DECORRELATION ESTIMATES FOR RANDOM SCHRÖDINGER OPERATORS WITH NON RANK ONE PERTURBATIONS

Transcription

1 DECORRELATION ESTIMATES FOR RANDOM SCHRÖDINGER OPERATORS WITH NON RANK ONE PERTURBATIONS PETER D. HISLOP, M. KRISHNA, AND C. SHIRLEY Abstract. We prove decorrelation estimates for generalized lattice Anderson models on Z d constructed with finite-ran perturbations in the spirit of Klopp [9]. These are applied to prove that the local eigenvalue statistics ξe ω and ξ ω E, associated with two energies E and E in the localization region and satisfying E E > 4d, are independent. That is, if I, J are two bounded intervals, the random variables ξe(i) ω and ξ ω E (J), are independent and distributed according to a compound Poisson distribution whose Lévy measure has finite support. We also prove that the extended Minami estimate implies that the eigenvalues in the localization region have multiplicity at most the ran of the perturbation. The method of proof contains new ingredients that simplify the proof of the ran one case [9, 18, 20], extends to models for which the eigenvalues are degenerate, and applies to models for which the potential is not sign definite [19] in dimensions d 1. Contents 1. Statement of the problem and results Asymptotic independence and decorrelation estimates Contents 5 2. Estimates on weighted sums of eigenvalues Properties of the weighted trace Variational formulae Dependence of the weighted eigenvalue averages on the random variables 7 3. Proof of Proposition Reduction via the extended Minami estimate Estimates on the joint probability Reduction of length scale using localization Key estimate on the log-scale Estimate of P{Ω i,j 0 ( l, 1, 2 )} Asymptotically independent random variables: Proof of Theorem Bounds on eigenvalue multiplicity Decorrelation estimates for the discrete alloy-type model 17 PDH was partially supported by NSF through grant DMS during the time some of this wor was done. MK was partially supported by IMSc Project 12-R&D-IMS The authors than F. Klopp for discussions on eigenvalue statistics and decorrelation estimates.

2 2 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY References 21 AMS 2010 Mathematics Subject Classification: 35J10, 81Q10, 35P20 Keywords: random Schrödinger operators, eigenvalue statistics, decorrelation estimates, independence, Minami estimate, compound Poisson distribution 1. Statement of the problem and results We consider random Schrödinger operators H ω = L + V ω on the lattice Hilbert space l 2 (Z d ) (or, for matrix-valued potentials, on l 2 (Z d ) C m ), and prove that certain natural random variables associated with the local eigenvalue statistics around two distinct energies E and E, in the region of complete localization Σ CL and with E E > 4d, are independent. From previous wor [10], these random variables distributed according to a compound Poisson distribution. The operator L is the discrete Laplacian on Z d, although this can be generalized. For these lattice models, the random potential V ω has the form (V ω f)(j) = i J ω i (P i f)(j), (1.1) where {P i } i J is a family of finite-ran projections with the same ran m 1, the set J is a sublattice of Z d, and i J P i = I. We assume that P i = U i P 0 Ui 1, for all i J, where U i is the unitary implementation of the translation group (U i f)() = f( + i), for i, Z d. The coefficients {ω i } are a family of independent, identically distributed (iid) random variables with a bounded density of compact support on a product probability space Ω with probability measure P. It follows from the conditions above that the family of random Schrödinger operators H ω is ergodic with respect to the translations generated by the sublattice J. One example on the lattice is the polymer model. For this model, the projector P i = χ Λ (i) is the characteristic function on the cube Λ (i) of side length 2 N centered at i Z d. The ran of P i is (2 + 1) d and the set J is chosen so that i J Λ (i) = Z d. Another example is a matrix-valued model for which P i, i Z d, projects onto the m -dimensional subspace C m, and J = Z d. The corresponding Schrödinger operator is H ω = L + i J ω i P i, (1.2) where L is the discrete lattice Laplacian on l 2 (Z d ), or I on l 2 (Z d ) C m or, more generally, A, where A is a Hermitian positive-definite m m matrix), respectively. In the following, we denote by H ω,l (or simply as H l omitting the ω) the matrices χ Λl H ω χ Λl and similarly H ω,l, H L by replacing l with L, for positive integers l and L. A lot is nown about the eigenvalue statistics for random Schrödinger operators on l 2 (R d ). When the projectors P i are ran one projectors, the local eigenvalue statistics in the localization regime has been proved to be given by a

3 DECORRELATION ESTIMATES FOR NON RANK ONE PERTURBATIONS 3 Poisson process by Minami [15] (see also Molchanov [16] for a model on R and Germinet-Klopp [6] for a comprehensive discussion and additional results). For the non ran one case, Tautenhahn and Veselić [19] proved a Minami estimate for certain models that may be described as wea perturbations of the ran one case. The general non finite ran case was studied by the first two authors in [10] who proved that, roughly speaing, the local eigenvalue statistics in the localization regime are compound Poisson point processes. This result also holds for random Schrödinger operators on R d. In this paper, we further refine these results for lattice models with non ran one projections and prove, roughly speaing, that the processes associated with two energies are independent. The method applies to Schrödinger operators with non simple eigenvalues, see the discussion at the end of section 5. Klopp [9] proved decorrelation estimates for ran one lattice models in any dimension. He applied them to show that the local eigenvalue point processes at distinct energies converge to independent Poisson processes (in dimensions d > 1 the energies need to be far apart as is the case for the models studied here). Shirley [18] extended the family of one-dimensional lattice models for which the decorrelation estimate may be proved to include alloy-type models with correlated random variables, hopping models, and certain one-dimensional quantum graphs. One of the advantages of the methods employed in this paper is that monotonicity is no longer needed. Consequently, we can treat the almost ran one models for which the potential is not sign definite. A class of such models was considered by Tautenhahn and Veseli`c [19] who proved a Minami estimate Asymptotic independence and decorrelation estimates. The main result is the asymptotic independence of random variables associated with the local eigenvalue statistics centered at two distinct energies E and E satisfying E E > 4d. We note that in one-dimension there are stronger results and the condition E E > 4d is not needed. Our results are inspired by the wor of Klopp [9] for the Anderson models on Z d and of Shirley [18] for related models on Z d. The condition E E > 4d requires that the two energies be fairly far apart. For example, if ω 0 [ K, K] so that the deterministic spectrum Σ = [ 2d K, 2d + K], the region of complete localization Σ CL is near the band edges ±(2d + K). In this case, one can consider E and E near each of the band edges. Our main result on asymptotic independence is the following theorem. Theorem 1.1. Let E, E Σ CL be two distinct energies with E E > 4d. Let ξ ω,e, respectively, ξ ω,e, be a limit point of the local eigenvalue statistics centered at E, respectively, at E. Then these two processes are independent. That is, for any bounded intervals I, J B(R), the random variables ξ ω,e (I) and ξ ω,e (J) are independent random variables distributed according to a compound Poisson process. We refer to [6] for a description of the region of complete localization Σ CL. For information on Lévy processes, we refer to the boos by Applebaum [2]

