Exact Sum Rules and Zeta Generating Formulas from the ODE/IM correspondence
Syo Kamata

TL;DR
This paper develops a spectral-zeta framework for quantum systems with PT-symmetric and Hermitian potentials, deriving exact sum rules and zeta generating formulas via the ODE/IM correspondence, revealing structural properties and information loss phenomena.
Contribution
It introduces a novel spectral-zeta approach using the ODE/IM correspondence to derive exact sum rules and zeta generating formulas for PT-symmetric and Hermitian quantum potentials, highlighting structural non-invertibility.
Findings
Exact sum rules reproduce WKB results for Hermitian cases.
Identification of algebraic information loss and non-invertibility in spectral mappings.
Connection of PT and Hermitian spectra via spectral-zeta formulas, supporting the ABS conjecture.
Abstract
We develop a spectral-zeta framework for quantum mechanics with the -symmetric potential and the Hermitian potential , based on the fusion relations of the T-system. Using the ODE/IM correspondence, we construct exact sum rules (ESRs) and zeta generating formulas (ZGFs) for the spectral zeta functions (SZFs) . In contrast to recursive T-Q relations, the ZGFs provide fixed-source, closed-form mappings between different fusion sectors. For Hermitian , our ESRs reproduce exact WKB results, extending them systematically to sectors and (half-)integer . Our analysis reveals a phenomenon of \textit{algebraic information loss}, distinct from analytic ambiguity. The structure is governed by a selection rule ${\cal…
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Taxonomy
TopicsQuantum Mechanics and Non-Hermitian Physics · Mathematical functions and polynomials · Spectral Theory in Mathematical Physics
