . . 2 -1 0 0 0 . . . -1 1          .
Therefore the point spectrum of H (Z+) m,0,W is contained in ± m 2 + 4 sin 2 (kπ/2n 0 ), k = 0, . . . , n 0 -1 with total multiplicity 2n 0 .

We turn to the case of Z, we have this important symmetry of charge:

U H (Z) m,V,W U = -H (Z) m,(-SV2,-SV1) t ,(SW1,τ SW2) t , (3.4)
where U is given by (1.1).

Theorem 3.3. Take V ∈ ℓ ∞ (Z, R 2 ) and W ∈ ℓ ∞ (Z, C 2 ) with: lim n→±∞ V (n) = lim n→±∞ W (n) = 0, then σ ess (H (Z) m,V,W ) = - √ m 2 + 4, -m ∪ m, √ m 2 + 4 . Assuming also that there exists ν 1 , ν 2 ∈ Z + \ {0} such that W 1 (n) = -1 and W 2 (n) = 1, for all n ∈ Z, V | Z + -τ ν1 V | Z + ∈ ℓ 1 (Z + , R 2 ) and W | Z + -τ ν2 W | Z + ∈ ℓ 1 (Z + , C 2 ), (3.5)
or alternatively

V | Z --τ ν1 V | Z -∈ ℓ 1 (Z -, R 2 ) and W | Z --τ ν2 W | Z -∈ ℓ 1 (Z -, C 2 ), (3.6) then the spectrum of H (Z) m,V,W is purely absolutely continuous on - √ m 2 + 4, -m ∪ m, √ m 2 + 4 .
Remark 3.4. Note that the second line of (3.5) and (3.6) are equivalent by using the transformation U .

A Laplacian-like approach

4.1. Another form for the resolvent. The objective is to reduce the analysis of the operator

H (G) m,V,W
on ℓ 2 (G, C 2 ) to the one of two operators which are similar to a Laplacian. Take

λ ∈ H, V = (V 1 , V 2 ) t ∈ ℓ ∞ (G, R 2 ), and W = (W 1 , W 2 ) t ∈ ℓ ∞ (G, C 2 ), we define ∆ (G) 1,λ,V,W := d 1 λ + m -V 2 d * -(λ -m -V 1 )
and

∆ (G) 2,λ,V,W := d * 1 λ -m -V 1 d -(λ + m -V 2 ). (4.1)
Note that to lighten the notation we have dropped the dependency on m. However we keep the one in λ. We first check their invertibility.

Proposition 4.1. Let λ ∈ H, V = (V 1 , V 2 ) t ∈ ℓ ∞ (G, R 2 ), and W = (W 1 , W 2 ) t ∈ ℓ ∞ (G, C 2 ), then ∆ (G) 1,λ,V,W and ∆ (G) 2,λ,V,W are invertible. Proof. For b ∈ B ℓ 2 (G, C) , X, Y ∈ ℓ ∞ (G, R) and µ ∈ H let A µ,b,X,Y := b * (µ -X) -1 b + Y , then ℑ f, A µ,b,X,Y f = -ℑ(µ) 1 |µ -X| bf 2 ≤ 0, (4.2) for f ∈ ℓ 2 (G, C).
With the Numerical Range Theorem (e.g., [START_REF] Boussaid | Limiting absorption principle for some long range perturbations of Dirac systems at threshold energies[END_REF]Lemma B.1]) we derive that we have

H ⊂ ρ(A µ,b,X,Y ), the resolvent set of A µ,b,X,Y . Since ∆ (G) 1,λ,V,W = A λ, d * ,V2-m,V1+m -λ and ∆ (G) 2,λ,V,W = A λ, d,V1+m,V2-m -λ, we get ∆ (G) 1,λ,V and ∆ (G) 2,λ,V are invertible.
We give a kind of Schur's Lemma, so as to compute the inverse of the Dirac operator, see also [DES], [BoGo], and [JeNe] for some applications in the continuous setting.

Proposition 4.2. Let λ ∈ H, V = (V 1 , V 2 ) t ∈ ℓ ∞ (G, R 2 ), and W = (W 1 , W 2 ) t ∈ ℓ ∞ (G, C 2 ). Then : H (G) m,V,W -λ -1 = (∆ (G) 1,λ,V,W ) -1 0 0 (∆ (G) 2,λ,V,W ) -1 1 d 1 λ+m-V2 d * 1 λ-m-V1 1 .
Proof. We set (H

(G) m,V,W -λ)f = g. This gives: (V 1 -λ + m)f 1 + df 2 = g 1 d * f 1 + (V 2 -λ -m)f 2 = g 2 ,      f 1 = 1 V 1 -λ + m (g 1 -df 2 ) f 2 = 1 V 2 -λ -m (g 2 -d * f 1 )        f 1 = 1 V 1 -λ + m g 1 - d 1 V 2 -λ -m (g 2 -d * f 1 ) f 2 = 1 V 2 -λ -m g 2 -d * 1 V 1 -λ + m (g 1 -df 2 )        d 1 λ + m -V 2 d * -(λ -m -V 1 ) f 1 = g 1 + d 1 λ + m -V 2 g 2 d * 1 λ -m -V 1 d -(λ + m -V 2 ) f 2 = d * 1 λ -m -V 1 g 1 + g 2 Since ∆ (G) 1,λ,V,W and ∆ (G)
2,λ,V,W are invertible, we obtain the result.

4.2. Study of the truncated operator. Note that in (4.1), if we forget about the terms in λ, V , and W , we obtain a Laplacian on Z + . Moreover, motivated by the results of Sections 5.1 and 5.2, it is enough to focus the analysis on the study of ∆ (Z+) 2,λ,V,W . Therefore, we stress that we will not study ∆

(Z+)
1,λ,V,W at all. In fact, the latter leads to some technical complications and is less natural, i.e., it is not a direct analogue of the Laplacian on Z + .

As in [FHS], we reduce the problem to Z k for k ∈ Z + . We define the truncated operator d

= d(n) ∈ B ℓ 2 (Z n , C) by d = d + W 1 + W 2 τ. Now we define ∆ (n) λ,V,W ∈ B ℓ 2 (Z n , C) by ∆ (n) λ,V,W := d * 1 λ -m -V 1| Zn d -(λ + m -V 2| Zn )
Again to lighten notation, we drop the dependency in m. We point out that:

∆ (0) λ,V,W = ∆ (Z+) 2,λ,V,W . Proposition 4.3. Let λ ∈ H, V = (V 1 , V 2 ) t ∈ ℓ ∞ (Z + , R 2 ), and W = (W 1 , W 2 ) t ∈ ℓ ∞ (Z + , C 2 ), then ∆ (n) λ,V,W is invertible for all n ∈ Z + .
Proof. This is essentially the same proof as for Proposition 4.1.

We study the related Green function:

α n := δ n , ∆ (n) λ,V,W -1 δ n ,
where δ n (m) := 1 if and only if n = m and 0 otherwise. The objective is to bound α 0 independently of λ. We give the first property of α n .

Proposition 4.4. Take λ ∈ H, V ∈ ℓ ∞ (Z + , R 2 ), and W ∈ ℓ ∞ (Z + , C 2 ) then α n = δ n , ∆ (n) λ,V,W -1 δ n ∈ H, for all n ∈ Z + .
Proof. We have

ℑ(α n ) = ℑ(λ) 1 |λ -m -V 1| Zn | d ∆ (n) λ,V,W -1 δ n 2 + ∆ (n) λ,V,W -1 δ n 2 . So ℑ(α n ) > 0 because λ ∈ H and ∆ (n) λ,V,W -1 δ n = 0.
We follow the strategy of [FHS] and express α n with the help of α n+1 . The aim is to use a fixed point argument in order to recover some bounds on α 0 .

Proposition 4.5. Take λ ∈ H, V ∈ ℓ ∞ (Z + , R 2 ), W ∈ ℓ ∞ (Z + , C 2 ) with W 1 (k) = -1 for all k ∈ Z + , and n ∈ Z + . By setting Φ n (z) := ϕ an,bn,cn (z) = -a n -(b n + c n z) -1 -1 a n := λ + m -V 2 (n) ∈ H b n := (λ -m -V 1 (n))|1 + W 1 (n)| -2 ∈ H (4.3) c n := 1 -W 2 (n) 1 + W 1 (n) 2 ∈ R + ,
we obtain α n = Φ n (α n+1 ).

Proof. We define in ℓ 2 (Z n , C) and in ℓ 2 (Z n+1 , C)

f := ∆ (n) λ,V,W -1 δ n and g := ∆ (n+1) λ,V,W -1 δ n+1 , respectively. Clearly α n = f (n) and α n+1 = g(n + 1). By definition f is the unique solution in ℓ 2 (Z n , C) of ∆ (n) λ,V,W f = δ n , i.e., 1 + W 1 (k) λ -m -V 1 (k) ((1 + W 1 (k))f (k) + (-1 + W 2 (k))f (k + 1)) (4.4) + 1 -W 2 (k -1) λ -m -V 1 (k -1) ((1 -W 2 (k -1))f (k) + (-1 -W 1 (k -1))f (k -1)) -(λ + m -V 2 (k))f (k) = 0,
for all k ≥ n + 1 and

1 + W 1 (n) λ -m -V 1 (n) ((1 + W 1 (n))f (n) + (-1 + W 2 (n))f (n + 1)) -(λ + m -V 2 (n))f (n) = 1. (4.5)
We see that f | Z n+1 is solution of (4.4) for all k ≥ n + 2 and

1 + W 1 (n + 1) λ -m -V 1 (n + 1) ((1 + W 1 (n + 1))f (n + 1) + (-1 + W 2 (n + 1))f (n + 2)) -(λ + m -V 2 (n + 1))f (n + 1) = 1 -W 2 (n) λ -m -V 1 (n) ((1 + W 1 (n))f (n) + (-1 + W 2 (n))f (n + 1)) .
So we obtain that

∆ (n+1) λ,V,W f | Z n+1 = 1 -W 2 (n) λ -m -V 1 (n) ((1 + W 1 (n))f (n) + (-1 + W 2 (n))f (n + 1)) δ n+1 . Because ∆ (n+1) λ,V,W g = δ n+1 we have ∆ (n+1) λ,V,W f | Z n+1 = 1 -W 2 (n) λ -m -V 1 (n) ((1 + W 1 (n))f (n) + (-1 + W 2 (n))f (n + 1)) ∆ (n+1) λ,V,W g But ∆ (n+1) λ,V,W is invertible, so f | Z n+1 = 1 -W 2 (n) λ -m -V 1 (n) ((1 + W 1 (n))f (n) + (-1 + W 2 (n))f (n + 1)) g.
Note that

f (n + 1) = 1 -W 2 (n) λ -m -V 1 (n) ((1 + W 1 (n))f (n) + (-1 + W 2 (n))f (n + 1)) g(n + 1).
Straightforwardly, using (4.5) we conclude that

α n = f (n) = Φ n (g(n + 1)) = Φ n (α n+1 ).
4.3. An iterative process. The key to the process relies on the fact that Φ n is a strict contraction.

Proposition 4.6.

Given λ ∈ H, n ∈ Z + , V ∈ ℓ ∞ (Z + , C 2 ), and W ∈ ℓ ∞ (Z + , C 2 ) with W 1 (n) = -1 and W 2 (n) = 1. Then Φ n is a strict contraction. More precisely, we have d H (Φ n (z 1 ), Φ n (z 2 )) ≤ 1 1 + (ℑ(λ)) 2 (1 + W 1 ∞ ) -2 d H (z 1 , z 2 ) , (4.6)
for all z 1 , z 2 ∈ H and n ∈ Z + . Moreover we obtain

d H (Φ n (z 1 ), Φ n (z 2 )) ≤ (1 + W 1 ∞ ) 2 + ℑ(λ) (|λ| + m + V 2 ∞ ) 2 (ℑ(λ)) 4 , for all z 1 , z 2 ∈ H and n ∈ Z + .
Proof. Using Lemma 2.2, we obtain that Φ n = ϕ an,bn,cn is a strict contraction. More precisely, we get

d H (Φ n (z 1 ), Φ n (z 2 )) ≤ 1 1 + ℑ(a n )ℑ(b n ) d H (z 1 , z 2 ) ≤ 1 1 + (ℑ(λ)) 2 (1 + W 1 ∞ ) -2 d H (z 1 , z 2 ) ,
for all z 1 , z 2 ∈ H, and

d H (Φ n (z 1 ), Φ n (z 2 )) ≤ 1 ℑ(bn) + |a n | ℑ(a n ) 2 ≤ (1 + W 1 ∞ ) 2 + ℑ(λ) (|λ| + m + V 2 ∞ ) 2 (ℑ(λ)) 4 , for all z 1 , z 2 ∈ H.
Now we have an asymptotic property. That is an analogue of [FHS][Theorem 2.3]. It relies strongly on the fact that Φ n is a strict contraction.

