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

Abstract

AbstractAs 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.KeywordsEichler integralHeun ODENon-commutative harmonic oscillatorOscillator representationRabi modelSpectral zeta functionUniversal enveloping algebraZeta regularization.