Solution of the nonrelativistic wave equation in the tridiagonal representation approach

TL;DR

This paper introduces a method to solve the nonrelativistic wave equation using a tridiagonal matrix representation, leading to exact solutions via orthogonal polynomials and expanding the class of solvable quantum problems.

Contribution

It develops a tridiagonal representation approach that yields exact solutions for certain quantum systems and extends the class of problems beyond traditional methods.

Findings

Exact solutions for wave equations using orthogonal polynomials

Derivation of phase shifts and spectra from polynomial asymptotics

Expanded set of solvable quantum potentials

Abstract

We choose a complete set of square integrable functions as basis for the expansion of the wavefunction in configuration space such that the matrix representation of the nonrelativistic time-independent wave operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction in this basis. The recursion relation is then solved exactly in terms of orthogonal polynomials in the energy. Some of these polynomials are not found in the mathematics literature. The asymptotics of these polynomials give the phase shift of the continuous energy scattering states and the spectrum for the discrete energy bound states. Depending on the space and boundary conditions, the basis functions are written in terms of either the Laguerre or Jacobi polynomials. The tridiagonal requirement limits the number of potential functions that yield exact solutions of the wave equation. Nonetheless, the class of exactly solvable problems in this approach is larger than the conventional class (see Table 12). We also give very accurate results for cases where the wave operator matrix is not tridiagonal but its elements could be evaluated either exactly or numerically with high precision.