4 4 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY and by Bertoin [3]. Theorem 1.1 follows (see section 4) from the following decorrelation estimate. We assume that L > 0 is a positive integer, and that l := [L α ] is the greatest integer less than L α for an exponent 0 < α < 1. For polymer type models, we assume that m divides L and l. Proposition 1.1. We choose any 0 < α < 1 and β > 5 2, and length scales L and l := [L α ] as described above. For a pair of energies E, E Σ CL, the region of complete localization, with E := E E > 4d, and bounded intervals I, J R, we define I L (E) := L d I + E and J L (E ) := L d J + E as two scaled energy intervals centered at E and E, respectively. There exists L 0 = L 0 (α, β, d) > 0 and a constant C 0 = C 0 (L 0 ) such that for all L > L 0 we have P{(TrE Hω,l (I L (E)) 1) (TrE Hω,l (J L (E K ρ 2 )) 1)} C m 2m 0 E 4d (C log L) (1+β)d L (2 α)d. (1.3) The extended Minami estimate [10] (see section 3.1) implies that we only need to estimate the probability that there is a small number of eigenvalues in each interval: P{(TrE Hω,l (I L (E)) m ) (TrE Hω,l (J L (E )) m )} (1.4) In fact, we consider the more general estimate: P{(TrE Hω,L (I L (E)) = 1 ) (TrE Hω,L (J L (E )) = 2 )}, (1.5) where 1, 2 m are positive integers independent of L. We allow that there may be several eigenvalues in I L (E) and J L (E ) with nontrivial multiplicities. To deal with this, we introduce the mean trace of the eigenvalues E j (ω) of H ω,l in the interval I L (E): T l (E, 1, ω) := T r( H ω,l E Hω,l (I L (E)) ) T r ( E Hω,l (I L (E)) ) = 1 1 E j(ω), l (1.6) 1 where 1 := T r ( E Hω,l (I L (E)) ) is the number of eigenvalues, including multiplicity, of H ω,l in I L (E). Similarly, we define T l (E, 2, ω) := T r( H ω,l E Hω,l (J L (E )) ) T r ( E Hω,l (J L (E )) ) = 1 2 E j(ω), l (1.7) 2 where 2 := T r ( E Hω,l (J L (E )) ). We will show in section 2 that these weighted sums behave lie effective eigenvalues in each scaled interval I L (E) and J L (E ), respectively. As another application of the extended Minami estimate, we prove that the multiplicity of eigenvalues in Σ CL is at most the multiplicity of the perturbations m in dimensions d 1. The proof of this fact follows the argument of Klein and Molchanov [8]. For d = 1, Shirley [18] proved that the usual Minami estimate holds for the dimer model (m = 2) so the eigenvalues are almost surely simple. j=1 j=1

5 DECORRELATION ESTIMATES FOR NON RANK ONE PERTURBATIONS Contents. We present properties of the average of eigenvalues in section 2, including gradients estimates.the proof of the main technical result, Proposition 1.1, is presented in section 3. The proof of asymptotic independence is given in section 4. We show in section 5 that the argument of Klein-Molchanov [8] applies to higher ran perturbations and implies that the multiplicity of eigenvalues in Σ CL is at most m, the uniform ran of the perturbations. In section 6, we prove that the decorrelation estimates, and therefore asymptotic independence of local eigenvalue processes, hold for non sign definite models studied by Tautenhahn and Veseli`c [19]. This paper replaces the manuscript [11] by the first two authors, completing and improving the arguments, and extending the results. 2. Estimates on weighted sums of eigenvalues In this section, we present some technical results on weighted sums of eigenvalues of H ω,l defined in (1.6)-(1.7). These are used in section 4 to prove the main technical result (1.3). We recall that l = [L α ], for 0 < α < Properties of the weighted trace. When the total number of eigenvalues of H ω,l in I L (E) := L d I + E is 1, we have T (ω) := T l (E, 1 ) := T l (E, 1, ω) = 1 1 E j (ω), (2.1) 1 for eigenvalues E l j (ω) I L(E). Properties (1)-(3) below are valid for the similar expression obtained by replacing 1 with 2, the interval I with J, and the energy E with E. We will write j=1 T (ω) := T l (E, 2 ) := T (E, 2, ω). We write P E (ω) for the spectral projection E Hω,l (I L (E)) onto the eigenspace of H ω,l corresponding to the eigenvalues E l m(ω) in I L (E). Let γ E be a simple closed contour containing only these eigenvalues of H ω,l with a counterclocwise orientation. Since the mean of the eigenvalues may be expressed as T (ω) = 1 1 TrH ω,l P E (ω), and the projection has the representation P E (ω) = 1 R(z) dz, R(z) := (H ω,l z) 1. 2πi γ E The Hamiltonian H ω,l is analytic in the variables ω j. The projection P E (ω) is also analytic in ω provided the contour γ E remains isolated from the other eigenvalues of H ω,l. We wor on that part of the probability space for which the total multiplicity of the eigenspace Ran P E (ω) = 1, that is, on the set Ω( 1 ) := {ω TrP E (ω) = 1 }. If we place a security zone around I L (E) of width L d then the probability that H ω,l has no eigenvalues in this zone is larger than 1 (l/l) d by the Wegner estimate. On this set, it follows that T (ω) ω j = 1 2πi 1 γ E Tr{R(z)P j R(z)} zdz, (2.2)

6 6 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY where P j is the finite-ran projector associated with site j or bloc j, depending on the model. Evaluating the contour integral, we find that T (ω) ω j = 1 1 Tr{P E (ω)p j }. (2.3) Formula (2.3) shows that the eigenvalue average behaves lie an effective eigenvalue in the following sense: (1) T l (E, 1, ω) I L (E), so the average of the eigenvalue cluster in I L (E) belongs to I L (E). (2) The ω j -derivative of T (ω) is nonnegative as follows directly from (2.3). (3) Since j J Λ l P j = I Λl, it follows from this and (2.3) that the ω- gradient of the weighted trace is normalized: ω T (ω) l 1 = 1. Remar 1. It follows from property (1) above and the fact that the intervals I L (E) and J L (E) are O(L d ), that if E E > 4d, then T (ω) T (ω ) > 4d cl d, for some c > 0. We will use this result below Variational formulae. We can estimate the variation of the mean trace with respect to the random variables as follows. The ω-directional derivative is { 1 ω ω (T (ω) T (ω)) = ω j TrP E (ω)p j 1 } TrP E (ω)p j 1 2 j J Λ l { 1 = TrP E (ω)vλ ω l 1 } TrP E (ω)vλ ω 1 l 2 { 1 = TrP E (ω)h ω,l 1 TrP E (ω)h ω,l TrP E (ω)l + 1 } TrP E (ω)l. (2.4) 1 2 The absolute value of each trace involving the Laplacian L in (2.4) may be bounded above by 2d. If we assume that then we obtain from (2.4), T (ω) T (ω) E E 4d T (ω) T (ω) 4d ω ω (T (ω) T (ω)). (2.5) As the number of components of ω is bounded by l d and ω j K, it follows by Cauchy-Schwartz inequality that ω (T (ω) T (ω)) 2 E 4d K 1 (2l + 1) d/2. (2.6) We also obtain an l 1 lower bound: ω (T (ω) T (ω)) 1 E 4d K. (2.7)

