The Toeplitz matrix $e^{- \kappa |i-j|}$ and its application to a layered electron gas

TL;DR

This paper provides an explicit analytical solution for the eigen-spectrum of a specific Toeplitz matrix and applies it to analytically determine plasma modes in a layered electron gas system, bypassing traditional methods.

Contribution

It introduces a straightforward elementary method to solve the eigen-spectrum of the Toeplitz matrix $e^{-ppa|i-j|}$ and applies it to layered electron gas plasma modes.

Findings

Explicit eigen-spectrum solution for the Toeplitz matrix.

Analytical determination of plasma modes in layered electron gases.

Discussion of extensions to related matrix classes.

Abstract

We present an explicit solution of the eigen-spectrum Toeplitz matrix $C_{ij}= e^{- \kappa |i-j|}$ with $0\leq i,j \leq N$ and apply it to find analytically the plasma modes of a layered assembly of 2-dimensional electron gas. The solution is found by elementary means that bypass the Wiener-Hopf technique usually used for this class of problems. It rests on the observation that the inverse of $C_{ij}$ is effectively a nearest neighbor hopping model with a specific onsite energies which can in turn be diagonalized easily. Extensions to a combination of a Toeplitz and Hankel matrix, and to a generalization of $C_{ij}$, are discussed at the end of the paper.