Corollary 4.7. Take V ∈ ℓ ∞ (Z + , R 2 ), and W ∈ ℓ ∞ (Z + , C 2 ) with W 1 (n) = -1 and W 2 (n) = 1 for all n ∈ Z + . Then for all λ ∈ H and (ζ n ) n ∈ H Z+ we have d H -lim n→∞ Φ 0 • • • • • Φ n (ζ n ) = α 0 , i.e., lim n→∞ d H (Φ 0 • • • • • Φ n (ζ n ), α 0 ) = 0.
Proof. With Proposition 4.4, for all n ∈ Z + we have α n ∈ H. With Proposition 4.6 there exist δ ∈ (0, 1) and η > 0 such that d H (Φ n (z 1 ), Φ n (z 2 )) ≤ min (δd H (z 1 , z 2 ) , η) , for all n ∈ Z + and z 1 , z 2 ∈ H. So, using that α n = Φ n (α n+1 ) for all n ∈ Z + , we obtain that for n

∈ Z + d H (Φ 0 • • • • • Φ n (ζ n ), α 0 ) = d H (Φ 0 • • • • • Φ n (ζ n ), Φ 0 • • • • • Φ n (α n+1 )) ≤ δ n d H (Φ n (ζ n ), Φ n (α n+1 )) ≤ ηδ n . Therefore, d H -lim n→∞ Φ 0 • • • • • Φ n (ζ n ) = α 0 .
From now on, set ν := ν 1 • ν 2 . Now unlike in [FHS][Lemma 4.5] or in [FHS2][Proposition 3.4] we shall not rely directly on a fixed point of Φ n but on one of Φ n • • • • • Φ n+ν-1 . The proof is unfortunately more complicated but the improvement is real as we can treat potentials satisfying V -τ ν1 V ∈ ℓ 1 . Recall that with the approach of [FHS], one covers only the case ν = 1. We localize in energy and introduce:

K x1,x2,ε := (x 1 , x 2 ) + i(0, ε). (4.7) Proposition 4.8. Take x ∈ - √ m 2 + 4, -m ∪ m, √ m 2 + 4 , ν ∈ Z + \ {0}
, and assume that (3.2) and (3.3) hold true, then there exist

x 1 , x 2 ∈ R such that x ∈ (x 1 , x 2 ) and M 1 , ε > 0 so that d H (α 0 , i) = d H δ 0 , ∆ (0) λ,V,W -1 δ 0 , i ≤ M 1 for all λ ∈ K x1,x2,ε . In particular there exists M 2 > 0 such that δ 0 , ∆ (Z+) 2,λ,V,W -1 δ 0 ≤ M 2 for all λ ∈ K x1,x2,ε .
Proof. Using Lemma 2.4, there exist some polynomials

A, B 1 , B 2 , C ∈ R[X 1 , . . . , X 3ν ] such that Φ n • • • • • Φ n+ν-1 (z) = ϕ an,bn,cn • • • • • ϕ an+ν-1,bn+ν-1,cn+ν-1 (z) = - C(ω n,λ ) + B 2 (ω n,λ )z B 1 (ω n,λ ) + A(ω n,λ )z ,
for all λ ∈ H and n ∈ Z + , where

ω n,λ := (a n , b n , c n , . . . , a n+ν-1 , b n+ν-1 , c n+ν-1 ),
and where B 1 (ω) + A(ω)z = 0 for all ω ∈ (cl(H) 2 × R * + ) ν and z ∈ H. We now work in a neighbourhood of x. First notice that the fixed points of ϕ x+m,x-m,1 are given by:

- x -m 2 ± 1 2 i x -m x + m (4 + m 2 -x 2 ). (4.8) Then ϕ x+m,x-m,1 • • • • • ϕ x+m,x-m,1 ν times = - C(ω ∞,x ) + B 2 (ω ∞,x )• B 1 (ω ∞,x ) + A(ω ∞,x )• ,
where ω ∞,x := (x + m, x -m, 1, . . . , x + m, x -m, 1), has at least (4.8) as fixed points. As it is a homography it has exactly at most two fixed points. Note also that A(ω ∞,x ) = 0, because there are two different fixed points. Now we would like to study the fixed points of

R(ω, z) := - C(ω) + B 2 (ω)z B 1 (ω) + A(ω)z
with respect to z, for ω being in a neighbourhood of ω ∞,x . As the Inverse Function Theorem does not seem to apply we rely on a direct approach. Since A is continuous, there exists a neighbourhood Ω 1 of ω ∞,x such that A(ω) = 0 for all ω ∈ Ω 1 . We define on Ω 1

Z(ω) := - B 1 (ω) + B 2 (ω) 2A(ω) + 1 2 i 4 C(ω) A(ω) - B 1 (ω) + B 2 (ω) A(ω) 2 , (4.9)
where we have chosen the square root in order to guarantee that:

ℜ   4 C(ω) A(ω) - B 1 (ω) + B 2 (ω) A(ω) 2   ≥ 0, for all ω ∈ Ω 1 . A direct computation gives that Z(ω) is a fixed point of R(ω, •) on Ω 1 . Since A, B 1
, and B 2 are polynomials with real coefficients, we infer that ℑ(Z(ω ∞,x )) ≥ 0, by the choice of the square root.

On the other hand Z(ω ∞,x ) belongs to (4.8). Therefore we infer that

Z(ω ∞,x ) = - x -m 2 + 1 2 i x -m x + m (4 + m 2 -x 2 ) ∈ H. (4.10)
In particular, since A, B 1 , B 2 , and C are polynomials with real coefficients,

4 C(ω ∞,x ) A(ω ∞,x ) - B 1 (ω ∞,x ) + B 2 (ω ∞,x ) A(ω ∞,x ) 2 = x -m x + m (4 + m 2 -x 2 ) > 0.
Therefore there exists a neighbourhood

Ω 2 ⊂ Ω 1 of ω ∞,x such that 4 C(ω) A(ω) - B 1 (ω) + B 2 (ω) A(ω) 2 ∈ R -, for all ω ∈ Ω 2 .
We infer that we can take the principal value of the square root in the definition of (4.9) when ω ∈ Ω 2 .

In particular, Z ∈ C ∞ (Ω 2 , C). Hence, recalling (4.10), there exists a compact neighbourhood Ω 3 ⊂ Ω 2 of ω ∞,x and η 1 , M 1 > 0 such that ℑ(Z(ω)) > η 1 and |Z(ω)| ≤ M 1 , for all ω ∈ Ω 3 . Now there exist x 1 , x 2 ∈ R, ε > 0, and n 0 ∈ Z + such that x ∈ (x 1 , x 2 ) and ω n,λ ∈ Ω 3 , for all λ ∈ K x1,x2,ε and n ≥ n 0 . We define now (4.11) for all λ ∈ K x1,x2,ε and n ≥ n 0 . Moreover, by definition of Z we have

z n (λ) := Z(ω n,λ ) for all λ ∈ K x1,x2,ε and n ≥ n 0 . Notice that ℑ(z n (λ)) > η 1 and |z n (λ)| ≤ M 1 ,
Φ n • • • • • Φ n+ν-1 (z n (λ)) = z n (λ) (4.12) for all λ ∈ K x1,x2,ε and n ≥ n 0 . Next, there is M 2 > 0 such that (|a k -a k+ν | + |b k -b k+ν | + |c k -c k+ν |) ≤ M 2 ( V (k) -V (k + ν) + W (k) -W (k + ν) ) for all λ ∈ K x1,x2,ε and k ∈ Z + . Now since Z is C ∞ (Ω 3 , C) and Ω 3 is compact, there exists a Lipschitz constant M 3 > 0 such that |z n+ν (λ) -z n (λ)| ≤ M 3 ω n+ν,λ -ω n,λ ≤ M 3 (|a n+ν (λ) -a n (λ)| + |b n+ν (λ) -b n (λ)| + |c n+ν (λ) -c n (λ)| + • • • +|a n+2ν-1 (λ) -a n+ν-1 (λ)| + |b n+2ν-1 (λ) -b n+ν-1 (λ)| + |c n+2ν-1 (λ) -c n+ν-1 (λ)|) ≤ M 2 M 3 ν-1 k=0 ( V (n + k + ν) -V (n + k) + W (n + k + ν) -W (n + k) ) (4.13)
for all λ ∈ K x1,x2,ε and n ≥ n 0 . By Corollary 4.7, for all λ ∈ K x1,x2,ε , we have:

d H (i, α 0 ) = lim n→∞ d H (i, Φ 0 • • • • • Φ n0+ν(n+1)-1 (z n0+νn (λ))) ≤ lim n→∞ d H (i, Φ 0 (i)) + n0+ν-3 k=0 d H (Φ 0 • • • • • Φ k (i), Φ 0 • • • • • Φ k+1 (i)) + d H (Φ 0 • • • • • Φ n0+ν-2 (i), Φ 0 • • • • • Φ n0+ν-1 (z n0 )) + n-1 k=0 d H (Φ 0 • • • • • Φ n0+ν(k+1)-1 (z n0+νk ), Φ 0 • • • • • Φ n0+ν(k+2)-1 (z n0+ν(k+1) )) ≤ n0+ν-2 k=0 d H (i, Φ k (i)) + d H (i, Φ n0+ν-1 (z n0 )) (4.14) + k≥0 d H (z n0+νk , Φ n0+ν(k+1) • • • • • Φ n0+ν(k+2)-1 (z n0+ν(k+1) )) = n0+ν-2 k=0 d H (i, Φ k (i)) + d H (i, Φ n0+ν-1 (z n0 )) + k≥0 d H (z n0+νk , z n0+ν(k+1) ) (4.15) ≤ n0+ν-2 k=0 d H (i, Φ k (i)) + d H (i, Φ n0+ν-1 (z n0 )) + k≥0 z n0+νk -z n0+ν(k+1) (ℑ(z n0+νk )) 1/2 ℑ(z n0+ν(k+1) ) 1/2 . (4.16)
Here in (4.14) we have used the fact that Φ n is a contraction, in (4.15) we exploited (4.12), and in (4.16) we relied on (2.2).

Coming back to (4.3), one finds easily M 4 , η 4 > 0 such that max (|a n |, |b n |) ≤ M 4 and η 4 < c n ≤ M 4 , for all λ ∈ K x1,x2,ε and for all n ∈ Z + . Then, Lemma 2.3 ensures that

d H (Φ n (z), i) ≤ (M 4 + M 4 |z|) 2 η 4 ℑ(z) + 1 (M 4 |z| + M 4 ) M 4 + (η 4 ℑ(z)) -1 η 4 ℑ(z) . (4.17)
for all z ∈ H, λ ∈ K x1,x2,ε , and n ∈ Z + . Finally combining (4.16) and estimates (4.11), (4.13), and (4.17), we infer:

d H (α 0 , i) ≤(n 0 + ν -1) (2M 4 ) 2 η 4 + 1 2M 4 M 4 + η -1 4 √ η 4 + (M 4 + M 4 M 1 ) 2 η 4 η 1 + 1 (M 4 M 1 + M 4 ) M 4 + (η 4 η 1 ) -1 √ η 4 η 1 + M 2 M 3 η 1 ( V -τ ν V 1 + W -τ ν W 1 ) , for all λ ∈ K x1,x2,ε . The second point comes by recalling that ∆ (0) λ,V,W = ∆ (Z+)
2,λ,V,W .

The absolutely continuous spectrum

We recall the following standard result, e.g., [ReSi][Theorem XIII.19].

Theorem 5.1. Let H be a self-adjoint operator of H, let (x 1 , x 2 ) be an interval and f ∈ H. Suppose

lim sup ε↓0 + sup x∈(x1,x2) f, (H -(x + iε)) -1 f < +∞, (5.1)
then the measure f, 1 (•) (H)f is purely absolutely continuous w.r.t. the Lebesgue measure on (x 1 , x 2 ). 5.1. The case of Z + . In the previous section we have estimated the resolvent of ∆ (Z+) 2,λ,V,W . Keeping in mind Proposition 4.2, we explain how to transfer the result to H (Z+) m,V,W . We start by reducing the study to a unique vector.

Lemma 5.2. Given A ⊂ C\R bounded, V ∈ ℓ ∞ (Z + , R 2 ), and W ∈ ℓ ∞ (Z + , C 2 ) with W 1 (n) = -1 and W 2 (n) = 1 for all n ∈ Z + . Suppose that there exists C 1 > 0 such that 0 1 δ 0 , H (Z+) m,V,W -λ -1 0 1 δ 0 ≤ C 1 ,
for all λ ∈ A. Then for all x 1 , y 1 , x 2 , y 2 ∈ C and n 1 , n 2 ∈ Z + there exists C 2 > 0 such that

x 1 y 1 δ n1 , H (Z+) m,V,W -λ -1 x 2 y 2 δ n2 ≤ C 2 ,
for all λ ∈ A.

