In this paper we consider the space-fractional Schr\"odinger equation with a singular potential for wide class of fractional hypoelliptic operators. Such analysis can be conveniently realised in setting graded Lie groups. The is continuation and extension previous one where classical on $\mathbb R^n$ potentials was considered.

We provide a general framework of stationary scattering theory for one-dimensional quantum systems with nontrivial spatial asymptotics. As a byproduct we characterize reeectionless potentials in terms of spectral multiplicities and properties of the diagonal Green's function of the underlying Schrr odinger operator. Moreover, we prove that single (Crum-Darboux) and double commutation methods to...

We present a parallel solution of the Schrr odinger{Poisson system on distributed memory machines. The Schrr odinger{Poisson system is an evolution model for the numerical simulation of a collisionless electron plasma. We apply the Galerkin{Fourier method to the one{ dimensional system which results in a nonlinear system of ordinary differential equations. This initial value problem is solved b...

We consider the modulated harmonic wave in discrete series connected Josephson transmission line (JTL). formulate approach to modulation problems for equations based on calculus. check up by applying it Fermi-Pasta-Ulam-Tsingou type problem. Applying JTL, we obtain equation describing amplitude, which turns out be defocusing nonlinear Schr\"odinger (NLS) equation. compare profile of single soli...

Principal component analysis (PCA) has been applied to analyze random fields in various scientific disciplines. However, the explainability of PCA remains elusive unless strong domain-specific knowledge is available. This paper provides a theoretical framework that builds duality between eigenmodes field and eigenstates Schr\"odinger equation. Based on we propose algorithm replace expensive sol...

We obtain a class of exact solutions Bessel-type differential equation, which is six-parameter linear ordinary equation the second order with irregular (essential) singularity at origin. The are obtained using Tridiagonal Representation Approach (TRA) as bounded series square integrable functions written in terms Bessel polynomial on real line. expansion coefficients orthogonal polynomials para...

We prove a limit theorem on the convergence of mathematical expectations functionals sums independent random variables to Cauchy problem solution for nonstationary Schr¨odinger equation with symmetric fractional derivative operator order $$ \upalpha\ \upepsilon \bigcup_{m=3}^{\infty}\left(m-1,m\right) in righthand side.

This paper surveys recent work on quasi-exactly solvable Schrr odinger operators and Lie algebras of diierential operators.

We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schr\"odinger-Poisson system. More specifically, we use Crank-Nicolson as stepping mechanism, whilst nonlinearity is handled by means relaxation approach spirit \cite{Besse, KK} nonlinear Schr\"odinger equation. For spatial discretisation standard conforming finite element sch...

A non-isospectral linear problem for an integrable 2+1 generalization of the non Schr\"odinger equation, which includes dispersive terms third and fourth order, is presented. The classical symmetries Lax pair related reductions are carefully studied. We obtain several that yield in some cases problems 1+1 dimensions.