Transmutations for Darboux transformed operators with applications

TL;DR

This paper develops a method to explicitly construct transmutation operators for Darboux transformed Schr"odinger potentials, enabling new applications in Dirac systems and advancing operator transformation techniques.

Contribution

It provides a closed-form expression for the transmutation kernel after Darboux transformation, extending the theory of transmutation operators for Schr"odinger operators.

Findings

Explicit kernel formula for Darboux transformed potentials

Derived commutation relations between transmutation operators

Constructed transmutation operator for 1D Dirac systems

Abstract

We solve the following problem. Given a continuous complex-valued potential q_1 defined on a segment [-a,a] and let q_2 be the potential of a Darboux transformed Schr\"odinger operator. Suppose a transmutation operator T_1 for the potential q_1 is known such that the corresponding Schr\"odinger operator is transmuted into the operator of second derivative. Find an analogous transmutation operator T_2 for the potential q_2. It is well known that the transmutation operators can be realized in the form of Volterra integral operators with continuously differentiable kernels. Given a kernel K_1 of the transmutation operator T_1 we find the kernel K_2 of T_2 in a closed form in terms of K_1. As a corollary interesting commutation relations between T_1 and T_2 are obtained which then are used in order to construct the transmutation operator for the one-dimensional Dirac system with a scalar potential.