Discrete phase integral method for five-term recursion relations

TL;DR

This paper develops a discrete phase integral method for five-term recursion relations, common in quantum spin systems, deriving conditions for approximation validity and analyzing new types of turning points.

Contribution

It introduces a formalism for five-term recursion relations using phase integral methods, including new connection formulas for complex turning points.

Findings

Valid conditions for phase integral approximation are derived.

New types of turning points in classically forbidden regions are identified.

Connection formulas for these turning points are established.

Abstract

A formalism is developed to study certain five-term recursion relations by discrete phase integral (or Wentzel-Kramers-Brillouin) methods. Such recursion relations arise naturally in the study of the Schrodinger equation for certain spin Hamiltonians. The conditions for the validity of the phase integral approximation are derived. It is shown that in contrast to the three-term problem, it is now possible to get a turning points "under the barrier", i.e., in the classically forbidden region, as well as inside the classically allowed region. Further, no qualitatively new types of turning points arise in recursion relations with still higher numbers of terms. The phase integral approximation breaks down at the new turning points, requiring new connection formulas, which are derived.