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

We clarify the algebraic structure of continuous and discrete quasi-exactly solvable spectral problems by embedding them in 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 nonperiodic boundary conditions. Applications to the Azbel-Hofstadter problem are outlined.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 8–24, July, 1995.