Quantum transfer matrices for discrete and continuous quasi-exactly solvable problems

TL;DR

This paper explores the algebraic structure of quasi-exactly solvable spectral problems by embedding them into the quantum inverse scattering framework, linking Hamiltonians to quantum monodromy matrices and discussing applications like the Azbel-Hofstadter problem.

Contribution

It introduces a novel algebraic approach to analyze quasi-exactly solvable problems using quantum inverse scattering, connecting Hamiltonians with monodromy matrices.

Findings

Identification of quasi-exactly solvable Hamiltonians with quantum monodromy matrices

Application framework for the Azbel-Hofstadter problem

Clarification of algebraic structures in spectral problems

Abstract

We clarify the algebraic structure of continuous and discrete quasi-exactly solvable spectral problems by embedding them into the framework of the quantum inverse scattering method. The quasi-exactly solvable hamiltonians in one dimension are identified with traces of quantum monodromy matrices for specific integrable systems with non-periodic boundary conditions. Applications to the Azbel-Hofstadter problem are outlined.