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

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.