arXiv subjects

M. Krishna

Publications and source records attributed to M. Krishna.

17 recordsLinked to original sources

arXiv2025

Dynamical localization and delocalization for random Schrodinger operators with $\delta$-interactions in $\mathbb{R}^3$

We prove that the random Schrodinger operators on $\mathbb{R}^3$ with independent, identically distributed random variables and single-site potentials given by $\delta$-functions on $\mathbb{Z}^3$, exhibit both dynamical localization and dynamical delocalization with probability one. That is, there are regions in the deterministic spectrum that exhibit dynamical localization, the nonspreading of wave packets, and regions in the deterministic spectrum where the models also exhibit nontrivial quantum transport, almost surely. These models are the first examples of ergodic, random Schrodinger operators exhibiting both dynamical localization and delocalization in dimension three or higher. The nontrivial transport is due to the presence of delocalized generalized eigenfunctions at positive energies $E > \pi^2$. The general idea of the proof follows [Hislop, Kirsch, Krishna (2024)] in which lower bounds on moments of the position operator are constructed using these generalized eigenfunctions. A new result of independent interest is a proof of the Combes-Thomas estimate on exponential decay of the Green's function for Schrodinger operators with infinitely-many $\delta$-potentials.

math-ph↗

arXiv2024

Decorrelation in Local Statistics for random operators

In this paper we study the local spectral statistics in the localised region of various random operator models, including the $d$-dimensional the Anderson model and random Schr\"odinger operators. It is already established, in the above models, that at an energy $E$, in the localised energy region of the spectrum, where the density of states $n(E) > 0$, the local eigenvalue statistics $X_E$ is a Poisson processes with intensity $n(E) \mathcal{L}$, $\mathcal{L}$ being the Lebesgue measure on $\mathbb{R}$. The question of independence of $X_E, X_{E^\prime}$ for distinct energies was partially solved in the literature. We solve it completely for all the models for which the Minami technique works.

math.SP↗

arXiv2024

Eigenfunctions and Quantum Transport with Applications to Trimmed Schrodinger Operators

We provide a simple proof of dynamical delocalization, that is, time-increasing lower bounds on quantum transport for discrete, one-particle Schrodinger operators on $\ell^2 (\mathbb{Z}^d)$, provided solutions to the Schrodinger equation satisfy certain growth conditions. The proof is based on basic resolvent identities and the Combes-Thomas estimate on the exponential decay of the Green's function. As a consequence, we prove that generalized eigenfunctions for energies outside the spectrum of $H$ must grow exponentially in some directions. We also prove that if $H$ has any absolutely continuous spectrum, then the Schrodinger operator exhibits dynamical delocalization. We apply the general result to $\Gamma$-trimmed Schrodinger operators, with periodic $\Gamma$, and prove dynamical delocalization for these operators. These results also apply to the $\Gamma$-trimmed Anderson model, providing a random, ergodic model exhibiting both dynamical localization in an energy interval and dynamical delocalization.

math-ph↗

arXiv2021

On the local eigenvalue statistics for random band matrices in the localization regime

We study the local eigenvalue statistics $\xi_{\omega,E}^N$ associated with the eigenvalues of one-dimensional, $(2N+1) \times (2N+1)$ random band matrices with independent, identically distributed, real random variables and band width growing as $N^\alpha$, for $0 < \alpha < \frac{1}{2}$. We consider the limit points associated with the random variables $\xi_{\omega,E}^N [I]$, for $I \subset \mathbb{R}$, and $E \in (-2,2)$. For Gaussian distributed random variables with $0 \leq \alpha < \frac{1}{7}$, we prove that this family of random variables has nontrivial limit points for almost every $E \in (-2,2)$, 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 $\xi_{\omega,E}^N [I]$ and associated quantities related to the intensities, as $N$ tends towards infinity, and employs known localization bounds of \cite{schenker, peled, et. al.}, and the strong Wegner and Minami estimates \cite{peled, et. al.}. 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 $0 < \alpha < \frac{1}{2}$, we prove that any nontrivial limit points of the random variables $\xi_{\omega,E}^N [I]$ are distributed according to Poisson distributions.

math-ph↗

arXiv2020

Analyticity of density of states for the Cauchy distribution

We compute the density of states for the Cauchy distribution for a large class of random operators and show it is analytic in a strip about the real axis.

math-ph↗

arXiv2020

Quantum Lattice Wave Guides with Randomness -- Localisation and Delocalisation

In this paper we consider Schr\"{o}dinger operators on $M \times \mathbb{Z}^{d_2}$, with $M=\{M_{1}, \ldots, M_{2}\}^{d_1}$ (`quantum wave guides') with a `$\Gamma$-trimmed' random potential, namely a potential which vanishes outside a subset $\Gamma$ which is periodic with respect to a sub lattice. We prove that (under appropriate assumptions) for strong disorder these operators have \emph{pure point spectrum } outside the set $\Sigma_{0}=\sigma(H_{0,\Gamma^{c}})$ where $H_{0,\Gamma^{c}} $ is the free (discrete) Laplacian on the complement $\Gamma^{c} $ of $\Gamma $. We also prove that the operators have some \emph{absolutely continuous spectrum} in an energy region $\mathcal{E}\subset\Sigma_{0}$. Consequently, there is a mobility edge for such models. We also consider the case $-M_{1}=M_{2}=\infty$, i.~e.~ $\Gamma $-trimmed operators on $\mathbb{Z}^{d}=\mathbb{Z}^{d_1}\times\mathbb{Z}^{d_2}$. Again, we prove localisation outside $\Sigma_{0} $ by showing exponential decay of the Green function $G_{E+i\eta}(x,y) $ uniformly in $\eta>0 $. For \emph{all} energies $E\in\mathcal{E}$ we prove that the Green's function $G_{E+i\eta} $ is \emph{not} (uniformly) in $\ell^{1}$ as $\eta$ approaches $0$. This implies that neither the fractional moment method nor multi scale analysis \emph{can} be applied here.

