Wronskian differential formula for k-confluent SUSY QM

TL;DR

This paper extends the Wronskian differential formula to k-confluent SUSY quantum mechanics, enabling more efficient transformations by avoiding indefinite integrals, with explicit formulas for third and fourth orders and applications to specific potentials.

Contribution

It develops a generalized Wronskian differential formula for k-confluent SUSY transformations, expanding the method's applicability beyond second order.

Findings

Derived explicit formulas for third- and fourth-order cases.

Applied the generalized formula to free particle and Lamé potentials.

Demonstrated the method's effectiveness in specific quantum systems.

Abstract

The confluent SUSY QM usually involves a second-order SUSY transformation where the two factorization energies converge to a single value. In order to achieve it, one generally needs to solve an indefinite integral, which limits the actual systems to which it can be applied. Nevertheless, not so long ago, an alternative method to achieve this transformation was developed through a Wronskian differential formula [Phys Lett. A 3756 (2012) 692]. In the present work, we consider the k-confluent SUSY transformation, where k factorization energies merge into a single value, and we develop a generalized Wronskian differential formula for this case. Furthermore, we explicitly work out general formulas for the third- and fourth-order cases and we present as examples the free particle and the single-gap Lam\'e potentials.