Proof. Let

f := H (Z+) m,V,W -λ -1 0 1 δ 0 .
We have clearly

|f 2 (0)| = 0 1 δ 0 , H (Z+) m,V,W -λ -1 0 1 δ 0 ≤ C 1 , for all λ ∈ A. By definition, f is the unique solution in ℓ 2 (Z + , C 2 ) of (m + V 1 (n) -λ)f 1 (n) + (1 + W 1 (n))f 2 (n) + (-1 + W 2 (n))f 2 (n + 1) = 0 (1 + W 1 (n))f 1 (n) + (-1 + W 2 (n -1))f 1 (n -1) + (-m + V 2 (n) -λ)f 2 (n) = 0
for all n ≥ 1 and of

(m + V 1 (0) -λ)f 1 (0) + (1 + W 1 (0))f 2 (0) + (-1 + W 2 (0))f 2 (1) = 0 (1 + W 1 (0))f 1 (0) + (-m + V 2 (0) -λ)f 2 (0) = 1.
So by induction on n ∈ Z + there exists D n > 0 such that

|f i (n)| ≤ D n ,
for all λ ∈ A and i ∈ {1, 2}. Therefore we obtain that

0 1 δ 0 , H (Z+) m,V,W -λ -1 x 2 y 2 δ n2 = x 2 y 2 δ n2 , H (Z+) m,V,W -λ -1 0 1 δ 0 = |x 2 1 (n 2 ) + y 2 f 2 (n 2 )| ≤ (|x 2 | + |y 2 |)D n2 =: C 2 .
for all λ ∈ A. Now let

g := H (Z+) m,V,W -λ -1 x 2 y 2 δ n2 ,
we have

|g 2 (0)| = 0 1 δ 0 , H (Z+) m,V,W -λ -1 x 2 y 2 δ n2 ≤ C 2 .
for all λ ∈ A. By definition, g is the unique solution in ℓ 2 (Z + , C 2 ) of

(m + V 1 (n) -λ)g 1 (n) + (1 + W 1 (n))g 2 (n) + (-1 + W 2 (n))g 2 (n + 1) = x 2 δ n2 (n) (1 + W 1 (n))g 1 (n) + (-1 + W 2 (n -1))g 1 (n -1) + (-m + V 2 (n) -λ)g 2 (n) = y 2 δ n2 (n)
for all n ≥ 1 and of

(m + V 1 (0) -λ)g 1 (0) + (1 + W 1 (0))g 2 (0) + (-1 + W 2 (0))g 2 (1) = x 2 δ n2 (0) (1 + W 1 (0))g 1 (0) + (-m + V 2 (0) -λ)g 2 (0) = y 2 δ n2 (0).
So by induction for all n ∈ Z + there exists D ′ n > 0 such that |g i (n)| ≤ D ′ n , for all λ ∈ A and i ∈ {1, 2}. Therefore we obtain that

x 1 y 1 δ n1 , H (Z+) m,V,W -λ -1 x 2 y 2 δ n2 = |x 1 g 1 (n 1 ) + y 1 g 2 (n 1 )| ≤ (|x 1 | + |y 1 |)D ′ n1 =: C 3 ,
for all λ ∈ A. This concludes the proof.

We are now in position to conclude with our main result.

Proof of Theorem 3.1. Since D

(Z+) m -H (Z+)
m,V,W is compact, then Weyl Theorem gives the first point. We prove the second one. Take x ∈ -√ m 2 + 4, -m ∪ m, √ m 2 + 4 . By Propositions 4.2 and 4.8 we have that there exists ε, C > 0 and x 1 , x 2 ∈ R such that x ∈ (x 1 , x 2 ) and Lemma 5.3.

0 1 δ 0 , H (Z+) m,V,W -λ -1 0 1 δ 0 = δ 0 , (∆ (Z+) 2,λ,V,W ) -1 δ 0 ≤ C for all λ ∈ K x1,
Take λ ∈ H, V ∈ ℓ ∞ (Z, R 2 ), W ∈ ℓ ∞ (Z, C 2 ) with W 1 (0) = -1 and W 2 (-1) = 1. Set: Φ(z 1 , z 2 ) := -a -(b + cz 1 ) -1 -(b ′ + c ′ z 2 ) -1 -1 a := λ + m -V 2 (0) ∈ H, b := (λ -m -V 1 (0))|1 + W 1 (0)| -2 ∈ H, b ′ := (λ -m -V 1 (-1))|1 -W 2 (-1)| -2 ∈ H, c := 1 -W 2 (0) 1 + W 1 (0) 2 ∈ R * + , c ′ := 1 + W 1 (-1) 1 -W 2 (-1) 2 ∈ R * + ,
we have

δ 0 , ∆ (Z) 2,λ,V,W -1 δ 0 = Φ δ 0 , ∆ (0) λ,(τ V ) | Z + ,(τ W ) | Z + δ 0 , δ 0 , ∆ (0) λ,(τ 2 SV1,τ SV2) t | Z + ,(τ -τ 2 SW2,-τ 2 SW1) t | Z + δ 0 .
Proof. We define

f := ∆ (Z) 2,λ,V,W -1 δ 0 , g := ∆ (0) λ,(τ V ) | Z + ,(τ W ) | Z + -1 δ 0 and h := ∆ (0) λ,(τ 2 SV1,τ SV2) t | Z + ,(τ -τ 2 SW2,-τ 2 SW1) t | Z + -1 δ 0 . Clearly δ 0 , ∆ (Z) 2,λ,V,W -1 δ 0 = f (0), δ 0 , ∆ (0) λ,(τ V ) | Z + ,(τ W ) | Z + δ 0 = g(0) δ 0 , ∆ (0) λ,(τ 2 SV1,τ SV2) t | Z + ,(τ -τ 2 SW2,-τ 2 SW1) t | Z + δ 0 = h(0)
.

By definition f is the unique solution in ℓ 2 (Z, C) of ∆ (Z) 2,λ,V,W f = δ 0 , i.e., 1 + W 1 (n) λ -m -V 1 (n) ((1 + W 1 (n))f (n) + (-1 + W 2 (n))f (n + 1)) + 1 -W 2 (n -1) λ -m -V 1 (n -1) ((1 -W 2 (n -1))f (n) + (-1 -W 1 (n -1))f (n -1)) (5.2) -(λ + m -V 2 (n))f (n) = δ 0 (n), for all n ∈ Z. Let f ′ := (τ f ) | Z + , we see that f ′ is solution of 1 + W 1 (n + 1) λ -m -V 1 (n + 1) ((1 + W 1 (n + 1))f ′ (n) + (-1 + W 2 (n + 1))f ′ (n + 1)) + 1 -W 2 (n) λ -m -V 1 (n) ((1 -W 2 (n))f ′ (n) + (-1 -W 1 (n))f ′ (n -1)) -(λ + m -V 2 (n + 1))f ′ (n) = 0
for all n ≥ 1 and

1 + W 1 (1) λ -m -V 1 (1) ((1 + W 1 (1))f ′ (0) + (-1 + W 2 (1))f ′ (1)) -(λ + m -V 2 (1))f ′ (0) = 1 -W 2 (0) λ -m -V 1 (0) ((1 + W 1 (0))f (0) + (-1 + W 2 (0))f (1)) .
So we obtain that

∆ (0) λ,(τ V ) | Z + ,(τ W ) | Z + f ′ = 1 -W 2 (0) λ -m -V 1 (0) ((1 + W 1 (0))f (0) + (-1 + W 2 (0))f (1)) δ 0 = 1 -W 2 (0) λ -m -V 1 (0) ((1 + W 1 (0))f (0) + (-1 + W 2 (0))f (1)) ∆ (0) λ,(τ V ) | Z + ,(τ W ) | Z + g. Since ∆ (0) λ,(τ V ) | Z + ,(τ W ) | Z + is invertible, we get 1 -W 2 (0) λ -m -V 1 (0) ((1 + W 1 (0))f (0) + (-1 + W 2 (0))f (1)) g(0) = f ′ (0) = f (1). (5.3) Now let f ′′ := (τ Sf ) | Z + We see that f ′′ is solution of 1 + W 1 (-n -1) λ -m -V 1 (-n -1) ((1 + W 1 (-n -1))f ′′ (n) + (-1 + W 2 (-n -1))f ′′ (n -1)) + 1 -W 2 (-n -2) λ -m -V 1 (-n -2) ((1 -W 2 (-n -2))f ′′ (n) + (-1 -W 1 (-n -2))f ′′ (n + 1)) -(λ + m -V 2 (-n -1))f ′′ (n) = 0 for all n ≥ 1 and 1 -W 2 (-2) λ -m -V 1 (-2) ((1 -W 2 (-2))f ′′ (0) + (-1 -W 1 (-2))f ′′ (1)) -(λ + m -V 2 (-1))f ′′ (0) = 1 + W 1 (-1) λ -m -V 1 (-1) ((1 -W 2 (-1))f (0) + (-1 -W 1 (-1))f (-1)) . By setting ∆ := ∆ (0) λ,(τ 2 SV1,τ SV2) t | Z + ,(τ -τ 2 SW2,-τ 2 SW1) t | Z + , we obtain ∆f ′′ = 1 + W 1 (-1) λ -m -V 1 (-1) ((1 -W 2 (-1))f (0) + (-1 -W 1 (-1))f (-1)) δ 0 = 1 + W 1 (-1) λ -m -V 1 (-1) ((1 -W 2 (-1))f (0) + (-1 -W 1 (-1))f (-1)) ∆h.
Since ∆ is invertible, we infer

1 + W 1 (-1) λ -m -V 1 (-1) ((1 -W 2 (-1))f (0) + (-1 -W 1 (-1))f (-1)) h(0) = f ′′ (0) = f (-1). (5.4)
Straightforwardly, using (5.2) for n = 0, (5.3) and (5.4) we have that f (0) = Φ(g(0), h(0)).

Now with this Lemma we can obtain that

δ 0 , ∆ (Z) 2,λ,V,W -1 δ 0 is bounded independently of λ with the Proposition 4.8. Corollary 5.4. Take x ∈ - √ m 2 + 4, -m ∪ m, √ m 2 + 4 , V ∈ ℓ ∞ (Z, R 2 ), and W ∈ ℓ ∞ (Z, C 2 ) with W 1 (n) = -1 and W 2 (n) = 1,
for all n ∈ Z. Suppose that there exists ν ∈ Z + \ {0} such that (3.5) or (3.6) holds true. Then there exist C, ε > 0 and x 1 , x 2 ∈ R such that x ∈ (x 1 , x 2 ) and

δ 0 , ∆ (Z) 2,λ,V,W -1 δ 0 ≤ C for all λ ∈ K x1,x2,ε . Proof. Let α λ := δ 0 , ∆ (0) λ,(τ V ) | Z + ,(τ W ) | Z + δ 0 ∈ H, α ′ λ := δ 0 , ∆ (0) λ,(τ 2 SV1,τ SV2) t | Z + ,(τ -τ 2 SW2,-τ 2 SW1) t | Z + δ 0 ∈ H.
With Lemma 5.3 we have

δ 0 , ∆ (Z) 2,λ,V,W -1 δ 0 = a -(b + cα λ ) -1 -(b ′ + c ′ α ′ λ ) -1 -1 ≤ min ℑ -(b + cα λ ) -1 -1 , ℑ -(b ′ + c ′ α ′ λ ) -1 -1 . Now if V | Z + -τ ν1 V | Z + , W | Z + -τ ν2 W | Z + ∈ ℓ 1 (Z + , C 2 ) with Proposition 4.8 there exist C 1 , ε > 0 and x 1 , x 2 ∈ R such that x ∈ (x 1 , x 2 ) and d H (α λ , i) ≤ C 1 for all λ ∈ K x1,x2,ε , so there exists C 2 > 0 such that δ 0 , ∆ (Z) 2,λ,V,W -1 δ 0 ≤ ℑ -(b + cα λ ) -1 -1 ≤ C 2 for all λ ∈ K x1,x2,ε . Now if V | Z --τ ν1 V | Z -, W | Z --τ ν2 W | Z -∈ ℓ 1 (Z -, C 2 ) with Proposition 4.8 there exist C 1 , ε > 0 and x 1 , x 2 ∈ R such that x ∈ (x 1 , x 2 ) and d H (α ′ λ , i) ≤ C 1 for all λ ∈ K x1,x2,ε , so there exists C 2 > 0 such that δ 0 , ∆ (Z) 2,λ,V,W -1 δ 0 ≤ ℑ -(b ′ + c ′ α ′ λ ) -1 -1 ≤ C 2 for all λ ∈ K x1,x2,ε .
Finally we conclude with the help of the symmetry of charge.

