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  • Open Access

Analytical singular-value structure of analytic-continuation kernels from Slepian information theory

Masayuki Ohzeki*

  • Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan; Department of Physics, Institute of Science Tokyo, Tokyo 152-8551, Japan; Research and Education Institute for Semiconductors and Informatics, Kumamoto University, Kumamoto 860-8555, Japan; and Sigma-i Co., Ltd., Tokyo 108-0075, Japan

  • *Contact author: mohzeki@tohoku.ac.jp
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APS Open Sci. 1, 000145 – Published 21 September, 2026

DOI: https://doi.org/10.1103/8yw5-dhd4

Abstract

Analytic continuation from imaginary-time Green’s functions to real-frequency spectra is a central ill-posed inverse problem in quantum many-body physics. We show that the thermal kernel admits an analytical generalized singular-value structure once its purely dynamical part is separated from the statistical weight imposed by the heat bath. The dynamical kernel is the imaginary-bandwidth continuation of Slepian’s finite Fourier transform and is governed by the same Sturm-Liouville algebra that yields prolate spheroidal wave functions. Fermionic and bosonic statistics then enter as gauge transformations of the frequency-space inner product, giving a generalized singular system and self-adjoint effective potentials in the corresponding weighted space. The Shannon number, Nc=βωmax/π, fixes the upper information capacity of this pure Laplace channel. Finally, the optimal sampling points are obtained as eigenvalues of a Legendre colleague matrix, giving a deterministic compressed-sensing grid without iterative root searches.

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APS Open Sci. 1, 000145– Published 21 September, 2026
Vol. 1
  • Received 26 May 2026
  • Revised 11 July 2026
  • Accepted 21 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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