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We study the local eigenvalue statistics ξNω,E<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup></math>&nbsp;associated with the eigenvalues of one-dimensional, (2N+1)×(2N+1)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math>&nbsp;random band matrices with independent, identically distributed, real random variables and band width growing as Nα<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>N</mi><mi>α</mi></msup></math>, for 0&lt;α&lt;12<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>. We consider the limit points associated with the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>, for I⊂R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>⊂</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math>, and E∈(−2,2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∈</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></math>. For random band matrices with Gaussian distributed random variables and for 0≤α&lt;17<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>7</mn></mfrac></math>, we prove that this family of random variables has nontrivial limit points for almost every E∈(−2,2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∈</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></math>, and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>&nbsp;and associated quantities related to the intensities, as N tends towards infinity, and employs known localization bounds of (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker in Commun Math Phys 290:1065–1097, 2009), and the strong Wegner and Minami estimates (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019). Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for 0&lt;α&lt;12<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>, we prove that any nontrivial limit points of the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>&nbsp;are distributed according to Poisson distributions.We study the local eigenvalue statistics ξNω,E<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup></math>&nbsp;associated with the eigenvalues of one-dimensional, (2N+1)×(2N+1)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math>&nbsp;random band matrices with independent, identically distributed, real random variables and band width growing as Nα<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>N</mi><mi>α</mi></msup></math>, for 0&lt;α&lt;12<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>. We consider the limit points associated with the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>, for I⊂R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>⊂</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math>, and E∈(−2,2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∈</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></math>. For random band matrices with Gaussian distributed random variables and for 0≤α&lt;17<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>7</mn></mfrac></math>, we prove that this family of random variables has nontrivial limit points for almost every E∈(−2,2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∈</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></math>, and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>&nbsp;and associated quantities related to the intensities, as N tends towards infinity, and employs known localization bounds of (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker in Commun Math Phys 290:1065–1097, 2009), and the strong Wegner and Minami estimates (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019). Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for 0&lt;α&lt;12<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>, we prove that any nontrivial limit points of the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>&nbsp;are distributed according to Poisson distributions.

Department of Mathematics

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Journal of Statistical Physics

On the Local Eigenvalue Statistics for Random Band Matrices in the Localization Regime

In this paper, we consider Schrödinger operators on M×Zd2<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>×</mo><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub></mrow></msup></math>, with M={M1,…,M2}d1<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>M</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>M</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><msup><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub></mrow></msup></math>&nbsp;(‘quantum wave guides’) with a ‘Γ<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Γ</mi></math>-trimmed’ random potential, namely a potential which vanishes outside a subset Γ<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Γ</mi></math>&nbsp;which is periodic with respect to a sub-lattice. We prove that (under appropriate assumptions) for strong disorder these operators have pure point spectrum outside the set Σ0=σ(H0,Γc)<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mtext>Σ</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mtext>σ</mtext></mrow><mo stretchy="false">(</mo><msub><mi>H</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>,</mo><msup><mi mathvariant="normal">Γ</mi><mrow class="MJX-TeXAtom-ORD"><mi>c</mi></mrow></msup></mrow></msub><mo stretchy="false">)</mo></math>&nbsp;where H0,Γc<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>,</mo><msup><mi mathvariant="normal">Γ</mi><mrow class="MJX-TeXAtom-ORD"><mi>c</mi></mrow></msup></mrow></msub></math>&nbsp;is the free (discrete) Laplacian on the complement Γc<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">Γ</mi><mrow class="MJX-TeXAtom-ORD"><mi>c</mi></mrow></msup></math>&nbsp;of Γ<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Γ</mi></math>. We also prove that the operators have some absolutely continuous spectrum in an energy region E⊂Σ0<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">E</mi></mrow></mrow><mo>⊂</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext>Σ</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>. Consequently, there is a mobility edge for such models. We also consider the case −M1=M2=∞<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><msub><mi>M</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>M</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">∞</mi></math>, i.e.&nbsp; Γ<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Γ</mi></math>-trimmed operators on Zd=Zd1×Zd2<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow></msup><mo>=</mo><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub></mrow></msup><mo>×</mo><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub></mrow></msup></math>. Again, we prove localisation outside Σ0<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mtext>Σ</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>&nbsp;by showing exponential decay of the Green function GE+iη(x,y)<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mi>E</mi><mo>+</mo><mi>i</mi><mi>η</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math>&nbsp;uniformly in η&gt;0<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi><mo>&gt;</mo><mn>0</mn></math>. For all energies E∈E<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">E</mi></mrow></mrow></math>&nbsp;we prove that the Green’s function GE+iη<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mi>E</mi><mo>+</mo><mi>i</mi><mi>η</mi></mrow></msub></math>&nbsp;is not (uniformly) in ℓ1<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ℓ</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msup></math>&nbsp;as η<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi></math>&nbsp;approaches 0. This implies that neither the fractional moment method nor multi-scale analysis can be applied here.