Proof of Theorem 3.3. Since D

(Z) m -H (Z)
m,V,W is compact, then Weyl Theorem gives the first point. We turn to the second one. Let n ∈ Z, let x ∈ -√ m 2 + 4, -m ∪ m, √ m 2 + 4 . Proposition 4.2 and Corollary 5.4 ensure that there exist ε 1 , C 1 > 0 and x 1 , x 2 ∈ R such that x ∈ (x 1 , x 2 ) and

0 1 δ n , H (Z) m,V,W -λ -1 0 1 δ n = 0 1 δ 0 , H (Z) m,τ n V,τ n W -λ -1 0 1 δ 0 = δ 0 , (∆ (Z)
2,λ,τ n V,τ n W ) -1 δ 0 ≤ C 1 , for all λ ∈ K x1,x2,ε1 . Then, Theorem 5.1 yields that the measure

0 1 δ n , 1 (•) (H (Z+) m,V,W ) 0 1 δ n
is purely absolutely continuous on (x 1 , x 2 ). Now we use U , see (3.4). Let

V ′ := (-Sτ n V 2 , -Sτ n V 1 ) t and W ′ := Sτ n W 1 , Sτ n-1 W 2 t ,
there exist ε 2 , C 2 > 0 and x 3 , x 4 ∈ R such that x ∈ (x 3 , x 4 ) and

1 0 δ n , H (Z) m,V,W -λ -1 1 0 δ n = 1 0 δ 0 , H (Z) m,τ n V,τ n W -λ -1 1 0 δ 0 = 0 δ 0 , H (

The case of the Laplacian

In this section we explain briefly how to adapt our proofs in order to prove Theorem 1.1. For the sake of clarity, we stick to the case ∆ + V and compare our proof directly to [FHS].

We start by the case of Z + .

Theorem 6.1. Take V ∈ ℓ ∞ (Z + , R) and τ ∈ Z + such that:

lim n→+∞ V (n) = 0, V -τ ν V ∈ ℓ 1 (Z + , R), (6.1)
then the spectrum of ∆ + V is purely absolutely continuous on (0, 4).

Apart from Proposition 6.3, our presentation is very close to the one of [FHS]. We start with the truncated case. Set:

α n := δ n , ∆ (n) + V | Zn -λ -1 δ n ∈ H,
where ∆ (n) is the Laplacian on Z n , see (2.1). As in Proposition 4.5 we have α n = Φ n (α n+1 ) with

Φ n (z) := ϕ λ-V (n),1,1 (z) = -λ -V (n) -(1 + z) -1 -1
.

Note that Φ n is a contraction of H. However, unlike in Proposition 4.6, this is not a strict contraction. However, Φ n • Φ n+1 is a strict contraction, see also [FHS][Proposition 2.1]. This infers: Proposition 6.2. Take V ∈ ℓ ∞ (Z + , R), then for all λ ∈ H and (ζ n ) n ∈ H Z+ we have

d H -lim n→∞ Φ 0 • • • • • Φ n (ζ n ) = α 0 .
Proof. See the proof of Corollary 4.7 and [FHS][Theorem 2.3]. Now unlike in [FHS][Lemma 4.5] or in [FHS2][Proposition 3.4] but as in Proposition 4.8 we use the fixed point of Φ n • • • • • Φ n+ν-1 . We obtain: Proposition 6.3. Take x ∈ (0, 4), V ∈ ℓ ∞ (Z + , R), and ν ∈ Z + \ {0} with lim n→+∞ V (n) = 0 and V -τ ν V ∈ ℓ 1 (Z + , R). Then there exist x 1 , x 2 ∈ R such that x ∈ (x 1 , x 2 ) and M 1 , ε > 0 so that d H (α 0 , i) ≤ M 1 for all λ ∈ K x1,x2,ε .

Recall that K x1,x2,ε is defined in (4.7).

Proof. This is the same proof as in Proposition 4.8. We study the fixed points of Φ n • • • • • Φ n+ν in a neighbourhood of ω ∞,x := (x, 1, 1, . . . , x, 1, 1). The fixed points of ϕ x,1,1 are

- 1 2 ± 1 2 i 4 x -1,
The rest remains the same.

Finally Theorem 5.1 concludes the proof of Theorem 6.1. We turn to the case of the line. As in Lemma 5.3, we reduce the problem to the case of Z + because

δ 0 , ∆ (Z) + V -λ -1 δ 0 = λ -V (0) -(1 + α λ ) -1 -(1 + α ′ λ ) -1 -1 ≤ 1 ℑ(-(1 + α λ ) -1 )
,

where

α λ := δ 1 , ∆ (Z1) + V | Z 1 -λ -1 δ 1 ∈ H α ′ λ := δ -1 , ∆ (-Z1) + V | -Z -1 -λ -1 δ -1 ∈ H,
with ∆ (Z1) and ∆ (-Z1) the Laplacian on Z 1 and -Z 1 respectively. This gives Theorem 1.1.

  x2,ε . Then Theorem 5.1 and Lemma 5.2 conclude by density.5.2. The case of Z. Now we express ∆

  λ,V ′ ,W ′ ) -1 δ 0 ≤ C 2 for all λ ∈ K x3,x4,ε2. Again Theorem 5.1 gives that the continuous (x 3 , x 4 ). Finally, remembering that x is arbitrary and by an argument of density, we infer that H(Z+) m,V,W has pure ac spectrum -√ m 2 + 4, -m ∪ m, √ m 2 + 4 .


Michel Bonnefont
email: michel.bonnefont@math.u-bordeaux.fr

AND Sylvain Golénia
email: sylvain.golenia@math.u-bordeaux.fr

Matthias Keller
email: m.keller@uni-jena.de

EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS

Keywords: Mathematics Subject Classification. 47A10, 34L20, 05C63, 47B25, 47A63 discrete Laplacian, locally finite graphs, eigenvalues, asymptotic, planarity, sparse, functional inequality

We consider Schrödinger operators on sparse graphs. The geometric definition of sparseness turn out to be equivalent to a functional inequality for the Laplacian. In consequence, sparseness has in turn strong spectral and functional analytic consequences. Specifically, one consequence is that it allows to completely describe the form domain. Moreover, as another consequence it leads to a characterization for discreteness of the spectrum. In this case we determine the first order of the corresponding eigenvalue asymptotics.

Introduction

The spectral theory of discrete Laplacians on finite or infinite graphs has drawn a lot of attention for decades. One important aspect is to understand the relations between the geometry of the graph and the spectrum of the Laplacian. Often a particular focus lies on the study of the bottom of the spectrum and the eigenvalues below the essential spectrum.

Certainly the most well-known estimates for the bottom of the spectrum of Laplacians on infinite graphs are so called isoperimetric estimates or Cheeger inequalities. Starting with [D1] in the case of infinite graphs, these inequalities were intensively studied and resulted in huge body of literature, where we here mention only [START_REF] Bauer | The dual Cheeger constant and spectra of infinite graphs[END_REF][START_REF] Bauer | Cheeger inequalities for unbounded graph Laplacians[END_REF][START_REF] Dodziuk | Elliptic operators on infinite graphs, Analysis, geometry and topology of elliptic operators[END_REF][START_REF] Dodziuk | Combinatorial Laplacians and isoperimetric inequality[END_REF][START_REF] Fujiwara | Laplacians on rapidly branching trees[END_REF][START_REF] Mohar | Isoperimetics inequalities, growth and the spectrum of graphs[END_REF][START_REF] Mohar | Some relations between analytic and geometric properties of infinite graphs[END_REF][START_REF] Keller | Essential spectrum of the Laplacian on rapidly branching tessellations[END_REF][START_REF] Keller | Unbounded Laplacians on graphs: basic spectral properties and the heat equation[END_REF][START_REF] Wojciechowski | Stochastic completeness of graphs[END_REF]. In certain more specific geometric situations the bottom of the spectrum might be estimated in terms of curvature, see [START_REF] Bauer | Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator[END_REF][START_REF] Higuchi | Combinatorial Curvature for Planar Graphs[END_REF][START_REF] Jost | Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs[END_REF][START_REF] Keller | Essential spectrum of the Laplacian on rapidly branching tessellations[END_REF][START_REF] Keller | Curvature, geometry and spectral properties of planar graphs[END_REF][START_REF] Keller | Cheeger constants, growth and spectrum of locally tessellating planar graphs[END_REF][START_REF] Lin | Ricci curvature and eigenvalue estimate on locally finite graphs[END_REF][START_REF] Woess | A note on tilings and strong isoperimetric inequality[END_REF]. There are various other more recent approaches such as Hardy inequalities in [G] and summability criteria involving the boundary and volume of balls in [KLW].

In this work we focus on sparse graphs to study discreteness of spectrum and eigenvalue asymptotics. In a moral sense, the term sparse means that there are not 'too many' edges, however, throughout the years various different definitions were investigated. We mention here [EGS, L] as seminal works which are closely related to our definitions. As it is impossible to give a complete discussion of the development, we refer to some selected more recent works such as [START_REF] Alon | Sums and products along sparse graphs[END_REF][START_REF] Breuer | Singular continuous spectrum for the Laplacian on certain sparse trees[END_REF][START_REF] Lee | Pebble game algorithms and sparse graphs[END_REF][START_REF] Mohar | Some relations between analytic and geometric properties of infinite graphs[END_REF] and references therein which also illustrates the great variety of possible definitions. Here, we discuss three notions of sparseness that result in a hierarchy of very general classes of graphs.

Let us highlight the work of Mohar [M3], where large eigenvalues of the adjacency matrix on finite graphs are studied. Although our situation of infinite graphs with unbounded geometry requires fundamentally different techniques -functional analytic rather than combinatorial -in spirit our work is certainly closely related.

The techniques used in this paper owe on the one hand to considerations of isoperimetric estimates as well as a scheme developed in [G] for the special case of trees. In particular, we show that a notion of sparseness is a geometric characterization for an inequality of the type

(1 -a) deg -k ≤ ∆ ≤ (1 + a) deg +k
for some a ∈ (0, 1), k ≥ 0 which holds in the form sense (precise definitions and details will be given below). The moral of this inequality is that the asymptotic behavior of the Laplacian ∆ is controlled by the vertex degree function deg (the smaller a the better the control).

Furthermore, such an inequality has very strong consequences which follow from well-known functional analytic principles. These consequences include an explicit description of the form domain, characterization for discreteness of spectrum and eigenvalue asymptotics.

Let us set up the framework. Here, a graph G is a pair (V , E ), where V denotes a countable set of vertices and E : V × V → {0, 1} is a symmetric function with zero diagonal determining the edges. We say two vertices x and y are adjacent or neighbors whenever E (x, y) = E (y, x) = 1. In this case, we write x ∼ y and we call (x, y) and (x, y) the (directed) edges connecting x and y. We assume that G is locally finite that is each vertex has only finitely many neighbors. For any finite set W ⊆ V , the induced subgraph G W := (W , E W ) is defined by setting E W := E | W ×W , i.e., an edge is contained in G W if and only if both of its vertices are in W .

We consider the complex Hilbert space ℓ 2 (V ) := {ϕ : V → C such that x∈V |ϕ(x)| 2 < ∞} endowed with the scalar product ϕ, ψ := x∈V ϕ(x)ψ(x), ϕ, ψ ∈ ℓ 2 (V ).

For a function g : V → C, we denote the operator of multiplication by g on ℓ 2 (V ) given by ϕ → gϕ and domain D(g) := {ϕ ∈ ℓ 2 (V ) | gϕ ∈ ℓ 2 (V )} with slight abuse of notation also by g.

Let q : V → [0, ∞). We consider the Schrödinger operator ∆ + q defined as

D(∆ + q) := ϕ ∈ ℓ 2 (V ) | v → w∼v (ϕ(v) -ϕ(w)) + q(v)ϕ(v) ∈ ℓ 2 (V ) (∆ + q)ϕ(v) := w∼v (ϕ(v) -ϕ(w)) + q(v)ϕ(v).
The operator is non-negative and selfadjoint as it is essentially selfadjoint on C c (V ), the set of finitely supported functions V → R, (confer [Woj1, Theorem 1.3.1], [START_REF] Keller | Dirichlet forms and stochastic completeness of graphs and subgraphs[END_REF]Theorem 6]). In Section 2 we will allow for potentials whose negative part is form bounded with bound strictly less than one. Moreover, in Section 4 we consider also magnetic Schrödinger operators. As mentioned above sparse graphs have already been introduced in various contexts with varying definitions. In this article we also treat various natural generalizations of the concept. In this introduction we stick to an intermediate situation.

Definition. A graph G := (V , E ) is called k-sparse if for any finite set W ⊆ V the induced subgraph G W := (W , E W ) satisfies 2|E W | ≤ k|W |,
where |A| denotes the cardinality of a finite set A and we set

|E W | := 1 2 |{(x, y) ∈ W × W | E W (x, y) = 1}|,
that is we count the non-oriented edges in G W .

Examples of sparse graphs are planar graphs and, in particular, trees. We refer to Section 6 for more examples.

For a function g : V → R and a finite set W ⊆ V , we denote

g(W ) := x∈W g(x)
.

Moreover, we define

lim inf |x|→∞ g(x) := sup W ⊂V finite inf x∈V \W g(x), lim sup |x|→∞ g(x) := inf W ⊂V finite sup x∈V \W g(x).
For two selfadjoint operators T 1 , T 2 on a Hilbert space and a subspace

D 0 ⊆ D(T 1 )∩ D(T 2 ) we write T 1 ≤ T 2 on D 0 if T 1 ϕ, ϕ ≤ T 2 ϕ, ϕ for all ϕ ∈ D 0 .
Moreover, for a selfadjoint semi-bounded operator T on a Hilbert space, we denote the eigenvalues below the essential spectrum by λ n (T ), n ≥ 0, with increasing order counted with multiplicity.

