Related papers: Singular Schr\"odinger operators and Robin billiar…
This note is devoted to optimal spectral estimates for Schr\"odinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent…
We prove sharp lower bounds on the spectral gap of 1-dimensional Schr\"odinger operators with Robin boundary conditions for each value of the Robin parameter. In particular, our lower bounds apply to single-well potentials with a centered…
We review recent advances in the spectral theory of Schr\"odinger operators with decaying potentials. The area has seen spectacular progress in the past few years, stimulated by several conjectures stated by Barry Simon starting at the 1994…
In this paper, we prove that on a compact manifold with isolated conical singularity the spectrum of the Schr\"odinger operator $-4\Delta+R$ consists of discrete eigenvalues with finite multiplicities, if the scalar curvature $R$ satisfies…
In this paper we investigate the spectral expansion for the one-dimensional Schrodinger operator with a periodic complex-valued potential. For this we consider in detail the spectral singularities and introduce new concepts as essential…
The spectral properties of two-dimensional Schr\"odinger operators with $\delta'$-potentials supported on star graphs are discussed. We describe the essential spectrum and give a complete description of situations in which the discrete…
We investigate a class of generalized Schr\"{o}dinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a smooth curve $\Gamma$. We find a strong-coupling asymptotic expansion of the discrete spectrum in case when…
A number of results on radial positive definite functions on ${\mathbb R^n}$ related to Schoenberg's integral representation theorem are obtained. They are applied to the study of spectral properties of self-adjoint realizations of two- and…
We establish uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schr\"odinger operators. Our results…
Spectral components of one-dimensional Schr\"odinger operator with complex potential are investigated. An effective upper bound for the total number of eigenvalues and spectral singularities is established. For dissipative Schr\"odinger…
We investigate spectral properties of a discrete random displacement model, a Schr\"odinger operator on $\ell^2(\Z^d)$ with potential generated by randomly displacing finitely supported single-site terms from the points of a sublattice of…
We consider Schr\"odinger operators on a bounded domain $\Omega\subset \mathbb{R}^3$, with homogeneous Robin or Dirichlet boundary conditions on $\partial\Omega$ and a point (zero-range) interaction placed at an interior point of $\Omega$.…
We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schr\"odinger operator on a star-graph with a finite number of…
We prove an asymptotic expansion for the eigenvalues and eigenfunctions of Schr\"{o}dinger-type operator with a confining potential and with principle part a periodic elliptic operator in divergence form. We compare the spectrum to the…
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…
In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schr\"odinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an…
We investigate the spectral properties of self-adjoint Schr\"odinger operators with attractive $\delta$-interactions of constant strength $\alpha > 0$ supported on conical surfaces in ${\mathbb R}^3$. It is shown that the essential spectrum…
We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schroedinger operator on a star-shaped…
We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum…
We consider singular self-adjoint extensions for one-dimensional Schr\"{o}dinger operator for two-component wave function within the framework of the distribution theory for discontinuous test functions…