Related papers: Schr\"odinger operators with complex potential but…
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…
We propose an exact method for solving a one-dimensional Schr\"odinger equation. An arbitrary potential is represented by the collection of short-width potentials. For building the collection scheme, a new solvable potential is introduced.…
In this paper, we introduce a new family of functions to construct Schr\"odinger operators with embedded eigenvalues. This particularly allows us to construct discrete Schr\"odinger operators with arbitrary prescribed sets of eigenvalues.
We consider non-self-adjoint electromagnetic Schr\"odinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic…
Using purely physical arguments it is claimed that for ID Schrodinger operators with complex PT- Symmatric potentials having a purely real attractive potential well and a purely imaginary repulsive part,bound state eigenvalues will be…
The absolutely continuous spectrum of one-dimensional Schr\"odinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic…
We review some of the work by the authors on this topic. In particular, we discuss the recent paper "Fine properties of the IDS and a quantitative separation property of the Dirichlet eigenvalues".
A new form to construct complex superpotentials that produce real energy spectra in supersymmetric quantum mechanics is presented. This is based on the relation between the nonlinear Ermakov equation and a second order differential equation…
The celebrated Cwikel-Lieb_Rozenblum inequality gives an upper estimate for the number of negative eigenvalues of Schroedinger operators in dimension three and higher. The situation is much more difficult in the two dimensional case. There…
The Self-Adjoint Extension in the Schrodinger equation for potentials behaved as an attractive inverse square at the origin is critically reviewed. Original results are also presented. It is shown that the additional solutions must be…
It has been observed that a quantum theory need not to be Hermitian to have a real spectrum. We study the non-Hermitian relativistic quantum theories for many complex potentials, and we obtain the real relativistic energy eigenvalues and…
Neat stuff about eigenfunctions, transfer matrices, and a.c. spectrum of one-dimensional Schrodinger operators
We study the one-dimensional Schr\"odinger operators $$ S(q)u:=-u"+q(x)u,\quad u\in \mathrm{Dom}\left(S(q)\right), $$ with $1$-periodic real-valued singular potentials $q(x)\in H_{\operatorname{per}}^{-1}(\mathbb{R},\mathbb{R})$ on the…
We study the resonances of (generally, non-selfadjoint) Schr\"odinger operators with matrix-valued square-well potentials. We compute explicitly the Jost function and derive complex transcendental equations for the resonances. We prove…
This paper mainly addresses the strong unique continuation property for the electromagnetic Schr\"{o}dinger operator with complex-valued coefficients. Appropriate multipliers with physical backgrounds have been introduced to prove a priori…
We study Schr\"odinger operators $H=-\Delta+V$ in $L^2(\Omega)$ where $\Omega$ is $\mathbb R^d$ or the half-space $\mathbb R_+^d$, subject to (real) Robin boundary conditions in the latter case. For $p>d$ we construct a non-real potential…
By the Moutard transformation method we construct two-dimensional Schrodinger operators with real smooth potential decaying at infinity and with a multiple positive eigenvalue. These potentials are rational functions of spatial variables…
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 this paper we consider the space-fractional Schr\"odinger equation with a singular potential for a wide class of fractional hypoelliptic operators. Such analysis can be conveniently realised in the setting of graded Lie groups. The paper…
We consider two dimensional real-valued analytic potentials for the Schr\"odinger equation which are periodic over a lattice $L$. Under certain assumptions on the form of the potential and the lattice $L$, we can show there is a large class…