Number Theory Seminar
Hidden modularity behind the non-commutative harmonic oscillators
Speaker: Masato Wakayama (NTT Institute for Fundamental Mathematics)Date: Thursday, 15 October 2026 - 12:30Place: Aula 520, Módulo 17, Departamento de Matemáticas, UAM
Abstract:
The non-commutative harmonic oscillator (NCHO) is a mathematical model, which is a matrix valued generalization of the (quantum) harmonic oscillator, introduced more than 25 years ago. It was shown recently that the eigenvalue problems of the NCHO and Two-photon quantum Rabi model, which describes the interaction of a two-level system interacting with a bosonic mode, are equivalent.
This talk discusses the special values of the spectral zeta function—the Dirichlet series for the eigenvalues—associated with the NCHO. The special values at small integers have explicit connections to specific modular forms, Eichler forms, and families of elliptic curves; furthermore, numbers analogous to Apéry numbers (called Apéry-like numbers) emerge, exhibiting congruence relations like those of the Apéry numbers. In the case of large integers, while the analysis requires combinatorial arguments, modularity may likewise be identified through these special values. An overview of the findings obtained to date and outline several conjectures and open problems will be addressed.
The study on this topic has evolved through a long-standing collaboration with Kazufumi Kimoto, while the latter on large integer case is based on recent joint work with Wadim Zudilin.
