A Class of Exactly Solvable Real and Complex $PT$ Symmetric Reflectionless Potentials

TL;DR

This paper investigates the classification of exactly solvable PT-invariant reflectionless potentials, revealing a specific count of such potentials with N bound states, including complex extensions.

Contribution

It introduces a formula for counting the total number of exactly solvable complex PT-invariant reflectionless potentials with N bound states, focusing on rational extensions.

Findings

Total number of potentials is 2[(2N-1)m+N]

Includes complex and real PT-invariant reflectionless potentials

Provides a systematic approach to classify these potentials

Abstract

We consider the question of the number of exactly solvable complex but PT-invariant reflectionless potentials with $N$ bound states. By carefully considering the $X_m$ rationally extended reflectionless potentials, we argue that the total number of exactly solvable complex PT-invariant reflectionless potentials are $2[(2N-1)m+N]$.