Annales Henri Poincare

Localisation and Delocalisation for a Simple Quantum Wave Guide with Randomness

We prove decorrelation estimates for generalized lattice Anderson models on Zd constructed with finite-rank perturbations in the spirit of Klopp [12]. These are applied to prove that the local eigenvalue statistics ! ωE and ! ωE0 , associated with two energies E and E0 in the localization region and satisfying jE E0j > 4d, are independent. That is, if I; J are two bounded intervals, the random variables ! ωE.I / and ! ωE0.J /, are independent and distributed according to a compound Poisson distribution whose Levy 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 rank of the perturbation. The method of proof contains new ingredients that simplify the proof of the rank one case [12, 19, 21], extends tomodels forwhich the eigenvalues are degenerate, and applies tomodels forwhich the potential is not sign definite [20] in dimensions d > 1. © 2021 European Mathematical Society.

Journal of Spectral Theory

Decorrelation estimates for random Schrödinger operators with non rank one perturbations

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on . We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular, we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory. In particular, for higher rank Anderson models with pure point spectrum, with the randomness having support equal to , there is a uniform lower bound on spectral multiplicity and in case this is larger than one, the local statistics is not Poisson. © 2020 World Scientific Publishing Company.

Reviews in Mathematical Physics

Global multiplicity bounds and spectral statistics for random operators

In this paper we solve a long standing open problem for Random Schrödinger operators on L2(Rd) with i.i.d single site random potentials. We allow a large class of free operators, including magnetic potential, however our method of proof works only for the case when the random potentials satisfy a complete covering condition. We require that the supports of the random potentials cover Rd and the bump functions that appear in the random potentials form a partition of unity. For such models, we show that the Density of States (DOS) is m times differentiable in the part of the spectrum where exponential localization is valid, if the single site distribution has compact support and has Hölder continuous m+ 1 st derivative. The required Hölder continuity depends on the fractional moment bounds satisfied by appropriate operator kernels. Our proof of the Random Schrödinger operator case is an extensions of our proof for Anderson type models on ℓ2(G) , G a countable set, with the property that the cardinality of the set of points at distance N from any fixed point grows at some rate in Nα, α&gt; 0. This condition rules out the Bethe lattice, where our method of proof works but the degree of smoothness also depends on the localization length, a result we do not present here. Even for these models the random potentials need to satisfy a complete covering condition. The Anderson model on the lattice for which regularity results were known earlier also satisfies the complete covering condition. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

Communications in Mathematical Physics

Regularity of the Density of States of Random Schrödinger Operators

We prove that the local eigenvalue statistics at energy E in the localization regime for Schrödinger operators with random point interactions on Rd, for d = 1, 2, 3, is a Poisson point process with the intensity measure given by the density of states at E times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schrödinger operators in the continuum. The special structure of resolvent of Schrödinger operators with point interactions facilitates the proof of the Minami estimate for these models. © 2020 Author(s).

Journal of Mathematical Physics

Eigenvalue statistics for Schrödinger operators with random point interactions on R d, d = 1, 2, 3

In this paper we consider Anderson models with a large number of sites with zero interaction. For such models we study the spectral statistics in the region of complete localization. We show that Poisson statistics holds for such energies, by proving the Minami estimate. © European Mathematical Society.

Spectral statistics for anderson models with sporadic potentials

In this paper, we prove a Szegö type limit theorem on ℓ 2 (Z d ). We take the self-adjoint operator H= - Δ + V on ℓ 2 (Z d ) , where (Δ u) (n) = ∑ | n - k | = 1 (u(k) - u(n)) and the operator V is the multiplication by a positive sequence { V(n) , n∈ Z d } with V(n) → ∞ as | n| → ∞. We take the orthogonal projection π λ onto the subspace, in ℓ 2 (Z d ) , spanned by eigenfunctions of H with eigenvalues ≤ λ. Let B be a zeroth order self-adjoint pseudo-difference operator associated with symbol b∈ S 1 , , ∞ (T d × Z d ). We then show for “nice functions” f, that limλ→∞Tr(f(πλBπλ))Tr(πλ)=limλ→∞1(2π)d∑V(n)≤λ∫Tdf(b(x,n))dx∑V(n)≤λ1. © 2018, Springer Nature Switzerland AG.