7 DECORRELATION ESTIMATES FOR NON RANK ONE PERTURBATIONS Dependence of the weighted eigenvalue averages on the random variables. Suppose that H ω,l has 1 eigenvalues (including multiplicities) {Em(ω); l m = 1,..., 1 } in an interval I R. The corresponding weighted average is T l ω ( 1 ) := j=1 E l m j (ω) I. (2.8) We consider the tensor product operator H 1 ω,l on the Hilbert space H 1 l := 1 l 2 (Λ l ) defined as H 1 ω,l := j=1 I H ω,l I, (2.9) where the Hamiltonian H ω,l appears in the j th -position. From the normalized eigenfunctions ψ l m of H ω,l, we form the eigenfunctions ψ σ(1 ) := ψ l m 1 ψ l m 2 ψ l m 1, (2.10) where σ( 1 ) = (m 1,..., m 1 ). These functions are eigenfunctions of H 1 ω,l with eigenvalue Tω l ( 1 ) defined in (2.8): H 1 ω,l ψl σ( 1 ) = T l ω ( 1 )ψ l σ( 1 ). (2.11) It is important to now how the eigenvalue average Tω l ( 1 ) depends on the random variables ω i, ω j with i, j Λ l. As the matrix H 1 ω,l is of size (m Λ l ) 1 (m Λ l ) 1, the expression det((m Λ l ) 1 EI) is a real polynomial of degree m 1 in the pair of random variables ω i, ω j for i, j Λ l. We will use this in section 3 when we apply the Harnac Curve Theorem. 3. Proof of Proposition 1.1 In this section, we prove the technical result, Proposition 1.1. We let X l (I L (E)) := TrE Hω,l (I L (E)), X l (J L (E )) := TrE Hω,l (J L (E )), and consider the scale l = [L α ], for 0 < α < 1. Then, we show P{(X l (I L (E)) 1) (X l (J L (E K ρ 2 )) 1)} C m 2m 0 E 4d for positive numbers 0 < α < 1 and any β > 5 2. (C log L) (1+β)d L (2 α)d. (3.1) 3.1. Reduction via the extended Minami estimate. Let χ A (ω) be the characteristic function on the subset A Ω. In this section, we write J L (E) := L d J + E since we are dealing with one interval. We use an extended Minami estimate of the form E{χ {ω Xl (J L (E)) m +1}X l (J L (E))(X l (J L (E)) m ) 1} C M ( l L) 2d, as follows from [10].

8 8 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY Lemma 3.1. Under the condition that the projectors have uniform dimension m 1, we have P{X l (J L (E)) > m } C M ( l L) 2d. (3.2) Proof. Recalling that X l (J L (E)) {0} N, we have P{X l (J L (E)) > m } P{X l (J L (E)) m 1} = P{X l (J L (E))(X l (J L (E)) m ) 1} = P{χ {ω Xl (J L (E)) m +1}X l (J L (E))(X l (J L (E)) m ) 1} E{χ {ω Xl (J L (E)) m +1}X l (J L (E))(X l (J L (E)) m ) 1} C M ( l L) 2d, (3.3) by the extended Minami estimate Estimates on the joint probability. We return to considering two scaled intervals I L (E) and J L (E ), with E E. Because of (3.2), we have P{(X l (I L (E)) 1) (X l (J L (E )) 1)} P{(X l (I L (E)) m + 1) (X l (J L (E )) m + 1)} +P{(X l (I L (E)) m ) (X l (J L (E )) m + 1)} +P{(X l (I L (E)) m + 1) (X l (J L (E )) m )} +P{(X l (I L (E)) m ) (X l (J L (E )) m )} P{(X l (I L (E)) m ) (X l (J L (E )) m )} +C 0 ( l L) 2d. (3.4) The probability on the last line of (3.4) may be bounded above by P{(X l (I L (E)) m ) (X l (J L (E )) m )} m P{(X l (I L (E)) = 1 ) (X l (J L (E )) = 2 )}. (3.5) 1, 2 =1 Since m is independent of L, it suffices to estimate for 1 1, 2 m. P{(X l (I L (E)) = 1 ) (X l (J L (E )) = 2 )}, (3.6) 3.3. Reduction of length scale using localization. If we continue to wor at scale l = L α, we can prove Proposition (1.1) but only for α > 0 sufficiently small. In order to prove Proposition 1.1 for all 0 < α < 1, we must follow Klopp [9, section 2.2] and use localization in order to reduce the length scale to l := C log L, for C > 0. Once we wor with the length scale l, we will be able to prove Proposition 1.1 for all 0 < α < 1. The goal of this section is to bound (3.6) by a similar estimate involving the length scale l up to errors

