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役に立つ本や資料

可解構造を持つ物理模型（非相対論的水素原子模型が中心）を対象とした数理物理を考えるにあたって役立つ本や資料を列挙．

■力学的対称性

Bohm, A., Ne'Eman, Y., & Barut, A. O. (1989). Dynamical Groups and Spectrum Generating Algebras. World Scientific.
Guillemin, V., & Sternberg, S. (2006). Variations on a Theme by Kepler (Vol. 42). American Mathematical Society.
Wulfman, C. E. (2010). Dynamical Symmetry. World Scientific.
Kikoin, K., Kiselev, M., & Avishai, Y. (2011). Dynamical symmetries for nanostructures: implicit symmetries in single-electron transport through real and artificial molecules. Springer Science & Business Media.
Cordani, B. (2012). The Kepler problem: group theoretical aspects, regularization and quantization, with application to the study of perturbations (Vol. 29). Birkhäuser.
Lee, J. H. (2016). Geometry of the Kepler Problem and the Kepler-Lorentz Duality. Bachelor thesis of Amherst College. (link of pdf)
Bander, M., & Itzykson, C. (1966). Group theory and the hydrogen atom (I). Reviews of modern Physics, 38(2), 330. (link of pdf)
Bander, M., & Itzykson, C. (1966). Group theory and the hydrogen atom (II). Reviews of Modern Physics, 38(2), 346. (link of pdf)
Rowe, D. J., Carvalho, M. J., & Repka, J. (2012). Dual pairing of symmetry and dynamical groups in physics. Reviews of Modern Physics, 84(2), 711. (link)
Cariglia, M. (2014). Hidden symmetries of dynamics in classical and quantum physics. Reviews of Modern Physics, 86(4), 1283. (link)
m-a-o さんブログポスト『水素原子の表現論』，はてなブログ. (link)
国場敦夫 (2007)『ラプラス-ルンゲ-レンツベクトル--Gruppen Pestの始祖的例題』数理科学 45(7), 50-55, 2007-07 サイエンス社．(link of pdf)
島和久 (1969)『群の表現と量子力学』数理科学 7(12), 50-52，1969-12 ダイヤモンド社．

■解析力学

Marsden, J. E., & Ratiu, T. S. (2010). Introduction to Mechanics and Symmetry, Second Edition: A Basic Exposition of Classical Mechanical Systems (Texts in Applied Mathematics (17)) . Springer New York.

■量子力学

Schiff, L. I. (1968). Quantum Mechanics (Pure & Applied Physics). McGraw-Hill College.

■経路積分

Kleinert, H. (2009). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th edition. World Scientific. (link)
Grosche, C. (1993). An Introduction into the Feynman Path Integral. arXiv:hep-th/9302097v1. (link)

■変数分離

Moon, P.,& Spencer, D. E. (1971). Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions. Springer Berlin Heidelberg.
Miller, W., Kalnins, E. G., & Kress. J. M. (2018). Separation of Variables and Superintegrability: The symmetry of solvable systems. IOP Expanding Physics.
Kalnins, E. G., Miller, Jr, W., & Winternitz, P. (1976). The Group O(4), Separation of Variables and the Hydrogen Atom. SIAM Journal on Applied Mathematics, 30(4), 630-664. (link)
Boyer, C. P., Kalnins, E. G., & Miller, W. (1976). Symmetry and separation of variables for the Helmholtz and Laplace equations. Nagoya Mathematical Journal, 60, 35-80. (link)

■因数分解解法

Infeld, L., & Hull, T. E. (1951). The factorization method. Reviews of modern Physics, 23(1), 21. (link)

■超可積分系

Miller Jr, W., Post, S., & Winternitz, P. (2013). Classical and quantum superintegrability with applications. Journal of Physics A: Mathematical and Theoretical, 46(42), 423001. (link)

■KS変換、KS正則化

Kustaanheimo, P., Schinzel, A., Davenport, H., & Stiefel, E. (1965). Perturbation theory of Kepler motion based on spinor regularization. Journal für die reine und angewandte Mathematik, 1965(218), 204-219. (link)
Cornish, F. H. J. (1984). The hydrogen atom and the four-dimensional harmonic oscillator. Journal of Physics A: Mathematical and General, 17(2), 323. (link)
Saha, P. (2009). Interpreting the Kustaanheimo–Stiefel transform in gravitational dynamics. Monthly Notices of the Royal Astronomical Society, 400(1), 228-231. (link)
Vicens, J. R. (2016). Regularization in Astrodynamics: applications to relative motion, low-thrust missions, and orbit propagation. Doctoral dissertation, Universidad Politécnica de Madrid. (link of pdf)
De Vries, S. (2018). Reductions on the Kepler problem. Master's Thesis, University of Groningen. (link)

