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Spectral asymptotics for Kac–Murdock–Szeg˝o matrices

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Japan. J. Math. 13, 67–107 (2018) DOI: 10.1007/s11537-018-1640-2

Spectral asymptotics for Kac–Murdock–Szeg˝o matrices

Alain Bourget Allen Alvarez Loya Tyler McMillen

Received: 21 September 2016 / Revised: 17 October 2017 / Accepted: 29 January 2018 Published online: 2 March 2018

© The Mathematical Society of Japan and Springer Japan 2018 Communicated by: Toshiyuki Kobayashi

Abstract. Szeg˝o’s First Limit Theorem provides the limiting statistical distribution of the eigen- values of large Toeplitz matrices. Szeg˝o’s Second (or Strong) Limit Theorem for Toeplitz matrices gives a second order correction to the First Limit Theorem, and allows one to calculate asymp- totics for the determinants of large Toeplitz matrices. In this paper we survey results extending the First and Second Limit Theorems to Kac–Murdock–Szeg˝o (KMS) matrices. These are ma- trices whose entries along the diagonals are not necessarily constants, but modeled by functions.

We clarify and extend some existing results, and explain some apparently contradictory results in the literature.

Keywords and phrases:Toeplitz matrices, Kac–Murdock–Szeg˝o matrices, Szeg˝o’s Limit Theo- rem, Schrödinger operators, determinants

Mathematics Subject Classification (2010): 15B05, 47B06, 47B35, 35P20

A. BOURGET

Department of Mathematics, California State University (Fullerton), McCarthy Hall 154, Fullerton CA 92834, USA

(e-mail:[email protected]) A.A. LOYA

Department of Mathematics, California State University (Fullerton), McCarthy Hall 154, Fullerton CA 92834, USA

(e-mail:[email protected]) T. MCMILLEN

Department of Mathematics, California State University (Fullerton), McCarthy Hall 154, Fullerton CA 92834, USA

(e-mail:[email protected])

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