9 DECORRELATION ESTIMATES FOR NON RANK ONE PERTURBATIONS 9 vanishing as L. The ey to this reduction is the localization properties of the Hamiltonians given in (Loc). Definition 3.1. We say that the local Hamiltonian H ω,l satisfies (Loc) in an interval I Σ if (1) The finite-volume fractional moment criteria of [1] holds on the interval I for some constant C > 0 sufficiently large; (2) There exists ν > 0 such that, for any p > 0, there exists q > 0 and a length scale l 0 > 0 such that, for all l > l 0, the following hold with probability greater than 1 L p : (a) If ϕ l j (ω) is a normalized eigenvector of H ω,l with eigenvalue Ej l (ω) I, and (b) x l j (ω) is a maximum of x ϕl j (ω) in Λ l, then, for n Λ l, one has ϕ l j(ω)(x) l q e ν x xl j (ω). (3.7) The point x l j (ω) is called a localization center for ϕl j (ω) or for El j (ω). The main consequence in the present context of (Loc) is the following result on the localization of eigenvectors. Lemma 3.2. [9, Lemma 2.2] Let Ω 0 be the set of configuration ω for which (Loc) holds with probability greater than 1 L 2d. There exists a covering of Λ l = γ Γ Λ l(γ), where Λ l(γ) = Λ l + γ, such that for all ω Ω 0 {ω (X l (I L (E)) = 1 ) (X l (J L (E )) = 2 )}, we have that either (1) There exist γ, γ Γ so that Λ l(γ) Λ l(γ ) = and X l(γ) (ĨL(E)) 1 and X l(γ ) ( J L (E )) 1, or (2) For some γ Γ, we have X 5 l(γ) (ĨL(E)) = 1 and X 5 l(γ) ( J L (E )) = 2. Because of the localization properties of the eigenfunctions given in Lemma 3.2, we can reduce the estimate on scale l to one on scale l as presented in the following lemma. Lemma 3.3. For any 1, 2 {1,..., m }, we have P{(X l (I L (E)) = 1 ) (X l (J L (E )) = 2 )} ( ) l 2d ( ) d + P{(X L l l (ĨL(E)) 5 l(γ) = 1 ) (X ( J 5 l(γ) L (E )) = 2 )}. (3.8) Proof. According to Lemma 3.2, we have P{(X l (I L (E)) = 1 ) (X l (J L (E )) = 2 )} P{(X l (I L (E)) = 1 ) (X l (J L (E )) = 2 ) Ω 0 } + P{Ω\Ω 0 } L 2d + P 1 + P 2, (3.9)

10 10 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY where, according to Lemma 3.2, P 1 is the probability that option (1) occurs and P 2 is the probability that option (2) occurs. To estimate P 1, we use the independence of the Hamiltonians associated with Λ l(γ) and Λ l(γ ), together with Wegner s estimate on scale l, to obtain ( ) 2d P 1 l l P{(X l(γ) (ĨL(E)) 1) (X l(γ )( J L (E )) 1)} ( ) 2d l l P{X l(γ) (ĨL(E)) 1} P{X l(γ ) ( J L (E )) 1} ( ) 2d CW l l 2 ( ĨL(E) l d )( ĨL(E) l d ) ( ) l 2d CW 2. (3.10) L In order to estimate P 2, condition (2) implies ( ) d P 2 P{(X l l (ĨL(E)) 5 l(γ) = 1 ) (X ( J 5 l(γ) L (E )) = 2 )}. (3.11) Bounds (3.9) (3.11) imply the bound (3.8) Key estimate on the log-scale. The proof of the next ey Proposition 3.1 follows the ideas in [9]. Proposition 3.1. For 1, 2 = 1,..., m and for any β > 5 2, there exists a scale L 0 > 0, so that for any L > L 0, there exists a constant C 0 > 0 so that, we have P{(X 5 l(γ) (ĨL(E)) = 1 ) (X 5 l(γ) ( J L (E K ρ 2 )) = 2 )} C m 2m 0 E 4d (C log L) (2+β)d L 2d. (3.12) Proof. 1. We begin with some observation concerning the eigenvalue averages. We let Ω 0 ( l, 1, 2 ) denote the event Ω 0 ( l, 1, 2 ) := {ω (X l((ĩl(e))) = 1 ) (X l(( J L (E ))) = 2 )} Ω 0, (3.13) for 1, 2 = 1,..., m, and where we write l instead of 5 l(γ) to simplify the notation. We define the subset Λ l Λ l by := {(i, i) i Λ l}. For each pair of sites (i, j) Λ l Λ l\, the Jacobian determinant of the mapping ϕ : (ω i, ω j ) (T l(e, 1 ), T l(e, 2 )), given by: J ij (T l(e, 1 ), T l(e, 2 )) := ω T l(e, i 1 ) T l(e, ωj 1 ) T l(e ωi, 2 ) T l(e ωj, 2 ). (3.14) As we will show in section 3.5, the condition J ij (T l(e, 1 ), T l(e, 2 )) λ(l) > 0 implies that the average of the eigenvalues in ĨL(E) and J L (E ) effectively vary independently with respect to any pair of independent random variables (ω i, ω j ), for i j. We define the following events for pairs (i, j) Λ l Λ l\ : Ω i,j 0 ( l, 1, 2 ) := Ω 0 ( l, 1, 2 ) {ω J ij (T l(e, 1 ), T l(e, 2 )) λ(l)}, (3.15)

11 DECORRELATION ESTIMATES FOR NON RANK ONE PERTURBATIONS 11 where λ(l) > 0 is given by λ(l) := ( E 4d)K 1 (C log L) βd, (3.16) where the exponent β > 0 is chosen below. Following Klopp [9, pg. 242], we note in section 3.5 that the positivity of the Jacobian determinant insures that the map ϕ, restricted to a certain domain, is a diffeomorphism. In particular, this allows us to compute P{Ω i,j 0 ( l, 1, 2 )} as in Lemma We next bound P{Ω 0 ( l, 1, 2 )} in terms of P{Ω i,j 0 ( l, 1, 2 )} using [9, Lemma 2.5]. This lemma states that for (u, v) (R + ) 2n normalized so that u 1 = v 1 = 1, we have max j u j v j u v 2 1 4n 5 u v 2 1. (3.17) Applying this with n = (2l + 1) d, and u = ω T (ω) and v = ω T (ω), and recalling the positivity property mentioned in point (2) following from (2.3) and the normalization in point (3) of section 2.1, we obtain from (3.17) and (2.7): max J ij (T l(e), T l(e )) 2 i j Λ l ( 2 3 ) ω l (T l(e) T l(e )) 2 5d 1 ( ) E 4d 2 ( ) 2 3. (3.18) K l 5d We partition the probability space as {ω J ij λ(l) some (i, j) Λ l Λ l\ } {ω J ij < λ(l) (i, j) Λ l Λ l\ }, where we write J ij for the Jacobian J ij (T l(e), T l(e )). Suppose that the second event {ω J ij < λ(l) (i, j) Λ l Λ l\ } occurs, so that from (3.18), we have: ( E 4d λ(l) 2 = K(C log L) βd Taing l = C log L, this implies that ) 2 max J ij (T l(e), T l(e )) 2 i j Λ l ( 2 3 l 5d ) ω (T l(e) T l(e )) 2 1. (3.19) ω (T l(e) T l(e )) 2 1 C 1 (C log L) (5 2β)d. (3.20) So, provided β > 5/2, we find that the bound (3.20) implies that the ω T l(e) is almost collinear with ω T l(e ). This contradicts the lower bound (2.7) as long as E 4d > 0. Consequently, the probability of the second event is zero. 3. It follows from this, the partition of the probability space, and Lemma 3.4 that P{Ω 0 ( l, 1, 2 )} (i,j) Λ l Λ l\ l 2d K ρ 2 C m 2m 0 E 4d P{Ω i,j 0 ( l, 1, 2 )} (C log L) βd L 2d. (3.21) With l = C log L, we find the probability P{Ω 0 ( l, 1, 2 )} is bounded as K ρ 2 P{Ω 0 ( l, 1, 2 )} C m 2m 0 E 4d (C log L) (2+β)d L 2d, (3.22)