■Conformal 正則化

Moser, J. (1970). Regularization of Kepler's problem and the averaging method on a manifold. Communications on pure and applied mathematics, 23(4), 609-636. (link)
Kummer, M. (1982). On the regularization of the Kepler problem. Communications in Mathematical Physics, 84(1), 133-152. (link)
Cordani, B. (1986). Conformal regularization of the Kepler problem. Communications in mathematical physics, 103(3), 403-413. (link)

■超対称性量子力学（SUSY QM）

Cooper, F., Khare, A., & Sukhatme, U. (1995). Supersymmetry and quantum mechanics. Physics Reports, 251(5-6), 267-385. (link)

■コヒーレント状態

Glauber, R. J. (1963). The quantum theory of optical coherence. Physical Review, 130(6), 2529. (link)

■Complete Symmetry

■Conformal Mechanics

Saghatelian, A. (2014). Action-Angle Variables In Conformal Mechanics. PhD thesis. arXiv preprint arXiv:1410.6515. (link)

■モノドロミー

Chen, C., Ivory, M., Aubin, S., & Delos, J. B. (2014). Dynamical monodromy. Physical Review E, 89(1), 012919. (link)
Dullin, H. R., & Waalkens, H. (2018). Defect in the joint spectrum of hydrogen due to monodromy. Physical review letters, 120(2), 020507. (link)

■一次元水素原子

■MICZ-Kepler

■南部力学

■直交多項式

Szego, G. (1981). Orthogonal Polynomials. (Colloquium Publications) American Mathematical Soc..
Chihara, T. S. (2011). An Introduction to Orthogonal Polynomials. (Dover Books on Mathematics) Dover Publications.
青本和彦（2013）直交多項式入門．数学書房．
佐々木隆（2016）. 可解な量子力学系の数理物理 : 直交多項式の生み出す多様な展開，（臨時別冊・数理科学, . SGCライブラリ||SGC ライブラリ ; 122）．サイエンス社．

■調和解析

竹内勝（1975）．現代の球関数，(数学選書) ．岩波書店．
岡本清郷（1980）．等質空間上の解析学―リー群論的方法による序説，（紀伊国屋数学叢書〈19〉）．紀伊国屋書店．
野村隆昭（2018）．球面調和函数と群の表現．日本評論社．

■超幾何級数，超幾何関数

原岡喜重（2002）．超幾何関数 (すうがくの風景)．朝倉書店．

■楕円関数

梅村浩（2000）．楕円関数論―楕円曲線の解析学．東京大学出版会．

■特殊関数

Arfken, G. B., Weber, H. J., & Harris, F. E. (2012). Mathematical Methods for Physicists: A Comprehensive Guide. Academic Press.
Lebedev, N. N., translated by Silverman, R. A. (1972). Special Functions & Their Applications. Dover Books on Mathematics.
Watson, G. N. (1995). A Treatise on the Theory of Bessel Functions, Second Edition. Cambridge University Press.

■Lie群，Lie代数

松島与三（1956）．リー環論．共立出版．
Gilmore, R. (1974). Lie Groups, Lie Algebras, and Some of Their Applications. Wiley, New York.
Wybourne, B. G. (1974). Classical Groups for Physicists. Wiley, London.
Barut, A. O., & Raczka, R. (1987).Theory of Group Representations and Applications. World Scientific.
Gilmore, R. (2008). Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists. Cambridge University Press.
井ノ口順一（2018）．はじめて学ぶリー環―線型代数から始めよう．現代数学社．
島和久（1981）．連続群とその表現 (応用数学叢書)．岩波書店．
Mirman, R. (2005). Quantum Field Theory Conformal Group Theory Conformal Field Theory. iUniverse.

■群論

Alperin, J. L., & Bell, R. B. (1995). Groups and Representations. Springer, New York.

■線形代数

杉浦光夫，横沼健雄（2002）．ジョルダン標準形・テンソル代数 (岩波基礎数学選書)．岩波書店．

■微分方程式

■関数論・複素解析

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