Related papers: A Class of Matrix Schr\"odinger Bispectral Operato…

We study the local solvability of a class of operators with multiple characteristics. The class considered here complements and extends the one studied in [9], in that in this paper we consider some cases of operators with complex…

We give a spectral description of the semi-classical Schrodinger operator with a piecewise linear, complex valued potential. Moreover, using these results, we show how an arbitrarily small bounded perturbation of a non-self-adjoint operator…

The research on spectral inequalities for discrete Schrodinger Operators has proved fruitful in the last decade. Indeed, several authors analysed the operator's canonical relation to a tridiagonal Jacobi matrix operator. In this paper, we…

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…

A long standing question in the theory of orthogonal matrix polynomials is the matrix Bochner problem, the classification of $N \times N$ weight matrices $W(x)$ whose associated orthogonal polynomials are eigenfunctions of a second order…

It is by now well known that the wave functions of rational solutions to the KP hierarchy (those which can be achieved as limits of the pure n-soliton solutions) satisfy an additional eigenvalue equation for ordinary differential operators…

The aim of the present thesis is twofold: to study the problem of discreteness of the spectrum of Schr\"odinger operators with matrix-valued potentials in ${\mathbb R}^d$ (Chapter 1), and to prove new pointwise bounds for weighted Bergman…

A logarithmic type Lieb-Thirring inequality for two-dimensional Schroedinger operators is established. The result is applied to prove spectral estimates on trapped modes in quantum layers.

We will discuss the asymptotic behaviour of the eigenvalues of Schr\"{o}dinger operator with a matrix potential defined by Neumann boundary condition in $L_2^m(F)$, where $F$ is $d$-dimensional rectangle and the potential is a $m \times m$…

We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schr\"odinger eigenvalue equation $H\Psi \equiv (\Delta_2 +V)\Psi=E\Psi$ on any 2D Riemannian…

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…

In this note we summarize some of the properties found in several papers. We characterize spectral properties of the quantum mechanical hamiltonian of theories with fermionic degrees of freedom beyond semiclassical approximation. We obtain…

We demonstrate that it is possible to compute wave function normalization constants for a class of Schr\"odinger type equations by an algorithm which scales linearly (in the number of eigenfunction evaluations) with the desired precision P…

We prove the validity of regularizing properties of a double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in Schauder spaces by…

The problem of finding eigenvalue estimates for the Schr\"odinger operator turns out to be most complicated for the dimension 2. Some important results for this case have been obtained recently. We discuss these results and establish their…

We study location of eigenvalues of one-dimensional discrete Schr\"odinger operators with complex $\ell^{p}$-potentials for $1\leq p\leq \infty$. In the case of $\ell^{1}$-potentials, the derived bound is shown to be optimal. For $p>1$, two…

We define a class of pseudo-ergodic non-self-adjoint Schr\"odinger operators acting in spaces $l^2(X)$ and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a non-self-adjoint Anderson…

Spectral operators of matrices proposed recently in [C. Ding, D.F. Sun, J. Sun, and K.C. Toh, Math. Program. {\bf 168}, 509--531 (2018)] are a class of matrix valued functions, which map matrices to matrices by applying a vector-to-vector…

Exact solutions to the d-dimensional Schroedinger equation, d\geq 2, for Coulomb plus harmonic oscillator potentials V(r)=-a/r+br^2, b>0 and a\ne 0 are obtained. The potential V(r) is considered both in all space, and under the condition of…

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.