The next theorem is a special case of the more general Theorem 2.2 in Section 2. It illustrates our results in the case of sparse graphs introduced above and includes the case of trees, [START_REF] Golénia | Hardy inequality and eigenvalue asymptotic for discrete Laplacians[END_REF]Theorem 1.1], as a special case. While the proof in [G] uses a Hardy inequality, we rely on some new ideas which have their roots in isoperimetric techniques. The proof is given in Section 2.2.

Theorem 1.1. Let G := (V , E ) be a k-sparse graph and q : V → [0, ∞). Then, we have the following:

(a) For all 0 < ε ≤ 1,

(1 -ε)(deg +q) - k 2 1 ε -ε ≤ ∆ + q ≤ (1 + ε)(deg +q) + k 2 1 ε -ε , on C c (V ). (b) D (∆ + q) 1/2 = D (deg + q) 1/2 . (c)
The operator ∆ + q has purely discrete spectrum if and only if

lim inf |x|→∞ (deg + q)(x) = ∞.
In this case, we obtain

lim inf λ→∞ λ n (∆ + q) λ n (deg + q) = 1.
As a corollary, we obtain following estimate for the bottom and the top of the (essential) spectrum.

Corollary 1.2. Let G := (V , E ) be a k-sparse graph and q : V → [0, ∞). Define d := inf x∈V (deg +q)(x) and D := sup x∈V (deg +q)(x). Assume d < k ≤ D < +∞, then d -2 k 2 d - k 2 ≤ inf σ(∆ + q) ≤ sup σ(∆ + q) ≤ D -2 k 2 D - k 2 . Define d ess := lim inf |x|→∞ (deg +q)(x) and D ess := lim sup |x|→∞ (deg +q)(x). As- sume d ess < k ≤ D ess < +∞, then d ess -2 k 2 d ess - k 2 ≤ inf σ ess (∆+q) ≤ sup σ ess (∆+q) ≤ D ess -2 k 2 D ess - k 2 .
Proof of Corollary 1.2. The conclusion follows by taking ε = min

k 2d-k , 1 in (a) in of Theorem 1.1.
Remark 1.3. The bounds in Corollary 1.2 are optimal for the bottom and the top of the (essential) spectrum in the case of regular trees.

The paper is structured as follows. In the next section an extension of the notion of sparseness is introduced which is shown to be equivalent to a functional inequality and equality of the form domains of ∆ and deg. In Section 3 we consider almost sparse graphs for which we obtain precise eigenvalue asymptotics. Furthermore, in Section 4 we shortly discuss magnetic Schrödinger operators. Our notion of sparseness has very explicit but non-trivial connections to isoperimetric inequalities which are made precise in Section 5. Finally, in Section 6 we discuss some examples.

A geometric characterization of the form domain

In this section we characterize equality of the form domains of ∆ + q and deg +q by a geometric property. This geometric property is a generalization of the notion of sparseness from the introduction. Before we come to this definition, we introduce the class of potentials that is treated in this paper.

Let α > 0. We say a potential q : V → R is in the class K α if there is C α ≥ 0 such that q -≤ α(∆ + q + ) + C α , where q ± := max(±q, 0). For α ∈ (0, 1), we define the operator ∆ + q via the form sum of the operators ∆ + q + and -q -(i.e., by the KLMN Theorem, see e.g., [START_REF] Reed | Methods of Modern Mathematical Physics I, II, IV: Functional analysis. Fourier analysis, Self-adjointness[END_REF]Theorem X.17]). Note that ∆ + q is bounded from below and

D(|∆ + q| 1 2 ) = D((∆ + q + ) 1 2 ) = D(∆ 1 2 ) ∩ D(q 1 2 + ),
where |∆ + q| is defined by the spectral theorem. The last equality follows from [START_REF] Güneysu | A Feynman-Kac-Itō Formula for magnetic Schrödinger operators on graphs[END_REF]Theorem 5.6]. in the sense of functions and forms.

An other important class are the potentials

K 0 + := α∈(0,1) K α .
In our context of sparseness, we can characterize the class K 0 + to be the potentials whose negative part q -is morally o(deg +q + ), see Corollary 2.9. Let us mention that if q -is in the Kato class with respect to ∆ + q + , i.e., if we have lim sup t→0 + e -t(∆+q+) q -∞ = 0, then q := q +q -∈ K 0 + by [START_REF] Stollmann | Perturbation of Dirichlet forms by measures[END_REF]Theorem 3.1]. Next, we come to an extension of the notion of sparseness. For a set W ⊆ V , let the boundary ∂W of W be the set of edges emanating from W

∂W := {(x, y) ∈ W × V \ W | x ∼ y}.
Definition. Let G := (V , E ) be a graph and q : V → R. For given a ≥ 0 and k ≥ 0, we say that (G , q) is (a, k)-sparse if for any finite set W ⊆ V the induced subgraph

G W := (W , E W ) satisfies 2|E W | ≤ k|W | + a(|∂W | + q + (W )).
Remark 2.1. (a) Observe that the definition depends only on q + . The negative part of q will be taken in account through the hypothesis K α or K 0 + in our theorems. (b) If (G , q) is (a, k)-sparse, then (G , q ′ ) is (a, k)-sparse for every q ′ ≥ q. (c) As mentioned above there is a great variety of definitions which were so far predominantly established for (families of) finite graphs. For example it is asked that

|E | = C|V | in [EGS], |E W | ≤ k|W | + l in [L, LS], |E | ∈ O(|V |) in [AABL] and deg(W ) ≤ k|W | in [M3].
We now characterize the equality of the form domains in geometric terms.

Theorem 2.2. Let G := (V , E ) be a graph and q ∈ K α , α ∈ (0, 1). The following assertions are equivalent:

(i) There are a, k ≥ 0 such that (G , q) is (a, k)-sparse.

(ii) There are ã ∈ (0, 1) and k ≥ 0 such that on C c (V )

(1 -ã)(deg +q) -k ≤ ∆ + q ≤ (1 + ã)(deg +q) + k.
(iii) There are ã ∈ (0, 1) and k ≥ 0 such that on C c (V )

(1 -ã)(deg +q) -k ≤ ∆ + q. (iv) D(|∆ + q| 1/2 ) = D(| deg +q| 1/2 ).
Furthermore, ∆ + q has purely discrete spectrum if and only if

lim inf |x|→∞ (deg + q)(x) = ∞.
In this case, we obtain

1 -ã ≤ lim inf n→∞ λ n (∆ + q) λ n (deg +q) ≤ lim sup n→∞ λ n (∆ + q) λ n (deg +q) ≤ 1 + ã.
Before we come to the proof of Theorem 2.2, we summarize the relation between the sparseness parameters (a, k) and the constants (ã, k) in the inequality in Theorem 2.2 (ii).

Remark 2.3. Roughly speaking a tends to ∞ as ã tends to 1 -and a tends to 0 + as ã tends to 0 + and vice-versa. More precisely, Lemma 2.5 we obtain that for given ã and k the values of a and k can be chosen to be

a = ã 1 - ã and k = k 1 - ã .
Reciprocally, given a, k ≥ 0 and q : V → [0, ∞), Lemma 2.7 distinguishes the case where the graph is sparse a = 0 and a > 0. For a = 0 we may choose ã ∈ (0, 1) arbitrary and

k = k 2 1 ã -ã .
For an (a, k)-sparse graph with a > 0 the precise constants are found below in Lemma 2.7. Here, we discuss the asymptotics. For a → 0 + , we obtain

ã ≃ √ 2a and k ≃ k 2a ,
and for a → ∞ ã ≃ 1 -3 8a 2 and k ≃ 3k 4a .

In the case q ∈ K α , the constants ã, k from the case q ≥ 0 have to be replaced by constants whose formula can be explicitly read from Lemma A.3. For α → 0 + , the constant replacing ã tends to ã while the asymptotics of the constant replacing k depend also on the behavior of C α from the assumption

q -≤ α(∆ + q) + C α . Remark 2.4. (a) Observe that in the context of Theorem 2.2 statement (iv) is equivalent to (iv') D(|∆ + q| 1/2 ) = D((deg +q + ) 1/2
). Indeed, (ii) implies the corresponding inequality for q = q + . Thus, as q ∈ K α ,

D(|∆ + q| 1 2 ) = D((∆ + q + ) 1 2 ) = D((deg +q + ) 1 2 ).
(b) The definition of the class K 0 + is rather abstract. Indeed, Theorem 2.2 yields a very concrete characterization of these potentials, see Corollary 2.9 below. (c) Theorem 2.2 characterizes equality of the form domains. Another natural question is under which circumstances the operator domains agree. For a discussion on this matter we refer to [START_REF] Golénia | Hardy inequality and eigenvalue asymptotic for discrete Laplacians[END_REF]Section 4.1].

The rest of this section is devoted to the proof of the results which are divided into three parts. The following three lemmas essentially show the equivalences (i)⇔(ii)⇔(iii) providing the explicit dependence of (a, k) on (ã, k) and vice versa. The third part uses general functional analytic principles collected in the appendix.

The first lemma shows (iii)⇒(i).

Lemma 2.5. Let G := (V , E ) be a graph and q : V → R. If there are ã ∈ (0, 1)

and k ≥ 0 such that for all f in C c (V ), (1 -ã) f, (deg +q)f -k f 2 ≤ f, ∆f + qf , then (G , q) is (a, k)-sparse with a = ã 1 - ã and k = k 1 - ã . Remark 2.6.
We stress that we suppose solely that q : V → R and work with ∆| Cc(V ) + q| Cc(V ) . We do not specify any self-adjoint extension of the latter.

Proof. Let f ∈ C c (V ). By adding q -to the assumed inequality we obtain immediately

(1 -ã) f, (deg +q + )f -k f 2 ≤ f, ∆f + q + f .
Let W ⊆ V be a finite set and denote by 1 W the characteristic function of the set W . We recall the basic equalities

deg(W ) = 2|E W | + |∂W | and 1 W , ∆1 W = |∂W |.
Therefore, applying the asserted inequality with f = 1 W , we obtain

2|E W | ≤ k 1 - ã |W | + ã 1 - ã (|∂W | + q + (W )
) . This proves the statement.

The second lemma gives (i)⇒(ii) for q ≥ 0.

Lemma 2.7. Let G := (V , E ) be a graph and q :

V → [0, ∞). If there are a, k ≥ 0 such that (G , q) is (a, k)-sparse, then (1 -ã)(deg +q) -k ≤ ∆ + q ≤ (1 + ã)(deg +q) + k. on C c (V )
, where if (G , q) is sparse, i.e., a = 0, we may choose ã ∈ (0, 1) arbitrary and

k = k 2 1 ã -ã . In the other case, i.e. a > 0, we may choose ã = min 1 4 , a 2 + 2a + a 2 (1 + a) and k = max max 3 2 , 1 a -a k 2(1 + a) , 2k(1 -ã) . Proof. Let f ∈ C c (V ) be complex valued. Assume first that f, (deg +q)f < k f 2 . In this case, remembering ∆ ≤ 2 deg, we can choose ã ∈ (0, 1) arbitrary and k such that k ≥ 2(1 -ã)k.
So, assume f, (deg +q)f ≥ k f 2 . Using an area and a co-area formula (cf. [KL2, Theorem 12 and Theorem 13]) with

Ω t := {x ∈ V | |f (x)| 2 > t},
in the first step and the assumption of sparseness in the third step, we obtain

f,(deg +q)f -k f 2 = ∞ 0 deg(Ω t ) + q(Ω t ) -k|Ω t | dt = ∞ 0 2|E Ωt | + |∂Ω t | + q(Ω t ) -k|Ω t | dt ≤ (1 + a) ∞ 0 |∂Ω t | + q(Ω t )dt = (1 + a) 2 x,y,x∼y |f (x)| 2 -|f (y)| 2 + (1 + a) x q(x)|f (x)| 2 ≤ (1 + a) 2 x,y,x∼y |(f (x) -f (y))(f (x) + f (y))| + (1 + a) x q(x)|f (x)| 2 ≤ (1 + a) 2 x,y,x∼y |f (x) -f (y)| 2 + 2 x q(x)|f (x)| 2 1/2 × x,y,x∼y |f (x) + f (y)| 2 + 2 x q(x)|f (x)| 2 1/2 = (1 + a) f, (∆ + q)f 1 2 2 f, (deg +q)f -f, (∆ + q)f 1 2 ,
where we used the Cauchy-Schwarz inequality in the last inequality and basic algebraic manipulation in the last equality. Since the left hand side is non-negative by the assumption f, (deg +q)f ≥ k f 2 , we can take square roots on both sides.