12 12 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY Replacing l by 5 l(γ), changing the constant C 0, this completes the proof of Proposition 3.1. In summary, Proposition 3.1 shows that P{Ω 0 (5 l(γ), 1, 2 )} 0, as L 0. As a consequence of this and (3.9) (3.11), there exist constants C 0, C 1 > 0 such that for all L >> 0, P{(X l (I L (E)) = 1 ) (X l (J L (E )) = 2 )} ( ) 1 K ρ 2 C 1 L 1 α 2d + C m 2m (C log L) (1+β)d 0 E 4d L (2 α)d, (3.23) showing that P{(X l (I L (E)) = 1 ) (X l (J L (E )) = 2 )} 0 as L, for any integers 1, 2 = 1,..., m. This proves, up to the proof of the diffeomorphism property of ϕ, the main result (1.3) Estimate of P{Ω i,j 0 ( l, 1, 2 )}. Let Ω 0 ( l, 1, 2 ), 1, 2 = 1,..., m be the set of configurations described in (3.13). Similarly, for any pair of sites (i, j) Λ l Λ l\, the Jacobian determinant J ij (T l(e, 1 ), T l(e, 2 )) is defined in equation (3.14). We also defined events Ω i,j 0 ( l, 1, 2 ), for pairs (i, j) Λ l Λ l\, in (3.15): Ω i,j 0 ( l, 1, 2 ) := Ω 0 ( l, 1, 2 ) {ω J ij (T l(e, 1 ), T l(e, 2 )) λ(l)}, (3.24) where λ(l) > 0 has the value E 4d λ(l) :=, (3.25) K(C log L) dβ for some β > 5 2. We present an important technical lemma that is a simplification of [9, Lemma 2.6]. Lemma 3.4. For all L large, there is a finite constant C 0 > 0, independent of L, so that P{Ω ij 0 ( l, K ρ 2 1, 2 )} C m 2m (C log L) βd 0 E 4d L 2d, for any β > 5 2. Proof. 1. Since Ω ij 0 ( l, 1, 2 ) Ω is measurable and bounded, we can find a compact set K Ω ij 0 ( l, 1, 2 ) so that P{Ω i,j 0 ( l, 1, 2 )\K} C 1 L 2d. (3.26) We define the map ϕ : Ω i,j 0 ( l, 1, 2 ) R 2 by ϕ(ω i, ω j ) = (T l(e, 1, ω), T l(e, 2, ω)). This map is continuous so ϕ(k) ĨL(E) J L (E ) is compact. 2. For each p ϕ(k), we choose any element ω ij (p) K in the pre-image of p under ϕ 1 : ω ij (p) ϕ 1 (p) K Ω i,j 0 ( l, 1, 2 ). Because the Jacobian of ϕ is bounded below at each point of ϕ 1 (p), as follows from the definition of Ω i,j 0 ( l, 1, 2 ), the Inverse Function Theorem states that there are open balls

13 DECORRELATION ESTIMATES FOR NON RANK ONE PERTURBATIONS 13 U ωij (p) Ω i,j 0 ( l, 1, 2 ) and V p ĨL(E) J L (E ), with ω ij (p) U ωij (p) and p V p so that the restriction ϕ to U ωij (p) is a diffeomorphism with V p. As a consequence, the point ω ij (p) K is the unique point in U ωij (p) such that ϕ(ω ij ) = p so that such points are isolated points of K. It follows that ϕ 1 (p) is a discrete subset of K. 3. We can apply Harnac s Curve Theorem [7] in order to obtain an upper bound on the number of points in ϕ 1 (p). In section 2.3, we showed that T l ω ( 1 ) is a zero of a determinant constructed from the tensor product operator H 1. For fixed E R, this determinant defines the function f 1 ω, l E (ω i, ω j ) := det(h 1 E) that is a polynomial of degree at most ω, l m 1 in each random variable ω i and ω j. As such, the polynomial f 1 E (ω i, ω j ) may be extended to R 2. For two distinct energies e e, we consider the polynomial f 1, 2 e,e (ω 1, ω 2 ) := The Harnac Curve Theorem states that the is bounded [f 1 e (ω i, ω j )] 2 + [f 1 e (ω i, ω j )] 2. maximum number of connected components of the zero set of f 1, 2 e,e above by max(m 2 1, m 2 2 ). Each of those connected components lying in K is necessarily zero dimensional by the above diffeomorphism argument. Hence, since 1, 2 m, the number of points in the set ϕ 1 (p) in K is bounded above by m 2m, independent of L. 4. The sets {V p } p ϕ(k) cover ϕ(k). Since ϕ(k) is compact, there is a finite subcover {V pt } N t=1 so that ϕ(k) N t=1v pt ĨL(E) J L (E ). The restriction of ϕ to U ωij (p t), a diffeomorphism with V pt, is denoted by ϕ pt. We tae intersections and relative complements to obtain a finite collection {W m } of disjoint sets so that ϕ(k) Ñm=1W m = N t=1v pt, up to a set of Lebesgue measure zero, and where Ñ is a function of N, and each W m V pt, for some index t. We can choose W m so that it is in the domain of ϕ 1 p t. 5. We compute the P-measure of Ω i,j 0 ( l, 1, 2 ) by first computing the measure of ϕp 1 t (W m ): P{ϕ 1 p t (W m )} = ρ(ω i )ρ(ω j ) dω i dω j. (3.27) ϕ 1 p t (W m) Upon changing variables, we obtain ϕ 1 p t (W m) ρ(ω i )ρ(ω j ) dω i dω j Jacϕ 1 p t ρ 2 W m de de. (3.28) It follows from (3.25) that the Jacobian Jacϕ 1 p t satisfies the bound Jacϕ 1 p t (T l(ω ij ), T l (ω ij)) K(C log L)βd E 4d. (3.29)

