Remarks on Quantum Interaction Models by Lie Theory and Modular Forms via Non-commutative Harmonic Oscillators

  • Masato Wakayama
Part of the Mathematics for Industry book series (MFI, volume 5)


As typically the quantum Rabi model, particular attention has been paid recently to studying the spectrum of self-adjoint operators with non-commutative coefficients, not only in mathematics but also in theoretical/experimental physics, e.g. aiming at an application to quantum information processing. The non-commutative harmonic oscillator (NcHO) is a self-adjoint operator, which is a generalization of the harmonic oscillator, having an interaction term. The Rabi model is shown to be obtained by a second order element of the universal enveloping algebra of the Lie algebra \(\mathfrak {sl}_2\), which is arising from NcHO through the oscillator representation. Precisely, an equivalent picture of the model is obtained as a confluent Heun equation derived from the Heun operator defined by that element via another representation. Though the spectrum of NcHO is not fully known, it has a rich structure. In fact, one finds interesting arithmetics/geometry described by e.g. elliptic curves, modular forms in the study of the spectral zeta function of NcHO. In this article, we draw this picture, which may give a better understanding of interacting quantum models.


Eichler integral Heun ODE Non-commutative harmonic oscillator Oscillator representation Rabi model Spectral zeta function Universal enveloping algebra Zeta regularization. 


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© Springer Japan 2014

Authors and Affiliations

  1. 1.Institute of Mathematics for IndustryKyushu UniversityNishikuJapan

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