Baxter T-Q Equation for Shape Invariant Potentials. The Finite-Gap Potentials Case

TL;DR

This paper demonstrates that the classical separation of variables in finite-gap potentials leads to the Baxter T-Q relation upon quantization, linking integrable systems and quantum potentials.

Contribution

It establishes a connection between the classical separation of variables and the quantum Baxter T-Q relation for finite-gap potentials.

Findings

Classical separation of variables corresponds to the Baxter T-Q relation after quantization.

The approach applies Sklyanin's method to finite-gap potentials.

The work bridges classical integrable systems and quantum spectral problems.

Abstract

The Darboux transformation applied recurrently on a Schroedinger operator generates what is called a {\em dressing chain}, or from a different point of view, a set of supersymmetric shape invariant potentials. The finite-gap potential theory is a special case of the chain. For the finite-gap case, the equations of the chain can be expressed as a time evolution of a Hamiltonian system. We apply Sklyanin's method of separation of variables to the chain. We show that the classical equation of the separation of variables is the Baxter T-Q relation after quantization.