Matrix representation of the resolvent operator in square-integrable basis and physical application

TL;DR

This paper derives simple formulas for matrix elements of the resolvent operator in square-integrable bases, facilitating numerical computations and providing insights into eigenvector expressions, with a physical application demonstrating practical utility.

Contribution

It introduces new formulas for the resolvent operator's matrix elements applicable to any finite square-integrable basis, regardless of orthogonality, and relates eigenvectors to eigenvalues.

Findings

Formulas are suitable for numerical computation in various bases.

Eigenvectors can be expressed in terms of eigenvalues.

Application demonstrates practical usefulness of the formulas.

Abstract

We obtain simple formulas for the matrix elements of the resolvent operator (the Green's function) in any finite set of square integrable basis. These formulas are suitable for numerical computations whether the basis elements are orthogonal or not. A byproduct of our findings is an expression for the normalized eigenvectors of a matrix in terms of its eigenvalues. We give a physical application as an illustration of how useful these results can be.