V. A. Manukyan | Searcharxiv

arXiv2019

A Schrödinger potential involving $x^\frac{2}{3}$ and centrifugal-barrier terms conditionally integrable in terms of the confluent hypergeometric functions

The solution of the one-dimensional Schrödinger equation for a potential involving an attractive $x^\frac{2}{3}$ and a repulsive centrifugal-barrier $\sim x^{-2}$ terms is presented in terms of the non-integer-order Hermite functions. The potential belongs to one of the five bi-confluent Heun families. This is a conditionally integrable potential in that the strength of the centrifugal-barrier term is fixed. The general solution of the problem is composed using fundamental solutions each of which presents an irreducible linear combination of two Hermite functions of a scaled and shifted argument. The potential presents an infinitely extended confining well defined on the positive semi-axis and sustains infinitely many bound states.

quant-ph↗

arXiv2018

A new exactly integrable hypergeometric potential for the Schrödinger equation

We introduce a new exactly integrable potential for the Schrödinger equation for which the solution of the problem may be expressed in terms of the Gauss hypergeometric functions. This is a potential step with variable height and steepness. We present the general solution of the problem, discuss the transmission of a quantum particle above the barrier, and derive explicit expressions for the reflection and transmission coefficients.

quant-ph↗

arXiv2018

Confluent hypergeometric expansions of the solutions of the double-confluent Heun equation

Several expansions of the solutions of the double-confluent Heun equation in terms of the Kummer confluent hypergeometric functions are presented. Three different sets of these functions are examined. Discussing the expansions without a pre-factor, it is shown that two of these functions provide expansions the coefficients of which obey three-term recurrence relations, while for the third confluent hypergeometric function the corresponding recurrence relation generally involves five-terms. The latter relation is reduced to a three-term one only in the case when the double-confluent Heun equation degenerates to the confluent hypergeometric equation. The conditions for obtaining finite sum solutions via termination of the expansions are discussed. The possibility of constructing expansions of different structure using certain equations related to the double-confluent Heun equation is discussed. An example of such expansion derived using the equation obeyed by a function involving the derivative of a solution of the double-confluent Heun equation is presented. In this way, expansions governed by three- or more term recurrence relations for expansion coefficients can be constructed. An expansion with coefficients obeying a seven-term recurrence relation is presented. This relation is reduced to a five-term one if the additional singularity of the equation obeyed by the considered auxiliary function coincides with a singularity of the double-confluent Heun equation.

math.CA↗