Equivalence between the eigenvalue problem of non-commutative harmonic oscillators and existence of holomorphic solutions of Heun differential equations, eigenstates degeneration, and the Rabi model (Q2796763)

The paper is mainly devoted to the study of the spectral problem for the non-commutative harmonic oscillator (NcHO) NEWLINE\[NEWLINE\begin{aligned} Q_{\alpha, \beta}(x,D)&=A(-\frac{1}{2} \frac{d^{2}}{dx^{2}} + \frac{1}{2} x^{2})+J(x \frac{d}{dx} +\frac{1}{2}),\\ A&=\begin{bmatrix} \alpha&0\\0&\beta \end{bmatrix},\;J=\begin{bmatrix} 0&-1\\1&0 \end{bmatrix},\;\alpha\beta>1.\end{aligned}NEWLINE\]NEWLINE The maximal operator associated to \(Q_{\alpha, \beta}(x,D)\) is globally elliptic and selfadjoint. Therefore its spectrum consists of a sequence of eigenvalues of finite multiplicity converging to \(+\infty\).NEWLINENEWLINEThe first main result asserts that there exist linear bijections between the spaces of even (odd) eigenfunctions corresponding to an eigenvalue \(\lambda\) and the solutions of a Heun equation \(H_{\lambda}^{+}f=0\) \((H_{\lambda}^{-}f=0)\). The coefficients of the Heun equations are expressed in terms of \(\lambda, \alpha\) and \( \beta\).NEWLINENEWLINEThe second main result states that the multiplicity of each eigenvalue \(\lambda\) is at most two. It is proved that, when \(\alpha \neq \beta\), the multiplicity of \(\lambda\) is two if and only if either there are even or odd eigenfunctions of finite type (finite linear combinations of even (odd) twisted Hermite functions) or there are even and odd eigenfunctions of infinite type. Examples for doubly degenerations of the spectrum of NcHO are given.NEWLINENEWLINEThe final section consists in a discussion of a connection between a two degree element of the universal enveloping algebra of the special linear algebra \(\mathfrak{s}\mathfrak{l}_{2} \) arising from the NcHO through the oscillator representation and the quantum Rabi model.