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面白そうな論文

• 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に関する面白そうな論文を列挙する．いい加減なコメント付き．

■超可積分系

• S. Wojciechowski. Superintegrability of the Calogero-Moser System. Phys. Lett. A 95, 279 (1983).

• Evans, N. W. (1990). Superintegrability in classical mechanics. Physical Review A, 41(10), 5666.

• 片山登揚. "超可積分系についての考察 (力学系と微分幾何学)." 数理解析研究所講究録 1070 (1998): 52-68.

• W. Miller, ‘Second Order Superintegrable Systems in Three Dimensions’, SIGMA 1, 015 (2005).

• E.G. Kalnins, J.M. Kress and W. Miller Jr., ‘Classification of superintegrable systems in three dimensions’, Quantum Theory and Symmetries IV, ed. V.K. Dobrev, Heron Press, Sofia (2006).

• Tempesta, Piergiulio, and Giorgio Tondo. "Generalized Lenard chains, separation of variables, and superintegrability." Physical Review E 85.4 (2012): 046602. / https://arxiv.org/abs/1205.6937

• Miller Jr, W., Post, S., & Winternitz, P. (2013). Classical and quantum superintegrability with applications. Journal of Physics A: Mathematical and Theoretical, 46(42), 423001. / https://arxiv.org/abs/1309.2694
超可積分系のレビュー．

■双Hamiltonian構造

• Tempesta, P., & Tondo, G. (2012). Generalized Lenard chains, separation of variables, and superintegrability. Physical Review E, 85(4), 046602. / https://arxiv.org/abs/1205.6937
超可積分系と双Hamiltonian構造の関係について．Kepler系も例として出てくる．

■水素原子散乱状態，Rutherford散乱

• Perelomov, A. M., and V. S. Popov. "The Lorentz group as a dynamic symmetry group of the hydrogen atom." SOVIET PHYSICS JETP 23.1 (1966).
水素原子の散乱状態の対称性がローレンツ群の対称性と同じになることについて．

• Shatilov, D. A., and Z. K. Silagadze. "A shadow of the repulsive Rutherford scattering and Hamilton vector." European Journal of Physics 42.3 (2021): 035001. / https://arxiv.org/abs/2101.03543

■水素原子と変数分離

• Kalnins, E. G., W. Miller, Jr, and P. Winternitz. "The Group O(4), Separation of Variables and the Hydrogen Atom." SIAM Journal on Applied Mathematics 30.4 (1976): 630-664
非相対論的水素原子のシュレディンガー方程式は４種類の直交座標系で変数分離可能であるが，SO(4)群の表現論の観点でまとめられている．また，運動量空間中では同方程式は立体射影変換をすることで超球面S^3上の自由粒子のシュレディンガー方程式に帰着するが，こちらは６種類の直交座標で変数分離可能である．こちらについてもSO(4)群の表現論の観点でまとめられている．

• Kereselidze, T., Z. S. Machavariani, and G. Chkadua. "Explicit spheroidal wave functions of the hydrogen atom." The European Physical Journal D 63.1 (2011): 81-87.
偏長楕円座標系で変数分離した際の水素原子の波動関数について．

■水素原子波動関数の運動量表示

• Lombardi, John R. "The momentum representation of the hydrogen atom in paraboloidal coordinates." Chemical Physics 530 (2020): 110636. / https://arxiv.org/abs/1906.02375

• Kirchbach, M., and J. A. Vallejo. "Wave functions of the Hydrogen atom in the momentum representation." Journal of Physics A: Mathematical and Theoretical 56.12 (2023): 125302. / https://arxiv.org/abs/2208.13989

• Lombardi, J. R., and J. F. Ogilvie. "The hydrogen atom in the momentum representation; a critique of the variables comprising the momentum representation." Chemical Physics 538 (2020): 110886.

■水素原子スペクトルとモノドロミー欠陥

• Dullin, Holger R., and Holger Waalkens. "Defect in the joint spectrum of hydrogen due to monodromy." Physical Review Letters 120.2 (2018): 020507. / https://arxiv.org/abs/1612.00823
非相対論的水素原子のシュレディンガー方程式は４種類の直交座標系で変数分離可能である．そのうちの prolate spheroidal 座標系による変数分離解について，固有値たちを量子数でラベリングして量子数空間に並べると欠陥構造が見られる．他の座標系ではそんなことがない．

■水素原子と特殊関数

• Bump, D., Choi, K. K., Kurlberg, P., & Vaaler, J. (2000). A local Riemann hypothesis, I. Mathematische Zeitschrift, 233(1), 1-18.

