Resonances and antibound states of P\"oschl-Teller potential: Ladder operators and SUSY partners

TL;DR

This paper investigates the scattering properties of all variations of the P"oschl-Teller potential, revealing the nature of bound, antibound, and resonance states, and constructing ladder operators and SUSY partners for these potentials.

Contribution

It provides a comprehensive analysis of resonance and antibound states in P"oschl-Teller potentials, including explicit Gamow states and supersymmetric partner constructions.

Findings

Well and low barrier potentials have no resonance poles but infinite antibound states.

High barrier potential exhibits infinite resonance poles with explicit Gamow states.

Ladder operators connect bound, antibound, and resonance wave functions.

Abstract

We analyze the one dimensional scattering produced by all variations of the P\"oschl-Teller potential, i.e., potential well, low and high barriers. We show that the P\"oschl-Teller well and low barrier potentials have no resonance poles, but an infinite number of simple poles along the imaginary axis corresponding to bound and antibound states. A quite different situation arises on the P\"oschl-Teller high barrier potential, which shows an infinite number of resonance poles and no other singularities. We have obtained the explicit form of their associated Gamow states. We have also constructed ladder operators connecting wave functions for bound and antibound states as well as for resonance states. Finally, using wave functions of Gamow and antibound states in the factorization method, we construct some examples of supersymmetric partners of the P\"oschl-Teller Hamiltonian.