To shorten notation, we assume for the rest of the proof q ≡ 0 since the proof with q = 0 is completely analogous.

Reordering the terms, yields

(1 + a) 2 f, ∆f 2 -2(1 + a) 2 f, deg f f, ∆f + ( f, (deg -k)f ) 2 ≤ 0.
Resolving the quadratic expression above gives,

f, deg f - √ δ ≤ f, ∆f ≤ f, deg f + √ δ, with δ := f, deg f 2 -(1 + a) -2 ( f, (deg -k)f ) 2 . Using 4ξζ ≤ (ξ + ζ) 2 , ξ, ζ ≥ 0, for all 0 < λ < 1, we estimate δ as follows (1 + a) 2 δ = (2a + a 2 ) f, deg f 2 + k f 2 f, (2 deg -k)f ≤ (2a + a 2 ) f, deg f 2 + λ f, deg f + k 2 1 λ -λ f 2 2 ≤ λ 2 + 2a + a 2 f, deg f + k 2 1 λ -λ f 2 2 .
If a = 0, i.e., the k-sparse case, then we take λ = ã to get

δ ≤ k f 2 f, 2 deg f ≤ ã f, deg f + k 2 1 ã -ã f 2 2 .
As k/2ã ≥ 2(1ã)k, this proves the desired inequality with k = k/2ã. If a > 0, we take λ = min 1 2 , a to get

(1 + a) 2 δ = min 1 4 , a 2 + 2a + a 2 f, deg f + k 2 max 3 2 , 1 a -a f 2 2 .
Keeping in mind the restriction k ≥ 2(1ã)k for the case f, (deg +q)f < k f 2 , this gives the statement with the choice of (ã, k) in the statement of the lemma.

The two lemmas above are sufficient to prove Theorem 2.2 for the case q ≥ 0. An application of Lemma A.3 turns the lower bound of Lemma 2.7 into a corresponding lower bound. This straightforward argument does not work for the upper bound. However, the following surprising lemma shows that such a lower bound by deg automatically implies the corresponding upper bound. There is a deeper reason for this fact which shows up in the context of magnetic Schrödinger operators. We present the non-magnetic version of the statement here for the sake of being self-contained in this section. For the more conceptual and more general magnetic version, we refer to Lemma 4.4.

Lemma 2.8 (Upside-Down-Lemma -non-magnetic version). Let G := (V , E ) be a graph and q : V → R. Assume there are ã ∈ (0, 1), k ≥ 0 such that for all

f ∈ C c (V ), (1 -ã) f, (deg +q)f -k f 2 ≤ f, ∆f + qf , then for all f ∈ C c (V ), we also have f, ∆f + qf ≤ (1 + ã) f, (deg +q)f + k f 2 . Proof. By a direct calculation we find for f ∈ C c (V ) f, (2 deg -∆)f = 1 2 x,y∈V ,x∼y (2|f (x)| 2 + 2|f (y)| 2 ) -|f (x) -f (y)| 2 ) = 1 2 x,y,x∼y |f (x) + f (y)| 2 ≥ 1 2 x,y,x∼y ||f (x)| -|f (y)|| 2 = |f |, ∆|f | .
Adding q to the inequality and using the assumption gives after reordering

f, (∆ + q)f -2 f, (deg +q)f ≤ -|f |, (∆ + q)|f | ≤ -(1 -ã) |f |, (deg +q)|f | + k |f |, |f | = -(1 -ã) f, (deg +q)f + k f, f
which yields the assertion.

Proof of Theorem 2.2. The implication (i)⇒(iii) follows from Lemma 2.7 applied with q + and from Lemma A.3 with q. The implication (iii)⇒(ii) follows from the Upside-Down-Lemma above. Furthermore, (ii)⇒(i) is implied by Lemma 2.5. The equivalence (ii)⇔(iv) follows from an application of the Closed Graph Theorem, Theorem A.1. Finally, the statements about discreteness of spectrum and eigenvalue asymptotics follow from an application of the Min-Max-Principle, Theorem A.2.

Proof of Theorem 1.1. (a) follows from Lemma 2.7. The other statements follow directly from Theorem 2.2.

As a corollary we can now determine the potentials in the class K 0 + explicitly and give necessary and sufficient criteria for potentials being in K α , α ∈ (0, 1). Corollary 2.9. Let (G , q) be an (a, k)-sparse graph for some a, k ≥ 0.

(a) The potential q is in K 0 + if and only if for all α ∈ (0, 1) there is κ α ≥ 0 such that

q -≤ α(deg +q + ) + κ α . (b) Let α ∈ (0, 1) and ã = min(1/4, a 2 ) + 2a + a 2 /(1 + a) (as given by Lemma 2.7). If there is κ α ≥ 0 such that q -≤ α(deg +q + ) + κ α , then q ∈ K α/(1-ã) . On the other, hand if q ∈ K α , then there is κ α ≥ 0 such that q -≤ α(1 + ã)(deg +q + ) + κ α .
Proof. Using the assumption q -≤ α(deg +q -) + κ α and the lower bound of Theorem 2.2 (ii), we infer

q -≤ α(deg +q + ) + κ α ≤ α (1 -ã) (∆ + q + ) + α (1 -ã) k + κ α .
Conversely, q ∈ K α and the upper bound of Theorem 2.2 (ii) yields

q -≤ α(∆ + q + ) + C α ≤ α(1 + ã)(deg +q + ) + α k + C α .
Hence, (a) follows. For (b), notice that ã = min(1/4, a 2 ) + 2a + a 2 /(1 + a) by Lemma 2.7.

Almost-sparseness and asymptotic of eigenvalues

In this section we prove better estimates on the eigenvalue asymptotics in a more specific situation. Looking at the inequality in Theorem 2.2 (ii) it seems desirable to have ã = 0. As this is impossible when the degree is unbounded, we consider a sequence of ã that tends to 0. Keeping in mind Remark 2.3, this leads naturally to the following definition.

Definition. Let G := (V , E ) be a graph and q : V → R. We say (G , q) is almost sparse if for all ε > 0 there is (G , q), every graph (G , q ′ ) with q ′ ≥ q is almost sparse.

k ε ≥ 0 such that (G , q) is (ε, k ε )-sparse, i.e., for any finite set W ⊆ V the induced subgraph G W := (W , E W ) satisfies 2|E W | ≤ k ε |W | + ε (|∂W | + q + (W )) .
The main result of this section shows how the first order of the eigenvalue asymptotics in the case of discrete spectrum can be determined for almost sparse graphs.

Theorem 3.2. Let G := (V , E ) be a graph and q ∈ K 0 + . The following assertions are equivalent:

(i) (G , q) is almost sparse. (ii) For every ε > 0 there are k ε ≥ 0 such that on C c (V ) (1 -ε)(deg +q) -k ε ≤ ∆ + q ≤ (1 + ε)(deg +q) + k ε .
(iii) For every ε > 0 there are

k ε ≥ 0 such that on C c (V ) (1 -ε)(deg +q) -k ε ≤ ∆ + q.
Moreover, D((∆+q) 1/2 ) = D((deg +q) 1/2 ) and the operator ∆+q has purely discrete spectrum if and only if lim inf |x|→∞ (deg + q)(x) = ∞. In this case, we have

lim n→∞ λ n (∆ + q) λ n (deg +q) = 1.
Proof. The statement is a direct application of Theorem 2.2 if one keeps track of the constants given explicitly by Lemma 2.5, Lemma 2.7 and Lemma A.3.

Magnetic Laplacians

In this section, we consider magnetic Schrödinger operators. Clearly, every lower bound can be deduced from Kato's inequality. However, for the eigenvalue asymptotics we also need to prove an upper bound.

We fix a phase

θ : V × V → R/2πZ such that θ(x, y) = -θ(y, x).
For a potential q : V → [0, ∞) we consider the magnetic Schrödinger operator ∆ θ + q defined as ,y) ϕ(y)) + q(x)ϕ(x).

D(∆ θ + q) := ϕ ∈ ℓ 2 (V ) | v → x∼y (ϕ(x) -e iθ(x,y) ϕ(x)) + q(x)ϕ(x) ∈ ℓ 2 (V ) (∆ θ + q)ϕ(x) := x∼y (ϕ(x) -e iθ(x
A computation for ϕ ∈ C c (V ) gives ϕ, (∆ θ + q)ϕ = 1 2 x,y,x∼y ϕ(x)e iθ(x,y) ϕ(y)

2 + x q(x)|ϕ(x)| 2 .
The operator is non-negative and selfadjoint as it is essentially selfadjoint on C c (V ) (confer e.g. [G]). For α > 0, let K θ α be the class of real-valued potentials q such that q -≤ α(∆ θ + q + ) + C α for some C α ≥ 0. Denote

K θ 0 + = α∈(0,1) K θ α .
Again, for α ∈ (0, 1) and q ∈ K θ α , we define ∆ θ + q to be the form sum of ∆ θ + q + and -q -.

We present our result for magnetic Schrödinger operators which has one implication from the equivalences of Theorem 2.2 and Theorem 3.2.

Theorem 4.1. Let G := (V , E ) be a graph, θ be a phase and q ∈ K θ 0 + be a potential. Assume (G , q) is (a, k)-sparse for some a, k ≥ 0. Then, we have the following:

(a) There are ã ∈ (0, 1),

k ≥ 0 such that on C c (V ) (1 -ã)(deg +q) -k ≤ ∆ θ + q ≤ (1 + ã)(deg +q) + k. (b) D |∆ θ + q| 1/2 = D |deg + q| 1/2 . (c)
The operator ∆ θ + q has purely discrete spectrum if and only if

lim inf |x|→∞ (deg + q)(x) = ∞.
In this case, if (G , q) is additionally almost sparse, then

lim inf λ→∞ λ n (∆ θ + q) λ n (deg + q) = 1.
Remark 4.2. (a) The constants ã and k can chosen to be the same as the ones we obtained in the proof of Theorem 2.2, i.e., these constants are explicitly given combining Lemma 2.7 and Lemma A.3. (b) Statement (a) and (b) of the theorem above remain true for q ∈ K α , α ∈ (0, 1) since K α ⊆ K θ α by Kato's inequality below. We will prove the theorem by applying Theorem 2.2 and Theorem 3.2. The considerations heavily rely on Kato's inequality and a conceptual version of the Upside-Down-Lemma, Lemma 2.8, which shows that a lower bound for ∆+q implies an upper and lower bound on ∆ θ + q. Secondly, in Theorem 3.2 potentials in K 0 + are considered, while here we start with the class K θ 0 + . However, it can be seen that K 0 + = K θ 0 + in the case of (a, k)-sparse graph, see Lemma 4.5 below. As mentioned above a key fact is Kato's inequality, see e.g. [START_REF] Dodziuk | Kato's inequality and asymptotic spectral properties for discrete magnetic Laplacians, The ubiquitous heat kernel[END_REF]Lemma 2.1] or [START_REF] Güneysu | A Feynman-Kac-Itō Formula for magnetic Schrödinger operators on graphs[END_REF]Theorem 5.2.b].

Proposition 4.3 (Kato's inequality). Let G := (V , E ) be a graph, θ be a phase and q :

V → R. For all f ∈ C c (V), we have |f |, (∆|f | + q|f |) ≤ f, (∆ θ f + qf ) .
In particular, for all α > 0

K α ⊆ K θ α and K 0 + ⊆ K θ 0 + .
Proof. The proof of the inequality can be obtained by a direct calculation. The second statement is an immediate consequence.

The next lemma is a rather surprising observation. It is the magnetic version of the Upside-Down-Lemma, Lemma 2.8. Lemma 4.4 (Upside-Down-Lemma -magnetic version). Let G := (V , E ) be a graph, θ be a phase and q : V → R be a potential. Assume that there are ã ∈ (0, 1) and k ≥ 0 such that for all f ∈ C c (V ), we have

(1 -ã) f, (deg +q)f -k f 2 ≤ f, ∆f + qf then for all f ∈ C c (V ), we also have (1 -ã) f, (deg +q)f -k f 2 ≤ f, ∆ θ f + qf ≤ (1 + ã) f, (deg +q)f + k f 2 .
Proof. The lower bound follows directly from Kato's inequality and the lower bound from the assumption (since f, (deg +q

)f = |f |, (deg +q)|f | for all f ∈ C c (V )). Now, observe that for all θ ∆ θ = 2 deg -∆ θ+π .
So, the upper bound for ∆ θ + q follows from the lower bound of ∆ θ+π + q which we deduced from Kato's inequality.