14 14 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY Using this bound, the fact that the W m are disjoint, and the fact that there are at most m 2m isolated points in ϕ 1 (p), for p ϕ(k), we find ( ) Ñ P{K} ρ 2 m 2m max n Jacϕ 1 n W m Finally, we have K(C log L)βd E 4d K ρ 2 m 2m E 4d m=1 ρ 2 m 2m ĨL(E) J L (E ) (C log L) βd L 2d (3.30) P{Ω i,j 0 ( l, 1, 2 )} P{K} + P{Ω i,j 0 ( l, 1, 2 )\K}, (3.31) so the result follows from (3.30) and the fact that P{Ω i,j 0 ( l, 1, 2 )\K} is O(L 2d ). 4. Asymptotically independent random variables: Proof of Theorem 1.1 In this section, we give the proof of Theorem 1.1. To prove that ξe ω (I) and ξe ω (J) are independent, we recall that the limit points ξ ω E are the same as those obtained from a certain uniformly asymptotically negligible array ([10, Proposition 4.4]). To obtain this array, we construct a cover of Λ L by nonoverlapping cubes of side length 2l + 1 centered at points n p. We use l = [L α ], where (α, β) satisfy 0 < α < 1 and β > 5 2. The number of such cubes Λ l(n p ) is N L := [(2L + 1)/(2l + 1)] d. The local Hamiltonian is Hp,l ω. The associated eigenvalue point process at energy E is denoted by ηe,l,p ω. We define the point process ζe,λ ω L = N L p=1 ηω E,p,l. For a bounded interval I R, we define the local random variable ηe,l,p ω (I) := Tr(E Hp,l ω (I L(E))) and similarly ηe ω,l,p(j) for the scaled interval J L (E ). For p p, the random variables ηe,l,p ω (I) and ηω E,l,p (J) are independent for any energies E and E and any bounded intervals I and J. We compute P{(ζ ω E,Λ L (I) 1) (ζ ω E Λ L (J) 1)} = = N L p,p =1 N L p,p =1 where the error term is just the diagonal p = p contribution: E L (E, E, I, J) = N L p=1 P{(ηE,l,p ω (I) 1) (ηω E,l,p (J) 1)} P{ηE,l,p ω (I) 1}P{ηω E,l,p (J) 1} +E L (E, E, I, J), (4.1) [ P{(η ω E,l,p (I) 1) (ηe ω,l,p(j) 1)} P{η ω E,l,p (I) 1}P{ηω E,l,p (J) 1}]. (4.2)

15 DECORRELATION ESTIMATES FOR NON RANK ONE PERTURBATIONS 15 If we now assume that E E > 4d and E, E Σ CL, then the first term on the right side of (4.2) is bounded above by L d (log L) (1+β)d due to the decorrelation estimate (1.3). The bound on the second probability on the right of (4.2) is C 2 W L 2d(1 α). It is obtained from the square of the Wegner estimate P{η ω E,l,p (J) 1} C W (l/l) d = C W L d(1 α). is bounded Since N L (L/l) d = L (1 α)d, we find that the second term on the right of (4.2) above by CW 2 L d(1 α). Consequently, the error term E L (E, E, I, J) 0 as L. Since the set of limit points ζ ω and ξ ω are the same [10], this estimate proves that lim L P{(ζω E,Λ L (I) 1) (ζe ω,λ L (J) 1)} = P{ξE(I) ω 1}P{ξE ω (J) 1}, (4.3) establishing the asymptotic independence of the random variables ξe ω (I) and ξe ω (J) provided E E > 4d. 5. Bounds on eigenvalue multiplicity The extended Minami estimate may be used with the Klein-Molchanov argument [8] to bound the multiplicity of eigenvalues in the localization regime. The basic argument of Klein-Molchanov is the following. If H ω has at least m + 1 linearly independent eigenfunctions with eigenvalue E in the localization regime, so that the eigenfunctions exhibit rapid decay, then any finite volume operator H ω,l must have at least m + 1 eigenvalues close to E for large L. But, by the extended Minami estimate, this event occurs with small probability. The first lemma is a deterministic result based on perturbation theory. Lemma 5.1. Suppose that E σ(h) is an eigenvalue of a self adjoint operator H with multiplicity at least m + 1. Suppose that all the associated eigenfunctions decay faster than x σ, for some σ > d/2 > 0. We define ɛ L := CL σ+ d 2. Then for all L >> 0, the local Hamiltonian H L := χ ΛL Hχ ΛL has at least m +1 eigenvalues in the interval [E ɛ L, E + ɛ L ]. Proof. 1. Let {ϕ j j = 1,..., M} be an orthonormal basis of the eigenspace for H and eigenvalue E. We assume that the eigenvalue multiplicity M m + 1. We define the local functions ϕ j,l := χ ΛL ϕ j, for j = 1,..., M. These local functions satisfy: 1 ɛ L ϕ j,l 1, ϕ i,l, ϕ j,l ɛ L, i j. (5.1) It is easy to chec that these conditions imply that the family is linearly independent. Let V L denote the M-dimensional subspace of l 2 (Λ L ) spanned by these functions. 2. As in [8], it is not difficult to prove that the functions ϕ j,l are approximate eigenfunctions for H L : (H L E)ϕ j,l ɛ L ϕ j,l. (5.2) Furthermore, for any ψ L V L, we have (H L E)ψ L 2ɛ L ψ L.

16 16 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY 3. Let J L := [E 3ɛ L, E + 3ɛ L ]. We write P L for the spectral projector P L := χ JL (H L ) and Q L := 1 P L is the complementary projector. For any ψ V L, we have Q L ψ (3ɛ L ) 1 (H L E)Q L ψ (2/3) ψ. Since P L ψ 2 = ψ 2 Q L ψ 2 (5/9) ψ, it follows that P L : V L l 2 (Λ L ) is injective. Consequently, we have dim Ran P L = Tr(P L ) dim V L = M > m. Redefining the constant C > 0 in the definition of ɛ L, we find that H has at least m + 1 eigenvalues in [E ɛ L, E + ɛ L ]. The second lemma is a probabilistic one and the proof uses the extended Minami estimate. Lemma 5.2. Let I R be a bounded interval. For q > 2d, and any interval J I with J L q, we define the event E L,I,q := {ω Tr(χ J (H ω,l )) m J I, J L q }. (5.3) Then, the probability of this event satisfies P{E L,I,q } 1 C 0 L 2d q. (5.4) Proof. We cover the interval I by 2([L q I /2] + 1) subintervals of length 2L q so that any subinterval J of length L q is contained in one of these. We then have P{E c L,I,q} (L q I + 2)P{χ J (H ω,l ) > m }. (5.5) The probability on the right side is estimated from the extended Minami estimate so that This establishes (5.4). P{χ J (H ω,l ) > m } C M (L q L d ) 2 = C M L 2(d q), (5.6) P{E c L,I,q} C M (L q I + 2)L 2(d q) = C M ( I + 1)L 2d q. (5.7) Theorem 5.1. Let H ω be the generalized Anderson Hamiltonian described in section 1 with perturbations P i having uniform ran m. Then the eigenvalues in the localization regime have multiplicity at most m with probability one. Proof. We consider a length scale L = 2. It follows from (5.4) that the probability of the complementary event EL c,i,q is summable. By the Borel- Cantelli Theorem, that means for almost every ω there is a (q, ω) so that for all > (q, ω) the event E L,I,q occurs with probability one. Let us suppose that H ω an eigenvalue with multiplicity at least m + 1 in an interval I and that the corresponding eigenfunctions decay exponentially. Then, by Lemma 5.1, the local Hamiltonian H ω,l has at least m + 1 eigenvalues in the interval [E ɛ L, E + ɛ L ] where ɛ L = CL (β d 2 ), for any β > 5d/2. This contradicts the event E L,I,q which states that there are no more than m eigenvalues in any subinterval J I with J L q since we can find q > 2d so that β q 2 > q.

