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 $0\le i,j\le N$ 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.

12 pages, 1 figure

References in corpus (2)

• Manipulating infrared photons using plasmons in transparent graphene superlattices
• Quasi-flat band enables subradiant two-photon bound states

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