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arXiv · 2006.15436

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

Onuttom Narayan·B Sriram Shastry
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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.

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Onuttom Narayan, B Sriram Shastry. 2020-06-27. The Toeplitz matrix $e^{- \kappa |i-j|}$ and its application to a layered electron gas. https://doi.org/10.1088/1751-8121%2Fabee67

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