Calculation of the Density of States Using Discrete Variable Representation and Toeplitz Matrices

TL;DR

This paper introduces an exact method for calculating the density of states in systems with localized potentials by leveraging discrete variable representation and Toeplitz matrix properties to simplify the inversion process.

Contribution

It presents a novel approach that uses Toeplitz matrices and discrete variable representation for efficient density of states calculation.

Findings

Exact inversion of the operator E-H for localized potentials

Reduction of the problem to finite matrix inversion using Toeplitz properties

Provides a direct and precise method for density of states calculation

Abstract

A direct and exact method for calculating the density of states for systems with localized potentials is presented. The method is based on explicit inversion of the operator $E-H$. The operator is written in the discrete variable representation of the Hamiltonian, and the Toeplitz property of the asymptotic part of the obtained {\it infinite} matrix is used. Thus, the problem is reduced to the inversion of a {\it finite} matrix.