2023-06-03T15:17:47Z
https://soar-ir.repo.nii.ac.jp/oai
oai:soar-ir.repo.nii.ac.jp:00011688
2022-12-14T04:09:57Z
1169:1170
Reflectionless potentials for difference Schrodinger equations
Odake, Satoru
Sasaki, Ryu
scattering problems in discrete QM
solvable scattering problems
Heine' s hypergometric functions with
q
=1
connection formula for 2.1
with
q
=1
q
ultraspherical polynomials with
q
=1
quantum dilogarithm
discrete analogue of 1 cosh2x potential
As a part of the program 'discrete quantum mechanics', we present general reflectionless potentials for difference Schr dinger equations with pure imaginary shifts. By combining contiguous integer wave number reflectionless potentials, we construct the discrete analogues of the h(h＋1)/cosh²x 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 ∈R>o<br/> scattering theory for these potentials, we need the connection formulas for the basic hypergeometric function. a b c 2 1, 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 2.1 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.
Article
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 48(11):115204 (2015)
journal article
IOP PUBLISHING LTD
2015-03-20
application/pdf
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
11
48
115204
1751-8113
AA12185372
https://soar-ir.repo.nii.ac.jp/record/11688/files/Reflectionless_Potentials_for_Difference.pdf
eng
10.1088/1751-8113/48/11/115204
https://doi.org/10.1088/1751-8113/48/11/115204
This is an author-created, un-copyedited version of an article accepted for publication in JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher-authenticated version is available online at 10.1088/1751-8113/48/11/115204© 2015 IOP Publishing Ltd