Analytical Singular-Value Structure of Analytic-Continuation Kernels from Slepian Information Theory

• URL: http://arxiv.org/abs/2605.26586v1
• Date: Tue, 26 May 2026 06:16:24 GMT
• Title: Analytical Singular-Value Structure of Analytic-Continuation Kernels from Slepian Information Theory
• Abstract summary: Analytic continuation from imaginary-time Green's functions to real-frequency spectra is a central ill-posed inverse problem in quantum many-body physics.<n>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.
• Score: 0.913755431537592
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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, producing self-adjoint effective potentials but no numerical kernel diagonalization. The Shannon number, $N_c=β\wmax/π$, 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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