Reflectionless potentials for difference Schrödinger equations

DOI10.1088/1751-8113/48/11/115204zbMATH Open1312.39010arXiv1411.2307OpenAlexW3104468324MaRDI QIDQ5245693

Publication date: 14 April 2015

Published in: Journal of Physics A: Mathematical and Theoretical (Search for Journal in Brave)

Abstract: As a part of the program `discrete quantum mechanics,' we present general reflectionless potentials for difference Schr"odinger equations with pure imaginary shifts. By combining contiguous integer wave number reflectionless potentials, we construct the discrete analogues of the

h (h + 1) / c o s h^{2} x

potential with the integer

h

, which belong to the recently constructed families of solvable dynamics having the

q

-ultraspherical polynomials with

| q | = 1

as the main part of the eigenfunctions. For the general (

h i n m a t h b b R_{> 0}

) scattering theory for these potentials, we need the connection formulas for the basic hypergeometric function

_{2} p h i_{1} (g e n f r a c 0 p t a, b c | q; z)

with

| q | = 1

, which is not known. The connection formulas are expected to contain the quantum dilogarithm functions as the

| q | = 1

counterparts of the

q

-gamma functions. We propose a conjecture of the connection formula of the

_{2} p h i_{1}

function with

| q | = 1

. Based on the conjecture, we derive the transmission and reflection amplitudes, which have all the desirable properties. They provide a strong support to the conjectured connection formula.

Full work available at URL: https://arxiv.org/abs/1411.2307

zbMATH Keywords

scattering problems quantum dilogarithm \(q\)-ultraspherical polynomials with \(|q|=1\)connection formula for \(_2\phi_1\) with \(|q|=1\)difference Schrödinger equations discrete analogue of \(1/\cosh^2 x\) potential Heine's hypergeometric functions with \(|q|=1\)

Mathematics Subject Classification ID

Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) Gamma, beta and polygamma functions (33B15) Additive difference equations (39A10) Discrete version of topics in analysis (39A12)