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- Title
Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals.
- Authors
Kazufumi Kimoto; Masato Wakayama
- Abstract
The purpose of the present paper is to study the number theoretic properties of the special values of the spectral zeta functions of the non-commutative harmonic oscillator (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations, and deepen the understanding of the spectrum of the NcHO. We study first the general expression of special values of the spectral zeta function Q.s/of the NcHO at s = n .n = 2; 3; : : : / and then the generating and meta-generating functions for Apéry-like numbers defined through the analysis of special values Q.n/. Actually, we show that the generating function w2n of such Apéry-like numbers appearing (as the "first anomaly") in Q.2n/for n = 2 gives an example of automorphic integral with rational period functions in the sense of Knopp, but still a better explanation remains to be clarified explicitly for n > 2. This is a generalization of our earlier result on showing that w2 is interpreted as a €.2/-modular form of weight 1. Moreover, certain congruence relations over primes for "normalized" Apérylike numbers are also proven. In order to describe w2n in a similar manner as w2, we introduce a differential Eisenstein series by using analytic continuation of a classical generalized Eisenstein series due to Berndt. The differential Eisenstein series is actually a typical example of the automorphic integral of negative weight. We then have an explicit expression of w4 in terms of the differential Eisenstein series. We discuss also shortly the Hecke operators acting on such automorphic integrals and relating Eichler's cohomology group.
- Subjects
HARMONIC oscillators; INTEGRALS; NONCOMMUTATIVE function spaces; EISENSTEIN series; ZETA functions
- Publication
Annales de l'Institut Henri Poincaré D, 2023, Vol 10, Issue 2, p205
- ISSN
2308-5827
- Publication type
Article
- DOI
10.4171/AIHPD/129