17 DECORRELATION ESTIMATES FOR NON RANK ONE PERTURBATIONS 17 Further investigations on the simplicity of eigenvalues for Anderson-type models may be found in the article by Naboo, Nichols, and Stolz [17], Mallic [12], Mallic and Krishna [13], and Mallic and Narayanan [14]. Mallic [12] proves that the singular spectrum is simple for a class of Anderson models with higher ran perturbation extending the results of Naboo, Nichols, and Stolz [17]. Mallic and Krishna [13] prove that, for higher ran Anderson models with the single site potential having support in the whole real line, the Minami estimate implies simplicity of the pure point spectrum away from the continuous spectrum. They a also show that in the case of higher multiplicity spectrum the spectral statistics cannot be Poisson but must be compound Poisson. Mallic and Narayanan [14] prove that higher ran models on some graphs have eigenvalues of higher multiplicity. 6. Decorrelation estimates for the discrete alloy-type model In this section, we prove decorrelation estimates for the nonsign definite alloy model studied by Tautenhahn and Veseli`c [19]. As above, these imply the asymptotic independence of local eigenvlaue statistics associated with two energies in the localization regime sufficiently far apart. The discrete random Schrödinger operator acting on l 2 (Z d ) is described by H ω = L + V ω, (6.1) where L is the finite-difference Laplacian, and the random potential V ω is defined by V ω (m) := n Z d ω n a m n. (6.2) The potentials at two sites, V ω (m) and V ω (n), are independent only if n m > diam a. Furthermore, the ran of V ω (m) is supp a. The single-site potential a and random variables (ω ) satisfy the following hypotheses. Hypothesis 1. The single-site potential a is a real, compactly supported function a : Z d R with a 0 > 0 satisfying the condition 0 < a n a 0. (6.3) n Z d \{0} Given a single-site potential a, we define a parameter δ 0 by: a m δ := m 0 a 0 < 1, (6.4) Hypothesis 2. The single-site potential a is such that the parameter δ > 0. The Fourier transform â : T d = [0, 2π) d C, is defined by â(θ) := e iθ a, θ T d, Z d

18 18 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY Hypothesis 3. The Fourier transform â of the single-site potential a is never zero: â(θ) 0, for all θ T d. Hypothesis 4. The family of random variables (ω m ) are iid random variables with a common, compactly supported density ρ W 2,1 (R) with support ρ [ M, M], for some 0 < M <. We note that the usual ran one Anderson model corresponds to a m = a 0 δ m0 so a is supported at a single point and δ = 0. In the case considered here, we will always assume that δ > 0 and the single site potential a has compact support. In particular, there is no restriction on the sign of the terms a m. Let us write m := n a n a 0 (1 δ) > 0. (6.5) It follows from standard methods that the almost sure spectrum of H ω is equal to [ 2d, 2d] + m supp ω 0. In particular, the almost sure spectrum is a union of intervals and contains at least two intervals I 1, I 2 such that dist(i 1, I 2 ) > 4d + mc, for some 0 < c 2M, only depending on supp ω 0. We always assume that the constant (M, c, δ) satisfy the condition cm 1 (1 δ) 2 2δ(1 + δ) > 0. (6.6) Under this condition, we extract from [19, Corollary 3.4] the following Minami estimate (M): There exists C > 0 such that for all interval I R, we have P (Tr X l (I) 2) C I 2 l 2 (6.7) Although not explicitly stated in [19], the Minami estimate (6.7) and the method of Klein-Molchanov [8], presented in section 5, allow us to prove that the eigenvalues of the alloy model (6.1) (6.2) are almost surely simple. So although the ran of a is greater than one, the standard Minami estimate holds implying simplicity of the eigenvalues in the localization regime and Poisson statistics. We now turn to the proof of the decorrelation estimates, Proposition 1.1, for the random alloy model assuming (6.6). Because of the Minami estimate (6.7), we may tae m = 1. We tae E, E Σ CL and such that E E > 4d. We may restrict ourselves to those configurations ω such there is one eigenvalue in I L (E) and one in J L (E ) and such that the distance with the rest of the spectrum of H ω,l is greater than (L log L) d. By the Wegner estimate, this is possible with probability greater than 1 (l/(l log L)) d. Let us write Ej l(ω) and El (ω) these two eigenvalues with normalized eigenvectors u l j and ul. We note that ul j (m), ul (m) = 0, if m / Λ l. The results of section 2.1 hold with 1 = 1, 2 = 1, and T l (E, 1, ω) = Ej l(ω) and T l (E, 1, ω) = E l(ω). The first main difference appears in the variational formulas of section 2.2, in particular, the lower bound (2.7). In the alloy case, the gradients of the eigenvalues are not normalized. We prove the following lower bound:

19 DECORRELATION ESTIMATES FOR NON RANK ONE PERTURBATIONS 19 Lemma 6.1. There exists a finite constant K > 0, depending only on M = sup ω 0, and δ defined in (6.4), such that ω Ej l (ω) ω Ej l(ω) ωe l(ω) 1 ω E l(ω) K 1 1 Proof. By the Feynman-Hellmann formula we have ωn Ej(ω) l = a m u l j(m + n) 2 (6.8) m Z d from which it follows that ωn Ej(ω) l a 0 u l j(n) 2 a m u l j(n + m) 2. (6.9) This implies that the L 1 -norm of the gradient of Ej l (ω) satisfies ω Ej(ω) l 1 a 0 a m. (6.10) Therefore, one has and n Z d m 0 m 0 a m u l j(n + m) 2 m 0 a 0 (1 δ) ω E l j(ω) 1, ω E l (ω) 1 a 0 (1 + δ), (6.11) ω Ej(ω) l 1 ω E l (ω) 1 2 a m 2δa 0. (6.12) m 0 It also follows from the Feynman-Hellmann formula that so that ω ( ω E l j(ω) ω E l (ω)) = ([ El j(ω)]u j, u j ) ([ E l (ω)]u, u ) = ( u j, u j ) ( u, u ) + (E l (ω) El j(ω)), M ω E l j(ω) ω E l (ω) 1 ω ( ω E l j(ω) ω E l (ω)) E E 4d > mc, where m > 0 is defined in (6.5). We can now finally estimate ω Ej l(ω) ω Ej l(ω) ωe l(ω) 1 ω E l(ω) 1 = ωe l(ω) 1 ω Ej l(ω) ωej l(ω) 1 ω E l(ω) ω Ej l(ω) 1 ω E l(ω) 1 ] ω E l(ω) 1 [ ω Ej l(ω) ωe l(ω) + [ ω E l(ω) ] 1 ω E j 1 ω E l(ω) = ω Ej l(ω) 1 ω E l(ω), 1 (6.13)

