| Project/Area Number |
14540259
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
素粒子・核・宇宙線
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| Research Institution | Kyoto University |
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Principal Investigator |
SASAKI Ryu Kyoto University, Yukawa Institute for Theoretical Physics, Associate Professor, 基礎物理学研究所, 助教授 (20154007)
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| Co-Investigator(Kenkyū-buntansha) |
TAKASAKI Kanehisa Kyoto University, Graduate School of Human and Environmental Studies, Professor, 大学院・人間・環境学研究科, 教授 (40171433)
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| Project Period (FY) |
2002 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2002: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | integrable models / quantum inerrability / Calogero-Moser Systems / Ruijsenaars-Schneider Systems / elliptic quantum groups / classical equilibrium point / hypergeometric orthogonal polynomials / infinite dimensional Grassmannian manifold / ソリトン方程式 / 代数曲線 / Lax表示 |
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| Research Abstract |
A general theorem relating the $o(hbar)$ part of the quantum spectrum to the frequencies of coupled oscillations at the equilibrium is proved for the systems having discrete energy spectrum. The theorem is verified broadly and generally for the Calogero-Sutherland and Ruijsenaars-Schneider-van Diejen systems. The polynomials describing the equilibrium points of exactly and quasi-exactly solvable systems are now quite well understood. The classical orthogonal polynomials, the Hermite, Laguerre, and Jacobi polynomials and their deformation, the Meixner-Pollaczek, continuous (dual) Hahn, Wilson, and Askey-Wilson polynomials are well-known. We have proved that the latter are describing the equilibrium positions of the Ruijsenaars-Schneider-van Diejen systems, which are integrable deformation of the Calogero-Sutherland systems. Moreover, these deformed polynomials are the exact eigenfunctions of the 'discrete' single particle quantum mechanics with shape-invariant potentials. The equations determining these equilibrium positions and those determining the spectra of quasi-exactly solvable systems have a form very similar to the Bethe ansatz equation, which suggests an interesting direction of new research. As for exactly and quasi-exactly solvable multiparticle/spin systems with the most general elliptic potentials, the high-spin Belavin systems and the elliptic quantum groups associated with the solutions of the Ruijsenaars-Schneider systems together with the structure of their Bethe ansatz equations have seen substantial progress.
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