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

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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.