Related papers: Discretization of Natanzon potentials
We show that there exist pairs of non-isometric potentials for the 1D semiclassical Schr\"odinger operator whose spectra agree up to $O(h^\infty)$, yet their corresponding eigenvalues differ no less than exponentially. This result was…
Our paper investigates one-dimensional Schr\"odinger operators defined as closed operators on $L^2(\mathbb{R})$ or $L^2(\mathbb{R}_+)$ that are exactly solvable in terms of confluent functions (or, equivalently, Whittaker functions). We…
We consider a family of extensions of the Kepler-Coulomb potential on a $d$-dimensional sphere and analyze it in a deformed supersymmetric framework, wherein the starting potential is known to exhibit a deformed shape invariance property.…
This work deals with Schr\"odinger equations with quadratic and sub-quadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations. We shall give a representation of such…
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…
For the two-dimensional Schr\"odinger equation $$ [- \Delta +v(x)]\psi=E\psi,\ x\in \R^2,\ E=E_{fixed}>0 \ \ \ \ \ (*)$$ at a fixed positive energy with a fast decaying at infinity potential $v(x)$ dispersion relations on the scattering…
We obtain three new solvable, real, shape invariant potentials starting from the harmonic oscillator, P\"oschl-Teller I and P\"oschl-Teller II potentials on the half-axis and extending their domain to the full line, while taking special…
Two rectangular models described by the one-dimensional Schroedinger equation with sharply localized potentials are suggested. The potentials have a multi-layer thin structure being composed from adjacent barriers and wells. Their peculiar…
In space dimension $n\geq3$, we consider the electromagnetic Schr\"odinger Hamiltonian $H=(\nabla-iA(x))^2+V$ and the corresponding Helmholtz equation (\nabla-iA(x))^2u+u+V(x)u=f\quad \text{in}\quad \mathbb{R}^n, where the magnetic and…
For any central potential V in D dimensions, the angular Schroedinger equation remains the same and defines the so called hyperspherical harmonics. For non-central models, the situation is more complicated. We contemplate two examples in…
Advantage is taken of the arbitrariness in energy reference to consider anew integral transcriptions of Schrodinger's equation in the presence of potentials which at infinity acquire constant, nonvanishing values. It is found possible to…
We present, to the best of our knowledge, the first numerical algorithm for explicit, computable two-sided eigenvalue bounds for Schr\"odinger operators H = -Delta + V on R^N, N = 2,3, in the presence of both an unbounded potential and an…
A new manner for deriving the exact potentials is presented. By making use of conformal mappings, the general expression of the effective potentials deduced under su(1,1) algebra can be brought back to the general Natanzon hypergeometric…
In this paper we are concerned with nonlinear Schr\"odinger equations with random potentials. Our class includes continuum and discrete potentials. Conditions on the potential $V_{\omega}$ are found for existence of solutions almost sure…
In this article we focus on a semiclassical Schr\"odinger equation with matrix-valued potential presenting a symmetric conjoint crossing of three eigenvalues. The potential we consider is well-known in the chemical literature as a pseudo…
We introduce some new classes of time dependent functions whose defining properties take into account of oscillations around singularities. We study properties of solutions to the heat equation with coefficients in these classes which are…
The existence and $L^{\infty}$ estimate of positive solutions are discussed for the following Schr\"{o}dinger-Poisson system {ll} -\Delta u +(\lambda+\frac{1}{|y|^\alpha})u+\phi (x) u =|u|^{p-1}u, x=(y,z)\in \mathbb{R}^2\times\mathbb{R},…
Let $\mathcal{H}=-\Delta_{\mathbb{H}}+V$ be the Schr\"odinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}}$ is the full laplacian on $\mathbb{H}^n$ and $V$ is a positive smooth potential, bounded below and…
Within the hyperspherical framework, the solution of the time-independent Schroedinger equation for a n-particle system is divided into two steps, the solution of a Schroedinger like equation in the hyperangular degrees of freedom and the…
We find exact solutions of the time-dependent Schr\"odinger equation for a family of quasi-exactly solvable time-dependent potentials by means of non-unitary gauge transformations.