20 20 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY so that ω Ej l (ω) ω Ej l(ω) ωe l(ω) 1 ω E l(ω) 1 1 ωej l(ω) ωe l(ω) 1 ω E l(ω) 1 ω E l(ω) 1 ω Ej l(ω) 1 ω E l(ω) 1 ω Ej l(ω) 1 ω E l(ω) 1 mcm 1 (1 δ) 2δ(1 + δ)a 0 (1 + δ) 2 a 0 cm 1 (1 δ) 2 2δ(1 + δ) (1 + δ) 2 > 0, (6.14) giving an explicit formula for the constant K > 0 in the lemma. We also compute ωn Ej(ω) l = ωn E l (ω) = m > 0, (6.15) n Z d n Z d for the constant m > 0 defined in (6.5), and n Z d ωn E l j (ω) ωn E l j (ω) 1 + ω n E l (ω) ω E l (ω) 1 2m 2. (6.16) a 0 (1 δ) Therefore, it follows that ω Ej l (ω) ω Ej l(ω) + ωe l(ω) 1 ω E l(ω) 2. (6.17) 1 1 To complete the proof of the decorrelation estimate (1.3), we note that the reduction of section 3 holds for the alloy-type model. It remains for us to prove the analog of Proposition 3.1 for the alloy-type model. Proposition 6.1. Let E, E Σ CL be two distinct energies with E E > 4d. For any bounded intervals I, J R, we define I L (E) := L d I + E and J L (E ) := L d J + E, as above. We write X l (I L (E) := TrE Hω,l (I L (E)), and similarly X l (J L (E ). Then, for any β > 5 2, there exists a scale L 0 > 0, so that for any L > L 0, there exists a constant C 0 > 0 so that P{(X l (I L (E)) = 1) (X l (J L (E K ρ 2 (C log L) (2+β)d )) = 1)} C 0 E 4d L 2d. (6.18) Lemma 6.1 allows us to write the analog of ( ) 2 max J ij (Ej(ω), l E l 3 ω E i j Λ (ω))2 j l (ω) l l 5d ω Ej l(ω) ωe l(ω) 2 1 ω E l(ω) 1 1 ( ) 2 K 2 3, (6.19) where K > 0 is the constant defined in (6.14). Consequently, an estimate of the form (3.20) holds for the alloy-type model, and the probability that the normalized gradients are collinear is zero. l 5d

21 DECORRELATION ESTIMATES FOR NON RANK ONE PERTURBATIONS 21 With regard to Lemma 3.4, we mention that because of the support of the single-site function a, the determinant f E (ω j, ω ) = det(h ω,l E) is a polynomial of degree supp a in each random variable ω j and ω. Hence, the Harnac Curve Theorem states that the number of connected components in the zero set of f E is bounded above by supp a 2. By the argument in the proof of Lemma 3.4, the number of points in ϕ 1 (p), for any p K, is bounded above by supp a 2. As this number is independent of L, the proof concludes as in section 3.2. References [1] M. Aizenman, J. H. Schener, R. M. Friedrich, and D. Hundertmar, Finite-volume fractional-moment criteria for Anderson localization, Comm. Math. Phys., 224(2001), no. 1, (2001). [2] D. Applebaum, Lévy processes and stochastic calculus, Cambridge Studies in Advanced Mathematics 116, second edition, Cambridge: Cambridge University Press, [3] J. Bertoin, Lévy processes, Cambridge: Cambridge University Press, [4] Dhriti Ranjan Dolai, M. Krishna Poisson statistics in the Anderson model with singular randomness, Journal of Ramanujan Mathematical Society, Vol. 30(3), (2015). [5] F. Germinet, A. Klein, New characterizations of the region of complete localization for random Schrdinger operators, J. Stat. Phys. 122 (2006), no. 1, [6] F. Germinet, F. Klopp, Spectral statistics for the random Schrödinger operators in the localized regime, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 9, ; arxiv: [7] Axel Harnac, Ueber die Vieltheiligeit der ebenen algebraischen Curven, Math. Ann. 10(2), , MR [8] A. Klein, S. Molchanov, Simplicity of eigenvalues in the Anderson model, J. Stat. Phys. 122 (2006), no. 1, [9] F. Klopp, Decorrelation estimates for the eigenlevels of the discrete Anderson model in the localized regime, Comm. Math. Phys. 303 (2011), no. 1, [10] P. D. Hislop, M. Krishna, Eigenvalue statistics for random Schrödinger operators with non ran one perturbations, Comm. Math. Phys. 340 (2015), no. 1, ; arxiv: [11] P. D. Hislop, M. Krishna, Decorrelation estimates for random Schrdinger operators with non ran one perturbations, arxiv: [12] A. Mallic, Multiplicity bound of singular spectrum for higher ran Anderson models, J. Funct. Anal. 272 (2017), no. 12, [13] A. Mallic, M. Krishna, Global multiplicity bounds and Spectral Statistics Random Operators, arxiv: [14] A. Mallic, P. A. Narayanan, On multiplicity of some Anderson type operators with higher ran perturbation, arxiv: [15] N. Minami, Local fluctuation of the spectrum of a multidimensional Anderson tightbinding model, Commun. Math. Phys. 177 (1996), [16] S. A. Molchanov, The local structure of the spectrum of one-dimensional Schrödinger operator, Commun. Math. Phys. 78 (1981), [17] S. Naboo, R. Nichols, G. Stolz, Simplicity of eigenvalues in Anderson-type models, Ar. Mat. 51 (2013), no. 1, [18] C. Shirley, Decorrelation estimates for random discrete Schrödinger operators in one dimension and applications to spectral statistics, J. Stat. Phys. 158 (2015), no. 6, ; arxiv: v1. [19] M. Tautenhahn, I. Veselić, Minami s estimate: Beyond ran one perturbation and monotonicity, Ann. Henri Poincaré 15 (2014),

22 22 P. D. HISLOP, M. KRISHNA, AND C. SHIRLEY [20] T. P. Trinh, Decorrelation estimates for a 1D tight binding model in the localized regime, Ann. Henri Poincaré 15 (3)(2014), Department of Mathematics, University of Kentucy, Lexington, Kentucy , USA address: peter.hislop@uy.edu Ashoa University, Rai, Haryana, India address: rishna.maddaly@ashoa.edu.in Université Libre de Bruxelles, Belgique address: cshirley@ulb.ac.be