Related papers: Lifshits Tails for Squared Potentials
We review some results and proofs on eigenvalue bounds for random Schr\"odinger operators with complex-valued potentials. We also include new Schatten norm estimates for the resolvent and use them to obtain bounds for sums of eigenvalues.
We investigate spectral properties of limit-periodic Schr\"odinger operators in $\ell^2(\Z)$. Our goal is to exhibit as rich a spectral picture as possible. We regard limit-periodic potentials as generated by continuous sampling along the…
We continue the investigations of Kirsch, Metzger and the second-named author [J. Stat. Phys. 143, 1035--1054 (2011)] on spectral properties of a certain type of random block operators. In particular, we establish an alternative version of…
We address the fundamental questions concerning the operator \begin{eqnarray*} H^{\theta_0}\psi(x)=-\psi"(x)+V(x,\omega)\psi(x),\,\psi(0)\cos\theta_0-\psi'(0)\sin\theta_0=0. \end{eqnarray*} where the random potential $V$ has a variety of…
In this paper, we study the scattering theory of a class of continuum Schr\"{o}dinger operators with random sparse potentials. The existence and completeness of wave operators are proven by establishing the uniform boundedness of modified…
We prove an upper bound for the (differentiated) density of states of the Anderson model at the bottom of the spectrum. The density of states is shown to exhibit the same Lifshitz tails upper bound as the integrated density of states.
In contrast to the neatly bounded spectra of densely populated large random matrices, sparse random matrices often exhibit unbounded eigenvalue tails on the real and imaginary axis, called Lifshitz tails. In the case of asymmetric matrices,…
We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite…
We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. The discrete alloy-type model is a family of Schr\"odinger operators $H_\omega = -…
We consider Schr\"odinger operators in $\ell^2(\Z)$ whose potentials are defined via continuous sampling along the orbits of a homeomorphism on a compact metric space. We show that for each non-atomic ergodic measure $\mu$, there is a dense…
We first analyze the integrated density of states (IDS) of periodic Schr\"odinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with…
We study discrete random Schr\"odinger operators via the supersymmetric formalism. We develop a cluster expansion that converges at both strong and weak disorder. We prove the exponential decay of the disorder-averaged Green's function and…
In this paper we prove the existence of orbitally stable standing waves for the critical Schr\"{o}dinger operator, involving potential of the form $\left(\frac{N-2}{2}\right)^2|x|^{-2}$. The approach, being purely variational, is based on…
We study the ergodic properties of Delone-Anderson operators, using the framework of randomly coloured Delone sets and Delone dynamical systems. In particular, we show the existence of the integrated density of states and, under some…
Positivity, essential self-adjointness, and spectral properties of a class of Schroedinger operators with multipolar inverse-square potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities…
We consider discrete Schr\"odinger operators with periodic potentials on periodic graphs perturbed by guided non-positive potentials, which are periodic in some directions and finitely supported in other ones. The spectrum of the…
We study spectral properties of Schr\"odinger operators on $\RR^d$. The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in $\ZZ^d$, with the property that frequencies of finite patterns are…
We study the $L^1$-smoothing properties for a broad class of semigroups arising from the ground state transformation of Schr\"odinger semigroups with confining potentials associated with non-local L\'evy operators, for which (asymptotic)…
We present a simple model of the local order in amorphous organic semiconductors which naturally produces a spatially correlated exponential density of states (DOS). The dominant contribution to the random energy landscape is provided by…
In this article, we study the Schr\"odinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the…