Reflectionless Potentials for Difference Schrödinger Equations

Abstract

As a part of the program `discrete quantum mechanics,' we present general reflectionless potentials for difference Schrödinger equations with pure imaginary shifts. By combining contiguous integer wave number reflectionless potentials, we construct the discrete analogues of the $h(h+1)/{\cosh }^{2}x$ 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 ($h\in {\mathbb{R}}_{>0}$) scattering theory for these potentials, we need the connection formulas for the basic hypergeometric function ${}_2{\varphi }_1(\genfrac{}{}{0ex}{}{a,b}{c}|q;z)$ 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 ${}_2{\varphi }_1$ 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.

arXiv comment24 pages. Comments and references added. To appear in J. Phys. A

References in corpus (10)

• Supersymmetry and Quantum Mechanics
• Discrete Heisenberg-Weyl Group and Modular Group
• Quantum Dilogarithm
• Orthogonal Polynomials from Hermitian Matrices
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Modification of Crum's Theorem for `Discrete' Quantum Mechanics
• Crum's Theorem for `Discrete' Quantum Mechanics
• Casoratian Identities for the Wilson and Askey-Wilson Polynomials
• Scattering Amplitudes for Multi-indexed Extensions of Solvable Potentials
• Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice

Cited by in corpus (2)

• Reflectionless and invisible potentials in photonic lattices
• Non-Hermitian invisibility in tight-binding lattices