Related papers: Eigenvalue inequalities in terms of Schatten norm …
This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We use a recently developed technique, the so called…
In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if $-A$ generates a $C_0$-semigroup on a…
Two-sided estimates for higher order eigenvalues are presented for a class of non-local Schr\"odinger operators by using the jump rate and the growth of the potential. For instance, let $L$ be the generator of a L\'evy process with L\'evy…
It is shown that the ground-state eigenvalue of a semirelativistic Hamiltonian of the form H = sqrt(m^2+p^2) + V is bounded below by the Schroedinger operator m + beta p^2 + V, for suitable beta > 0. An example is discussed.
The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the $2M$-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new…
We review some recent results on eigenvalues of fractional Laplacians and fractional Schr\"odinger operators. We discuss, in particular, Lieb-Thirring inequalities and their generalizations, as well as semi-classical asymptotics.
Let $\Omega$ be a bounded domain in $\mathbb R^2$ with smooth boundary $\partial\Omega$, and let $\omega_h$ be the set of points in $\Omega$ whose distance from the boundary is smaller than $h$. We prove that the eigenvalues of the…
We consider a semi-classical Schr\"odinger operator, -h^2\Delta + V(x). Assuming that the potential admits a unique global minimum and that the eigenvalues of the Hessian are linearly independent over the rationals, we show that the…
We formulate the issue of minimality of self-adjoint operators on a Hilbert space as a semi-definite problem, linking the work by Overton in [1] to the characterization of minimal hermitian matrices. This motivates us to investigate the…
The Schatten-$p$ quasi-norm with $p\in(0,1)$ has recently gained considerable attention in various low-rank matrix estimation problems offering significant benefits over relevant convex heuristics such as the nuclear norm. However, due to…
We derive novel concentration inequalities for the operator norm of the sum of self-adjoint operators that do not explicitly depend on the underlying dimension of the operator, but rather an intrinsic notion of it. Our analysis leads to…
For one-dimensional Schroedinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. Our (non-semi-classical) approach results in substantial progress in achieving optimal conditions…
We prove general comparison theorems for eigenvalues of perturbed Schrodinger operators that allow proof of Lieb--Thirring bounds for suitable non-free Schrodinger operators and Jacobi matrices.
In this paper, we establish the following perturbation result concerning the singular values of a matrix: Let $A,B \in \mathbb{R}^{m\times n}$ be given matrices, and let $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ be a concave function…
In this paper, we compute the eigenvalue problem (EVP) for the semiclassical random Schr\"odinger operators, where the random potentials are parameterized by an infinite series of random variables. After truncating the series, we introduce…
In this article, we prove the finiteness of the number of eigenvalues for a class of Schr\"odinger operators $H = -\Delta + V(x)$ with a complex-valued potential $V(x)$ on $\bR^n$, $n \ge 2$. If $\Im V$ is sufficiently small, $\Im V \le 0$…
We study the existence of negative eigenvalues for two-dimensional Schr\"odinger operators with real-valued potentials in the weak coupling regime. In his pioneering paper [Simon 1976] from half a century ago, Simon was the first to…
We consider Schr\"odinger operators $H=- \d^2/\d r^2+V$ on $L^2([0,\infty))$ with the Dirichlet boundary condition. The potential $V$ may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum of $H$ is…
In this paper we present symbolic criteria for invariant operators on compact topological groups $G$ characterising the Schatten-von Neumann classes $S_{r}(L^{2}(G))$ for all $0<r\leq\infty$. Since it is known that for pseudo-differential…
The Agmon estimate for multi-dimensional discrete Schr\"{o}dinger operators is studied with emphasis on the microlocal analysis on the torus. We first consider the semiclassical setting where semiclassical continuous Schr\"{o}dinger…