Mathematics for Non-commutative Harmonic Oscillators and Representation Theory of alpha-determinants

1.通过引入余模形式的概念，描述了非对易谐振子的谱Zeta函数在S=4处的特殊值，它是Eichler积分的推广。在此基础上，我们引入了周期Eichler上同调群，并确定了SL_2(Z)的某些同余子群的结构。2.我们用Heun的常微分方程组给出了NcHO的特征值问题的完整描述(仅Ochiai在2001年给出了奇数情形)。作为副产品，利用Heun常微分方程组的单列表示，我们证明了NcHO本征值的重数至多为2。3.利用SL2的主级数和Heun常微分方程组的合流过程，我们描述了量子Rabi模型与NcHO之间的联系。

1.We described the special value at s=4 of the spectral zeta function of the non-commutative harmonic oscillator (NcHO) by introducing the notion of residual modular forms, which is a generalization of Eichler integrals. With this investigation, we introduced the periodic Eichler cohomology groups and determined the structure of certain congruence subgroups of SL_2(Z).2.We showed a complete description of the eigenvalue problem for NcHO, which is defined by a matrix-valued self-adjoint ordinary differential operator, in terms of Heun's ordinary differential equations (Only the odd case was given by Ochiai in 2001). As a by-product, using the monodromy representation of Heun's ODEs, we proved that the multiplicity of the eigenvalue of NcHO is at most two.3.We described a connection between the quantum Rabi model and NcHO, employing principal series of sl2 and confluent procedure of Heun ODE.