The $J$-matrix method

TL;DR

The paper reviews the J-matrix method, a technique for tridiagonalizing operators to analyze spectral properties, illustrated through examples like the Morse potential and Lame equation.

Contribution

It provides a comprehensive overview of the J-matrix method and demonstrates its application to specific differential operators.

Findings

Effective tridiagonalization of operators enables spectral analysis.

Application to Morse potential and Lame equation illustrates the method's utility.

Facilitates computation of eigenvalues and eigenfunctions.

Abstract

Given an operator L acting on a function space, the J-matrix method consists of finding a sequence y_n of functions such that the operator L acts tridiagonally on y_n with respect to n. Once such a tridiagonalization is obtained, a number of characteristics of such an operator L can be obtained. In particular, information on eigenvalues and eigenfunctions, bound states, spectral decompositions, etc. can be obtained in this way. We review the general set-up, and we discuss two examples in detail; the Schrodinger operator with Morse potential and the Lame equation.