相关论文: Borg-Type Theorems for Matrix-Valued Schr\"{o}ding…
The spectra of the Schr\"odinger operators with periodic potentials are studied. When the potential is real and periodic, the spectrum consists of at most countably many line segments (energy bands) on the real line, while when the…
We investigate existence and qualitative behaviour of solutions to nonlinear Schr\"odinger equations with critical exponent and singular electromagnetic potentials. We are concerned with magnetic vector potentials which are homogeneous of…
We consider Schr\"odinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of such operators consists of a finite number of bands. We determine trace formulas for the magnetic Schr\"odinger…
We study the semi-classical trace formula at a critical energy level for a Schr\"odinger operator on $\mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result,…
We study half-line Schr\"odinger operators with locally $H^{-1}$ potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last--Simon-type description of the absolutely continuous…
We consider Schr\"odinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schr\"odinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a…
In the present paper, we establish a reduction theorem for linear Schr\"odinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM technique. Moreover, it is proved that the…
Considering symmetric strictly convex potentials, a local relationship is inferred from the virial theorem, based on which a real log-concave function can be constructed. Using this as a weight function and in such a way that the virial…
In this article, we investigate systems of generalized Schr\"odinger operators and their fundamental matrices. More specifically, we establish the existence of such fundamental matrices and then prove sharp upper and lower exponential decay…
This article is devoted to the spectral analysis of the electro-magnetic Schr\"odinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum…
For periodic graph operators, we establish criteria to determine the overlaps of spectral band functions based on Bloch varieties. One criterion states that for a large family of periodic graph operators, the irreducibility of Bloch…
We establish semiclassical resolvent estimates for Schr\"odinger operators with long-range matrix-valued potentials. As an application we prove resonance free domains both in trapping and non-trapping situations. Our results generalize the…
We study the Bloch variety of discrete Schr\"odinger operators associated with a complex periodic potential and a general finite-range interaction, showing that the Bloch variety is irreducible for a wide class of lattice geometries in…
The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which…
We consider a semi-classical Schrodinger operator with a degenerate potential V(x,y) =f(x) g(y) . g is assumed to be a homogeneous positive function of m variables and f is a strictly positive function of n variables, with a strict minimum.…
In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schr\"odinger type operator $\mathcal{H}_V:=-\operatorname{div}(A\nabla)+V$, where $V$ is convex and Kato decomposable, using the trace class…
Non-self-adjoint Schrodinger operators which correspond to non-symmetric zero-range potentials are investigated. We show that various properties of these operators (eigenvalues, exceptional points, spectral singularities and the property of…
The spectral properties of the singular Schr\"odinger operator with complex-valued potential which takes values in a wider region than the half-plane, have been little studied. In general case, the operator is non-sectorial, and the…
We construct a potential $V$ on $\RR^d$, smooth away from one pole, and a sequence of quasi-modes for the operator $-\Delta+V$, which concentrate on this pole. No smoothing effect, Strichartz estimates nor dispersive inequalities hold for…
As a part of the program `discrete quantum mechanics,' we present general reflectionless potentials for difference Schr\"odinger equations with pure imaginary shifts. By combining contiguous integer wave number reflectionless potentials, we…