The lemma above allows to relate the classes K α and K θ α for (a, k)-sparse graphs. Lemma 4.5. For a, k ≥ 0 let G := (V , E ) be an (a, k)-sparse graph, θ be a phase and α > 0. Then,

K θ α ⊆ K α ′ , for α ′ = 1 + ã 1 - ã α,
where ã is given in Lemma 2.7. In particular,

K θ 0 + = K 0 + . Moreover, if (G , q) is almost-sparse, then K θ α ⊆ K α ′ , for all α ′ > α.
Proof. Let q ∈ K θ α . Applying Lemma 2.7, we get ∆ + q + ≥ (1ã)(deg +q + ) -k. Now, by the virtue of the Upside-Down-Lemma, Lemma 4.4, we infer

∆ θ + q + ≤ (1 + ã)(deg +q + ) + k ≤ 1 + ã 1 - ã (∆ + q + ) + 2 1 - ã k
which implies the first statement and K θ 0 + ⊆ K 0 + . The reverse inclusion K θ 0 + ⊇ K 0 + follows from Kato's inequality, Lemma 4.3. For almost sparse graphs a can be chosen arbitrary small and accordingly ã (from Lemma 2.7) becomes arbitrary small. Hence, the statement K θ α ⊆ K α ′ , for α ′ > α follows from the inequality above.

Proof of Theorem 4.1. Let q ∈ K θ 0 + . By Lemma 4.5, q ∈ K 0 + . Thus, (a) follows from Theorem 2.2 and Lemma 4.4. Using (a) statement (b) follows from an application of the Closed Graph Theorem, Theorem A.1 and statement (c) follows from an application of the Min Max Principle, Theorem A.2. Remark 4.6. Instead of using Kato's inequality one can also reproduce the proof of Lemma 2.7 using the following estimate

||f (x)| 2 -|f (y)| 2 | ≤ |(f (x) -e iθ(x,y) f (y))(f (x) + e -iθ(x,y) f (y))|.
So, we infer the key estimate:

f,(deg +q)f -k f 2 ≤ (1 + a) f, (∆ θ + q)f 1 2 f, (∆ θ+π + q)f 1 2 , = (1 + a) f, (∆ θ + q)f 1 2 2 f, (deg +q)f -f, (∆ θ + q)f 1 2 .
The rest of the proof is analogous.

It can be observed that unlike in Theorem 2.2 or Theorem 3.2 we do not have an equivalence in the theorem above. A reason for this seems to be that our definition of sparseness does not involve the magnetic potential. This direction shall be pursued in the future. Here, we restrict ourselves to some remarks on the perturbation theory in the context of Theorem 4.1 above.

Remark 4.7. (a) If the inequality Theorem 4.1 (a) holds for some θ, then the inequality holds with the same constants for -θ and θ ± π. This can be seen by the fact

∆ θ+π = 2 deg -∆ and f, ∆ θ f = f , ∆ -θ f while f, deg f = f , deg f for f ∈ C c (V ).
(b) The set of θ such that Theorem 4.1 (a) holds true for some fixed ã and k is closed in the product topology, i.e., with respect to pointwise convergence. This follows as f,

∆ θn f → f, ∆ θ f if θ n → θ, n → ∞, for fixed f ∈ C c (V ).
(c) For two phases θ and θ ′ let h(x) = max y∼x |θ(x, y)θ ′ (x, y)|. By a straight forward estimate lim sup |x|→∞ h(x) = 0 implies that for every ε > 0 there is C ≥ 0 such that

-ε deg -C ≤ ∆ θ -∆ θ ′ ≤ ε deg +C on C c (V ).
We discuss three consequences of this inequality:

First of all, this inequality immediately yields that if D(∆

1/2 θ ) = D(deg 1/2 ) then D(∆ 1/2 θ ′ ) = D(deg 1/2
) (by the KLMN Theorem, see e.g., [START_REF] Reed | Methods of Modern Mathematical Physics I, II, IV: Functional analysis. Fourier analysis, Self-adjointness[END_REF]Theorem X.17]) which in turn yields equality of the form domains of ∆ θ and ∆ θ ′ .

Secondly, combining this inequality with Theorem 3.2 we obtain the following: If lim sup |x|→∞ max y∼x |θ(x, y)| = 0 and for every ε > 0 there is k ε ≥ 0 such that

(1 -ε) deg -k ε ≤ ∆ θ ≤ (1 + ε) deg +k , ε
then the graph is almost sparse and in consequence the inequality in Theorem 4.1 (a) holds for any phase.

Thirdly, using the techniques in the proof of [START_REF] Golénia | Hardy inequality and eigenvalue asymptotic for discrete Laplacians[END_REF]Proposition 5.2] one shows that the essential spectra of ∆ θ and ∆ θ ′ coincide. With slightly more effort and the help of the Kuroda-Birman Theorem, [START_REF] Reed | Methods of Modern Mathematical Physics I, II, IV: Functional analysis. Fourier analysis, Self-adjointness[END_REF]Theorem XI.9] one can show that if h ∈ ℓ 1 (V ), then even the absolutely continuous spectra of ∆ θ and ∆ θ ′ coincide.

Isoperimetric estimates and sparseness

In this section we relate the concept of sparseness with the concept of isoperimetric estimates. First, we present a result which should be viewed in the light of Theorem 2.2 as it points out in which sense isoperimetric estimates are stronger than our notions of sparseness. In the second subsection, we present a result related to Theorem 3.2. Finally, we present a concrete comparison of sparseness and isoperimetric estimates. As this section is of a more geometric flavor we restrict ourselves to the case of potentials q : V → [0, ∞).

5.1. Isoperimetric estimates. Let U ⊆ V and define the Cheeger or isoperimetric constant of U by

α U := inf W ⊂U finite |∂W | + q(W ) deg(W ) + q(W ) .
In the case where deg(W ) + q(W ) = 0, for instance when W is an isolated point, by convention the above quotient is set to be equal to 0, Note that α U ∈ [0, 1).

The following theorem illustrates in which sense positivity of the Cheeger constant is linked with (a, 0)-sparseness. We refer to Theorem 5.4 for precise constants.

Theorem 5.1. Given G := (V , E ) a graph and q : V → [0, ∞). The following assertions are equivalent

(i) α V > 0. (ii) There is ã ∈ (0, 1) (1 -ã)(deg +q) ≤ ∆ + q ≤ (1 + ã)(deg +q).
(iii) There is ã ∈ (0, 1) such that

(1 -ã)(deg +q) ≤ ∆ + q.
The implication (iii)⇒(i) is already found in [START_REF] Golénia | Hardy inequality and eigenvalue asymptotic for discrete Laplacians[END_REF]Proposition 3.4]. The implication (i)⇒(ii) is a consequence from standard isoperimetric estimates which can be extracted from the proof of [KL2, Proposition 15].

Proposition 5.2 ([KL2]

). Let G := (E , V ) be a graph and q : V → [0, ∞). Then, for all U ⊆ V we have on C c (U ).

1 -1α 2 U (deg +q) ≤ ∆ + q ≤ 1 + 1α 2 U (deg +q). 5.2. Isoperimetric estimates at infinity. Let the Cheeger constant at infinity be defined as

α ∞ = sup K ⊆V finite α V \K . Clearly, 0 ≤ α V ≤ α U ≤ α ∞ ≤ 1 for any U ⊆ V .
As a consequence of Proposition 5.2, we get the following theorem.

Theorem 5.3. Let G := (E , V ) be a graph and q : V → [0, ∞) be a potential. Assume α ∞ > 0. Then, we have the following:

(a) For every ε > 0 there is k ε ≥ 0 such that on C c (V ) (1 -ε) 1 -1 -α 2 ∞ (deg +q) -k ε ≤ ∆ + q ≤ (1 + ε) 1 + 1 -α 2 ∞ (deg +q) + k ε . (b) D((∆ + q) 1/2 ) = D((deg +q) 1/2 ).
(c) The operator ∆ + q has purely discrete spectrum if and only if we have

lim inf |x|→∞ (deg + q)(x) = ∞. In this case, if additionally α ∞ = 1, we get lim inf λ→∞ λ n (∆ + q) λ n (deg + q) = 1.
Proof. (a) Let ε > 0 and K ⊆ V be finite and large enough such that

(1 -ε) 1 -1 -α 2 ∞ ≤ 1 -1 -α 2 V \K 1 + 1 -α 2 V \K ≤ (1 + ε) 1 + 1 -α 2 ∞ .
From Proposition 5.2 we conclude on

C c (V \ K ) (1 -ε) 1 -1 -α 2 ∞ (deg +q) ≤ 1 -1 -α 2 V \K (deg +q) ≤ ∆ + q ≤ 1 + 1 -α 2 V \K (deg +q) ≤ (1 + ε) 1 + 1 -α 2
∞ (deg +q) By local finiteness the operators 1 V \K (∆ + q)1 V \K and 1 V \K (deg +q)1 V \K are bounded (indeed, finite rank) perturbations of ∆ + q and deg +q. This gives rise to the constants k ε and the inequality of (a) follows. Now, (b) is an immediate consequence of (a), and (c) follows by the Min-Max-Principle, Theorem A.2. 5.3. Relating sparseness and isoperimetric estimates. We now explain how the notions of sparseness and isoperimetric estimates are exactly related.

First, we consider classical isoperimetric estimates.

Theorem 5.4. Let G := (V , E ) be a graph, a, k ≥ 0, and let q : V → [0, ∞) be a potential.

(a) α V ≥ 1 1 + a if and only if (G , q) is (a, 0)-sparse. (b) If (G , q) is (a, k)-sparse, then α V ≥ d -k d(1 + a)
,

where d := inf x∈V (deg +q)(x). In particular, α V > 0 if d > k. (c) Suppose that (G , q) is (a, k)-sparse graph that is not (a, k ′ )-sparse for all k ′ < k. Suppose also that there is d such that d = deg(x) + q(x) for all x ∈ V . Then α V = d -k d(1 + a) . Proof. Let W ⊂ V be a finite set. Recalling the identity deg(W ) = 2|E W | + |∂W | we notice that 1 1 + a ≤ |∂W | + q(W ) (deg +q)(W )
is equivalent to

2|E W | ≤ a(|∂W | + q(W ))
which proves (a). For (b), the definition of (a, k)-sparseness yields

|∂W | + q(W ) (deg +q)(W ) = 1 - 2|E W | (deg +q)(W ) ≥ 1 -a |∂W | + q(W ) (deg +q)(W ) -k |W | (deg +q)(W )
.

This concludes immediately.

For (c), the lower bound of α V follows from (b). Since (G , q) is not (a, k ′ )-sparse there is a finite W 0 ⊂ V such that

|∂W 0 | + q(W 0 ) (deg +q)(W 0 ) < 1 -a |∂W 0 | + q(W 0 ) (deg +q)(W 0 ) -k ′ |W 0 | (deg +q)(W 0 ) . Therefore α V < (d -k ′ )/(d(1 + a)).
We next address the relation between almost sparseness and isoperimetry and show two "almost equivalences".

Theorem 5.5. Let G := (V , E ) be a graph and let q : V → [0, ∞) be a potential.

(a) If α ∞ > 0, then (G , q) is (a, k)-sparse for some a > 0, k ≥ 0. On the other hand, if (G , q) is (a, k)-sparse for some a > 0, k ≥ 0 and (G , q) is almost sparse. On the other hand, if (G , q) is almost sparse and lim inf |x|→∞ (deg +q)

l := lim inf |x|→∞ (deg +q)(x) > k, then α ∞ ≥ l -k l(a + 1) > 0, if l is finite and α ∞ ≥ 1/(1 + a) otherwise. (b) If α ∞ = 1, then
(x) = ∞, then α ∞ = 1.
Proof. The first implication of (a) follows from Theorem 5.3 (a) and Theorem 2.2 (ii)⇒(i). For the opposite direction let ε > 0 and K ⊆ V be finite such that deg +q ≥ lε on V \ K . Using the formula in the proof of Theorem 5.4 above, yields for

W ⊆ V \ K |∂W | + q(W ) (deg +q)(W ) = 1 - 2|E W | (deg +q)(W ) ≥ 1 - k|W | + a(|∂W | + q(W )) (deg +q)(W ) ≥ 1 - k (l -ε) - a(|∂W | + q(W )) (deg +q)(W ) .
This proves (a). The first implication of (b) follows from Theorem 5.3 (a) and Theorem 3.2 (ii)⇒(i).

The other implication follows from (a) using the definition of almost sparseness. The previous theorems provides a slightly simplified proof of [K1] which also appeared morally in somewhat different forms in [START_REF] Dodziuk | Difference equations, isoperimetric inequality and transience of certain random walks[END_REF][START_REF] Woess | A note on tilings and strong isoperimetric inequality[END_REF]. We turn to graphs which admit a 2-cell embedding into S g , where S g denotes a compact orientable topological surface of genus g. (The surface S g might be pictured as a sphere with g handles.) Admitting a 2-cell embedding means that the graphs can be embedded into S g without self-intersection. By definition we say that a graph is planar when g = 0. Note that unlike other possible definitions of planarity, we do not impose any local compactness on the embedding. We treat the cases (b) and (c) simultaneously. Let G := (V , E ) be a graph which is connected 2-cell embedded in S g with g ≥ 0 (as remarked above planar graphs correspond to g = 0). Let G W := (W , E W ) be a finite induced subgraph of G which, clearly, also admits a 2-cell embedding into S g . The statement is clear for |W | ≤ 2. Assume |W | ≥ 3. Let F W be the faces induced by G W := (W , E W ) in S g . Here, all faces (even the outer one) contain at least 3 edges, each edge belongs only to 2 faces, thus,

2|E W | ≥ 3|F W |. Euler's formula, |W | -|E W | + |F W | = 2 -2g, gives then 2 -2g + |E W | = |W | + |F W | ≤ |W | + 2 3 |E W | that is |E W | ≤ 3|W | + 6(g -1) ≤ max(2g + 1, 3)|W |.
This concludes the proof.