math-ph↗

arXiv2019

Eigenvalue statistics for Schr\"odinger operators with random point interactions on $\mathbb{R}^d$, $d=1,2,3$

We prove that the local eigenvalue statistics at energy $E$ in the localization regime for Schr\"odinger operators with random point interactions on $\mathbb{R}^d$, 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 Schr\"odinger operators in the continuum. The special structure of resolvent of Schr\"odinger operators with point interactions facilitates the proof of the Minami estimate for these models.

math-ph↗

arXiv2018

Decorrelation estimates for random Schr\"odinger operators with non rank one perturbations

We prove decorrelation estimates for generalized lattice Anderson models on $Z^d$ constructed with finite-rank perturbations in the spirit of Klopp \cite{klopp}. These are applied to prove that the local eigenvalue statistics $\xi^\omega_{E}$ and $\xi^\omega_{E^\prime}$, 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 $\xi^\omega_{E}(I)$ and $\xi^\omega_{E'}(J)$, are independent and distributed according to a compound Poisson distribution whose L\'evy 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 \cite{klopp,shirley,trinh}, extends to models for which the eigenvalues are degenerate, and applies to models for which the potential is not sign definite \cite{tautenhahn-veselic1} in dimensions $d \geq 1$.

math-ph↗

arXiv2015

Decorrelation estimates for random Schr\"odinger operators with non rank one perturbations

math-ph↗

arXiv2014

Eigenvalue statistics for random Schrodinger operators with non rank one perturbations

We prove that certain natural random variables associated with the local eigenvalue statistics for generalized lattice Anderson models constructed with finite-rank perturbations are compound Poisson distributed. This distribution is characterized by the fact that the Levy measure is supported on at most a finite set determined by the rank. The proof relies on a Minami-type estimate for finite-rank perturbations. For Anderson-type continuum models on $\R^d$, we prove a similar result for certain natural random variables associated with the local eigenvalue statistics. We prove that the compound Poisson distribution associated with these random variables has a Levy measure whose support is at most the set of positive integers.

math-ph↗

arXiv2014

Poisson Statistics for Anderson Model with Singular Randomness

In this work we consider the Anderson model on the $d$-dimensional lattice with the single site potential having singular distribution, mainly $α$-Hölder continuous ones and show that the eigenvalue statistics is Poisson in the region of exponential localization.

math.SP↗

arXiv2012

Szego limit theorem on the lattice

In this paper, we prove a Szegö type limit theorem on $\ell^2(\ZZ^d)$. We consider operators of the form $H=Δ+V$, $V$ multiplication by a positive sequence $\{V(n), n \in \ZZ^d\}$ with $V(n) \rightarrow \infty, |n| \rightarrow \infty $ on $\ell^2(\ZZ^d)$ and $π_λ$ the orthogonal projection of $\ell^2(\mathbb{Z}^d)$ on to the space of eigenfunctions of $H$ with eigenvalues $\leq λ$. We take $B$ to be a pseudo difference operator of order zero with symbol $b(x,n), (x,n) \in \TT^d\times \ZZ^d$ and show that for nice functions $f$ $$ \lim_{λ\rightarrow \infty} Tr(f(π_λBπ_λ))/Tr(π_λ) = \lim_{λ\rightarrow \infty} \frac{1}{(2π)^d} \frac{\sum_{V(n) \leq λ} \int_{\TT^d} f(b(x,n)) ~ dx}{\sum_{V(n)\leqλ} 1}. $$

math-ph↗

arXiv2012

A Note on the Analyticity of Density of States

We consider the $d$-dimensional Anderson model, and we prove the density of states is locally analytic if the single site potential distribution is locally analytic and the disorder is large. We employ the random walk expansion of resolvents and a simple complex function theory trick. In particular, we discuss the uniform distribution case, and we obtain a sharper result using more precise computations. The method can be also applied to prove the analyticity of the correlation functions.

math-ph↗

arXiv2010

Direct integrals and spectral averaging

A one parameter family of selfadjoint operators gives rise to a corresponding direct integral. We show how to use the Putnam Kato theorem to obtain a new method for the proof of a spectral averaging result.

math-ph↗

arXiv2009

Lifshitz tails for the Interband Light Absorption Coefficient

In this paper we consider the Interband Light Absorption Coefficient for various models. We show that at the lower and upper edges of the spectrum the Lifshitz tails behaviour of the density of states implies similar behaviour for the ILAC at appropriate energies. The Lifshitz tails property is also exhibited at some points corresponding to the internal band edges of the density of states.

math-ph↗

arXiv1999

Spectra of Anderson type models with decaying randomness

In this paper we consider some Anderson type models, with decaying randomness and the free parts having long range tails. The randomness may decay at different rates in different directions, though in majority of directions we require some sort of short range decay. We show that there is pure a.c. spectrum and some pure point spectrum in such models.

math-ph↗

arXiv1993

Asymptotic analysis and spectrum of three anyons

The spectrum of anyons confined in harmonic oscillator potential shows both linear and nonlinear dependence on the statistical parameter. While the existence of exact linear solutions have been shown analytically, the nonlinear dependence has been arrived at by numerical and/or perturbative methods. We develop a method which shows the possibility of nonlinearly interpolating spectrum. To be specific we analyse the eigenvalue equation in various asymptotic regions for the three anyon problem.

cond-mat↗