Related papers: Two-dimensional von Neumann--Wigner potentials wit…
This paper studies the resonances of Schr\"odinger operators with bounded, compactly supported, real-valued potentials on d-dimensional Euclidean space, where d is even. If the potential V is non-trivial and d is not 4 then the meromorphic…
A transformation method is applied to the second order ordinary differential equation satisfied by orthogonal polynomials to construct a family of exactly solvable quantum systems in any arbitrary dimensional space. Using the properties of…
For a class of one-dimensional Schrodinger operators with polynomial potentials that includes Hermitian and PT-symmetric operators, we show that the zeros of scaled eigenfunctions have a limit disctibution in the complex plane as the…
We construct Darboux-Moutard type transforms for the two-dimensional conductivity equation. This result continues our recent studies of Darboux-Moutard type transforms for generalized analytic functions. In addition, at least, some of the…
We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous…
We consider the Schrodinger operator a given domain. Our goal is to study some optimization problems where an optimal (non-negative) potential V has to be determined in some suitable admissible classes and for some suitable optimization…
We develop direct and inverse scattering theory for one-dimensional Schroedinger operators with steplike potentials which are asymptotically close to different finite-gap periodic potentials on different half-axes. We give a complete…
The second order N-dimensional Schrodinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation. Exact bound state solutions are obtained using convolution or Faltungs…
The eigenvalue problem for one-dimensional Schr\"{o}dinger equation with the rational potential is numerically solved by the operator method. We show that the operator method, applied for solving the Schr\"{o}dinger equation with the…
We show that formal Schr\"odinger operators with singular potentials from the space W^{-1}_{2,unif}(R) can be naturally defined to give selfadjoint and bounded below operators, which depend continuously in the uniform resolvent sense on the…
Laptev and Safronov conjectured that any non-positive eigenvalue of a Schr\"odinger operator $-\Delta+V$ in $L^2(\mathbb R^\nu)$ with complex potential has absolute value at most a constant times…
By converting the rectangular basis potential V(x,y) into the form as V(r)+V(r, phi) described by the pseudo central plus noncentral potential, particular solutions of the two dimensional Schrodinger equation in plane-polar coordinates have…
Using the method of shape invariant potentials, a number of exact solutions of one dimensional effective mass Schrodinger equation are obtained. The solutions with equi-spaced spectrum are discussed in detail.
Exact bound state solutions and corresponding normalized eigenfunctions of the radial Schr\"odinger equation are studied for the pseudoharmonic and Mie-type potentials by using the Laplace transform approach. The analytical results are…
We consider the time-dependent non linear Schrodinger equations with a double well potential in dimensions d =1 and d=2. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest two eigenvalues…
Following the proof given by Froese and Herbst in [FH82] with another conjugate operator, we show for a class of real potential that possible eigenfunction of the Schr\"odinger operator has to decay sub-exponentially. We also show that, for…
The power of the disconjugacy properties of second-order differential equations of Schr\"odinger type to check the regularity of rationally-extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by…
We consider a Schr\"odinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive…
We study the spectral properties of a Schr\"odinger operator, in presence of a confining potential given by the distance squared from a fixed compact potential well. We prove continuity estimates on both the eigenvalues and the eigenstates,…
We estimate sums of functions of negative eigenvalues of Schr\"odinger-type operators whose kinetic energy vanishes on a codimension one submanifold. Our main technical tool is the Stein-Tomas theorem and some of its generalizations.