Related papers: 1-D Schr\"odinger operators with local point inter…
On a Lie group $G$, we investigate the discreteness of the spectrum of Schr\"odinger operators of the form $\mathcal{L} +V$, where $\mathcal{L}$ is a subelliptic sub-Laplacian on $G$ and the potential $V$ is a locally integrable function…
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 investigate the equivalence between dynamical localization and localization properties of eigenfunctions of Schr\"odinger Hamiltonians. We introduce three classes of equivalent properties and study the relationships between them. These…
Spectral properties of Schr\"odinger operators on compact metric graphs are studied and special emphasis is put on differences in the spectral behavior between different classes of vertex conditions. We survey recent results especially for…
In this paper, we develop a systematic framework to study the dispersion surfaces of Schr{\"o}dinger operators $ -\Delta + V$, where the potential $V \in C^\infty(\mathbb{R}^n,\mathbb{R})$ is periodic with respect to a lattice $\Lambda…
We revisit an archive submission by P. B. Denton, S. J. Parke, T. Tao, and X. Zhang, arXiv:1908.03795, on $n \times n$ self-adjoint matrices from the point of view of self-adjoint Dirichlet Schr\"odinger operators on a compact interval.
We explore the sparsity of Weyl-Titchmarsh $m$-functions of discrete Schr\"odinger operators. Due to this, the set of their $m$-functions cannot be dense on the set of those for Jacobi operators. All this reveals why an inverse spectral…
We show that a real eigenfunction of the Schr\"odinger operator changes sign near some point in $\mathbb{R}^n$ under a suitable assumption on the potential.
In dimension greater than or equal to three, we investigate the spectrum of a Schr{\"o}dinger operator with a $\delta$-interaction supported on a cone whose cross section is the sphere of co-dimension two. After decomposing into fibers, we…
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…
We give an explicit correspondence between the domains of the self-adjoint extensions of a one-dimensional Schr\"odinger differential operator with symmetric real-valued potential and the boundary conditions the functions in the resulting…
Non-self-adjoint Schrodinger operators A which correspond to non-symmetric zero-range potentials are investigated. For a given A, the description of non-real eigenvalues, spectral singularities and exceptional points are obtained; the…
The Schr\"odinger operator on a metric tree is a family of ordinary differential operators on its edges complemented by certain matching conditions at the vertices. The regular trees are highly symmetric. This allows one to construct an…
In this paper, we present an overview of the development of one of the most dynamic areas of mathematics today: local differential operators of non-integer order. The underlying question is whether we are witnessing a period of…
We complete the classical Schoenberg representation theorem for radial positive definite functions. We apply this result to study spectral properties of self-adjoint realizations of two- and three-dimensional Schr\"odinger operators with…
In the recent literature, various authors have studied spectral comparison results for Schr\"odinger operators with discrete spectrum in different settings including Euclidean domains and quantum graphs. In this note we derive such spectral…
We investigate random, discrete Schr\"odiner operators which arise naturally in the theory of random matrices, and depend parametrically on Dyson's Coulomb gas inverse temperature $\beta$. They belong to the class of "critical" random…
We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this…
This paper analyzes Sch\"odinger operators from viewpoint of correlation inequalities. We construct Griffiths inequalities for the ground state expectations by applying operator-theoretic correlation inequalities. As an example of such an…
The paper is devoted to the study of the essential spectrum of discrete Schr\"{o}dinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method. This method has been applied by one of the authors to describe the…