Energy Density Bands Engineering

TL;DR

This paper develops a method to analyze quantum systems by expanding wavefunctions into orthogonal polynomials related to energy density, revealing band structures and bound states through recursion relations.

Contribution

It introduces a novel approach linking energy density bands to orthogonal polynomial zeros via a tridiagonal Hamiltonian matrix representation.

Findings

Zeros of polynomials form separated energy bands.

Bound states are located at discrete zeros in the gaps.

Number of bands equals the periodicity of recursion coefficients.

Abstract

We expand the quantum mechanical wavefunction in a complete set of square integrable orthonormal basis such that the matrix representation of the Hamiltonian operator is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction whose solution is a set of orthogonal polynomials in the energy. The polynomials weight function is the energy density of the system constructed using the Green's function, which is written in terms of the Hamiltonian matrix elements. We study the distribution of zeros of these polynomials on the real energy line based exclusively on their three-term recursion relations. We show that the zeros are generally grouped into sets belonging to separated bands on the orthogonality interval. The number of these bands is equal to the periodicity (multiplicity) of the asymptotic values of the recursion coefficients and the location of their boundaries depend only on these asymptotic values. Bound states (if they exist) are located at discrete zeros found in the gaps between the density bands that are stable against variation in the order of the polynomial for very large orders. We give examples of systems with a single, double and triple energy density bands.