Next, we explain how to construct sparse graphs from existing sparse graphs. Remark 6.4. (a) We point out that there are bi-partite graphs which are not sparse. See for example [START_REF] Golénia | Hardy inequality and eigenvalue asymptotic for discrete Laplacians[END_REF]Proposition 4.11] or take an antitree, confer [START_REF] Keller | Volume Growth, Spectrum and Stochastic Completeness of Infinite Graphs[END_REF]Section 6], where the number of vertices in the spheres grows monotonously to ∞. (b) The last point of the lemma states that the k-sparseness is non-decreasing when we remove edges from the graph. This is not the case for the isoperimetric constant. 6.2. Examples of almost-sparse and (a, k)-sparse graph. We construct a series of examples which are perturbations of a radial tree. They illustrate that sparseness, almost sparseness and (a, k)-sparseness are indeed different concepts.

Lemma 6.3. Let G 1 := (V 1 , E 1 ) and G 2 := (V 2 , E 2 ) be two graphs. (a) Assume V 1 = V 2 , G 1 is k 1 -sparse and G 2 is k 2 -sparse. Then, G := (V , E ) with E := max(E 1 , E 2 ) is (k 1 + k 2 )-sparse. (b) Assume G 1 is k 1 -sparse and G 2 is k 2 -sparse. Then G 1 ⊕ G 2 := (V , E ) with where V := V 1 × V 2 and E ((x 1 , x 2 ), (y 1 , y 2 )) := δ {x1} (y 1 ) • E 2 (x 2 , y 2 ) + δ {x2} (y 2 ) • E 1 (x 1 , y 1 ), is (k 1 + k 2 )-sparse. (c) Assume V 1 = V 2 , G 1 is k-sparse and E 2 ≤ E 1 . Then, G 2 is k-sparse. Proof. For (a) let W ⊆ V be finite and note that |E W | ≤ |E 1,W | + |E 2,W |. For (b) let p 1 , p 2 the canonical projections from V to V 1 and V 2 . For finite W ⊆ V we observe |E W | = |E 1,p1(W ) | + |E 2,p2(W ) | ≤ k 1 |p 2 (W )| + k 2 |p 1 (W )| ≤ (k 1 + k 2 )|W |.
Let β = (β n ), γ = (γ n ) be two sequences of natural numbers. Let T = T (β) with T = (V , E T ) be a radial tree with root o and vertex degree β n at the n-th sphere, that is every vertex which has natural graph distance n to o has (β n -1) forward neighbors. We denote the distance spheres by S n . We let G (β, γ) be the set of graphs G := (V , E G ) that are super graphs of T such that the induced subgraphs G Sn are γ n -regular and

E G (x, y) = E T (x, y) for x ∈ S n , y ∈ S m , m = n.
Observe that G (β, γ) is non empty if and only if γ n n j=0 (β j -1) is even and γ n < |S n | = n j=0 (β j -1) for all n ≥ 0.

lim sup n→∞ γ n < ∞. (b) If a > 0, then G is (a ′ , k)-sparse for some k ≥ 0 if a ′ > a. Conversely, if G is (a ′ , k)-sparse for some k ≥ 0, then a ′ ≥ a.
Proof. Let ε > 0 and let N ≥ 0 be so large that

γ n ≤ (a + ε)β n , n ≥ N. Set C ε := N -1 n=0 deg G (S n ).
Let W be a non-empty finite subset of V . We calculate Hence, if k 0 = ∞, then G is not sparse. On the other hand, if k 0 < ∞, then G is (k 0 + 2)-sparse by Lemma 6.3 as T is 2-sparse by Lemma 6.2. This finishes the proof of (a). Finally, assume that G is (a ′ , k)-sparse with k ≥ 0. Then, for W = S n (-∞, λ ess 0 (T )). For λ 0 (T ) < λ ess 0 (T ) we denote the eigenvalues below λ ess 0 (T ) by λ n (T ), for 0 ≤ n ≤ n(T ), in increasing order counted with multiplicity.

2|E G W | + |∂ G W | = deg G (W ) = deg T (W ) + n≥0 |W ∩ S n |γ n ≤ deg T (W ) + (a + ε) n≥0 |W ∩ S n |β n + N -1 n=0 |W ∩ S n |γ n ≤ (1 + a + ε) deg T (W ) + C ε |W | = 2(1 + a + ε)|E T W | + (1 + a + ε)|∂ T W | + C ε |W | ≤ (2(1 + a + ε) + C ε )|W | + (1 + a + ε)|∂ T W |,
Theorem A.2. Let (Q 1 , D(Q 1 )) and (Q 2 , D(Q 2 )) be closed non-negative quadratic forms with a common form core D 0 and let T 1 and T 2 be the corresponding selfadjoint operators. Assume there are constants c 1 > 0, c 2 ∈ R such that on D 0

c 1 Q 2 -c 2 ≤ Q 1 .
Then, c 1 λ n (T 2 )-c 2 ≤ λ n (T 1 ), for 0 ≤ n ≤ min(n(T 1 ), n(T 2 )). Moreover, c 1 λ ess 0 (T 2 )c 2 ≤ λ ess 0 (T 1 ), in particular, σ ess (T 1 ) = ∅ if σ ess (T 2 ) = ∅ and in this case

c 1 ≤ lim inf n→∞ λ n (T 1 )
λ n (T 2 ) .

Proof. Letting n be the eigenfunctions of T j to λ 0 (T j ), . . . , λ n (T j ) we get c 1 λ n (T 2 )c 3 = inf 0 =ψ∈{ϕ

(2) 1 ,...,ϕ

(2)

n } ⊥ ∩D0 c 1 T 2 ψ, ψ ψ, ψ -c 3 ≤ inf 0 =ψ∈{ϕ (2) 1 ,...,ϕ (2)
n } ⊥ ∩D0

T 1 ψ, ψ ψ, ψ ≤ µ n (T 1 ) = λ n (T 1 )

This directly implies the first statement. By a similar argument the statement about the bottom of the essential spectrum follows, in particular, λ ess 0 (T 2 ) = ∞ implies lim n→∞ µ n (T 1 ) = ∞ and, thus, λ ess 0 (T 1 ) = ∞. In this case λ n (T 2 ) → ∞, n → ∞, which implies the final statement.

Finally, we give a lemma which helps us to transform inequalities under form perturbations.

Lemma A.3. Let (Q 1 , D(Q 1 )), (Q 2 , D(Q 2 )) and (q, D(q)) be closed symmetric non-negative quadratic forms with a common form core D 0 such that there are α ∈ (0, 1), C α ≥ 0 such that q ≤ αQ 1 + C on D 0 . If for a ∈ (0, 1) and k ≥ 0

(1 -a)Q 2 -k ≤ Q 1 on D 0 , then (1 -α)(1 -a) (1 -α(1 -a)) (Q 2 -q) - (1 -α)k + aC α (1 -α(1 -a)) ≤ Q 1 -q, on D 0 .
In particular, if a → 0 + , then (1α)(1a)/(1α(1a)) → 1 -and if α → 0 + , then (1α)(1a)/(1α(1a)) → (1a).

Proof. The assumption on q implies q ≤ α (1α)

(Q 1q) + C α (1α) .

We subtract (1a)q on each side of the lower bound in (1a)Q 2k ≤ Q 1 . Then, we get

(1 -a)(Q 2 -q) -k ≤ (Q 1 -q) + aq ≤ 1 -α(1 -a) (1 -α) (Q 1 -q) + aC α (1 -α)
and, thus, the asserted inequality follows.

  Remark 3.1. (a) Every sparse graph G is almost sparse. (b) For an almost sparse graph

  Remark 5.6. (a) We point out that without the assumptions on (deg +q) the converse implications do not hold. For example the Cayley graph ofZ is 2-sparse (cf. Lemma 6.2), but has α ∞ = 0. (b) Observe that α ∞ = 1 implies lim inf |x|→∞ (deg +q)(x) = ∞.Hence, (b) can be rephrased as the following equivalence: α ∞ = 1 is equivalent to (G , q) almost sparse and lim inf |x|→∞ (deg +q)(x) = ∞.

  Corollary 5.7. Let G := (V , E ) be a planar graph. (a) If for all vertices deg ≥ 7, then α V > 0. (b) If for all vertices away from a finite set deg ≥ 7, then α ∞ > 0. Proof. Combine Theorem 5.4 and Theorem 5.5 with Lemma 6.2. 6. Examples 6.1. Examples of sparse graphs. To start off, we exhibit two classes of sparse graphs. First we consider the case of graphs with bounded degree. Lemma 6.1. Let G := (V , E ) be a graph. Assume D := sup x∈V deg(x) < +∞, then G is D-sparse. Proof. Let W be a finite subset of V . Then, 2|E W | ≤ deg(W ) ≤ D|W |.

  Lemma 6.2. (a) Trees are 2-sparse. (b) Planar graphs are 6-sparse. (c) Graphs admitting a 2-cell embedding into S g with g ≥ 1 are 4g + 2-sparse. Proof. (a) Let G := (V , E ) be a tree and G W := (W , E W ) be a finite induced subgraph of G . Clearly |E W | ≤ |W | -1. Therefore, every tree is 2-sparse.

For

  (c) and W ⊆ V finite, we have |E 2,W | ≤ |E 1,W | which yields the statement.

Figure 1 .

 1 Figure 1. G with β = (3, 3, 4, . . .) and γ = (0, 2, 4, 5, . . .).

  where we used that trees are 2-sparse in the last inequality. Finally, since|∂ G W | ≥ |∂ T W |, we conclude 2|E G W | ≤ (2(1 + a + ε) + C ε ) |W | + (a + ε)|∂ G W |.This shows that the graph in (a) with a = 0 is almost sparse and that the graph in (b) with a > 0 is (a + ε, k ε )-sparse for ε > 0 and k ε = 2(1 + a + ε) + C ε . Moreover, for the other statement of (a) let k 0 = lim sup n→∞ γ n and note that for G Sn2|E Sn | = γ n |S n |.

  γ n |S n | = 2|E Sn | ≤ k|S n | + a ′ |∂ G S n | = k|S n | + a ′ β n |S n |Dividing by β n |S n | and taking the limit yields a ≤ a ′ . This proves (b). Remark 6.6. In (a), we may suppose alternatively that we have the complete graph on S n and the following exponential growth lim n→∞ |Sn| |Sn+1| = 0.

  operator T , we know by the Min-Max-Principle[START_REF] Reed | Methods of Modern Mathematical Physics I, II, IV: Functional analysis. Fourier analysis, Self-adjointness[END_REF] Chapter XIII.1] µ n (T ) = λ n (T ) if λ n (T ) < λ ess 0 (T ) and µ n (T ) = λ ess 0 (T ) otherwise, n ≥ 0. Assume n ≤ min{n(T 1 ), n(T 2 )} and let ϕ

Acknowledgement. MB was partially supported by the ANR project HAB (ANR-12-BS01-0013-02). SG was partially supported by the ANR project GeRaSic and SQFT. MK enjoyed the hospitality of Bordeaux University when this work started. Moreover, MK acknowledges the financial support of the German Science Foundation (DFG), Golda Meir Fellowship, the Israel Science Foundation (grant no. 1105/10 and no. 225/10) and BSF grant no. 2010214.

Appendix A. Some general operator theory

We collect some consequences of standard results from functional analysis that are used in the paper. Let H be a Hilbert space with norm • . For a quadratic form Q, denote the form norm by

The following is a direct consequence of the Closed Graph Theorem, (confer e.g. [START_REF] Weidmann | Lineare Operatoren in Hilberträumen 1[END_REF]Satz 4.7]).

) be closed non-negative quadratic forms with a common form core D 0 . Then, the following are equivalent:

Proof. If (ii) holds, then any • Q1 -Cauchy sequence is a • Q2 -Cauchy sequence. Thus, (ii) implies (i). On the other hand, consider the identity map j :

The map j is closed as it is defined on the whole Hilbert space (D(Q 1 ), • Q1 ) and, thus, bounded by the Closed Graph Theorem [START_REF] Reed | Methods of Modern Mathematical Physics I, II, IV: Functional analysis. Fourier analysis, Self-adjointness[END_REF]Theorem III.12] which implies (i).

For a selfadjoint operator T which is bounded from below, we denote the bottom of the spectrum by λ 0 (T ) and the bottom of the essential spectrum by λ ess 0 (T ). Let n(T ) ∈ N 0 ∪ {∞} be the dimension of the range of the spectral projection of