2013 Fiscal Year Final Research Report
Mathematics for Non-commutative Harmonic Oscillators and Representation Theory of alpha-determinants
| Project/Area Number |
21340011
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kyushu University |
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Principal Investigator |
WAKAYAMA Masato 九州大学, マス・フォア・インダストリ研究所, 教授 (40201149)
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| Project Period (FY) |
2009-04-01 – 2014-03-31
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| Keywords | 非可換調和振動子 / ゼータ関数 / 表現論 / α行列式 / 不変式論 / 保型形式 |
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| Research Abstract |
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.
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