• Dunkl, Charles F. "A Laguerre polynomial orthogonality and the hydrogen atom." Analysis and Applications 1.02 (2003): 177-188. / https://arxiv.org/abs/math-ph/0011021

• Coffey, Mark W. "Special functions and the Mellin transforms of Laguerre and Hermite functions." Analysis 27.1 (2007): 95-108. / https://arxiv.org/abs/math-ph/0612085v1
Laguerre函数をMellin変換するとそのゼロ点が Re s = 1/2 上に並ぶという話．リーマン予想に似ている．

• Coletti, C., D. Calderini, and V. Aquilanti. "d-Dimensional Kepler–Coulomb Sturmians and hyperspherical harmonics as complete orthonormal atomic and molecular orbitals." Advances in Quantum Chemistry. Vol. 67. Academic Press, 2013. 73-127.

■水素原子と南部力学

• Curtright, Thomas, and Cosmas Zachos. "Classical and quantum Nambu mechanics." Physical Review D 68.8 (2003): 085001. / https://arxiv.org/abs/hep-th/0212267
古典/量子南部力学に関するレビュー．水素原子についても言及あり．

• Zachos, Cosmas, and Thomas Curtright. "Branes, quantum Nambu brackets and the hydrogen atom." Czechoslovak journal of physics 54.11 (2004): 1393-1398. / https://arxiv.org/abs/math-ph/0408012
水素原子やKepler系を始め極大超可積分系における物理量の時間発展は南部括弧を使って記述することができる．

■水素原子経路積分，グリーン関数

• Wichmann, Eyvind H., and Ching‐Hung Woo. "Integral representation for the nonrelativistic Coulomb Green's function." Journal of Mathematical Physics 2.2 (1961): 178-180.
水素原子のグリーン関数．

• Schwinger, Julian. "Coulomb Green's function." Journal of Mathematical Physics 5.11 (1964): 1606-1608.
水素原子のグリーン関数．1940年代にSchwinger自身の講義ノートに書いていたらしいが，それを論文にしたもの．

• Duru, I. Hakki, and H. Kleinert. "Solution of the path integral for the H-atom." Physics Letters B 84.2 (1979): 185-188.

• Duru, I. H., and H. Kleinert. "Quantum Mechanics of H‐Atom from Path Integrals." Fortschritte der Physik 30.8 (1982): 401-435.

• Ho, Roger, and Akira Inomata. "Exact-path-integral treatment of the hydrogen atom." Physical Review Letters 48.4 (1982): 231.

• Inomata, Akira. "Alternative exact-path-integral treatment of the hydrogen atom." Physics Letters A 101.5-6 (1984): 253-257.

• Fujikawa, Kazuo. "Path integral of the hydrogen atom, Jacobi's principle of least action and one-dimensional quantum gravity." Nuclear physics B 484.1-2 (1997): 495-520. / https://arxiv.org/abs/hep-th/9602080v2

■相対論的水素原子

• Johnson, M. H., and B. A. Lippmann. "Relativistic Kepler problem." PHYSICAL REVIEW. Vol. 78. No. 3. ONE PHYSICS ELLIPSE, COLLEGE PK, MD 20740-3844 USA: AMERICAN PHYSICAL SOC, 1950.

• Kähler, E.: Die Dirac-Gleichung. Abh. Dtsch. Akad. Wiss. Berlin, Kl. Math. Phys. Tech. 1, 1–38 (1961).
幾何代数の手法を用いて水素原子のDirac方程式に相当する方程式を解く話がある．

• Dahl, J. P. "Physical interpretation of the Runge-Lenz vector." Physics Letters A 27.1 (1968): 62-63.
LRLベクトルの起源は特殊相対論にあると言っている．

• Biedenharn, Lawrence Christian. "The “Sommerfeld Puzzle” revisited and resolved." Foundations of Physics 13.1 (1983): 13-34.
Sommerfeldは電子がスピン1/2を持つことを知らずして，ディラック方程式から予測される水素原子のエネルギー準位を求めることができた．この謎に関して議論している．

• Dahl, Jens Peder, and Thomas Jøorgensen. "On the Dirac–Kepler problem: The Johnson–Lippmann operator, supersymmetry, and normal‐mode representations." International Journal of Quantum Chemistry 53.2 (1995): 161-181.

• Dahl, Jens Peder. "Physical origin of the Runge-Lenz vector." Journal of Physics A: Mathematical and General 30.19 (1997): 6831.
LRLベクトルの起源は特殊相対論にあると言っている．

• Boyer, Timothy H. "Unfamiliar trajectories for a relativistic particle in a Kepler or Coulomb potential." American Journal of Physics 72.8 (2004): 992-997. / https://arxiv.org/abs/physics/0405090v1

• Katsura, Hosho, and Hideo Aoki. "Exact supersymmetry in the relativistic hydrogen atom in general dimensions—supercharge and the generalized Johnson-Lippmann operator." Journal of mathematical physics 47.3 (2006): 032301. / https://arxiv.org/abs/quant-ph/0410174v1
相対論的水素原子に対するDirac方程式においてはLRLベクトルは保存されないが，代わりにJohnson-Lippmann演算子というものが保存されることが知られている．この演算子を任意の次元の水素原子Dirac方程式に拡張している．

• Vargas, Jose G. "The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus." Foundations of Physics 38.7 (2008): 610-647.

• Sakata, Y., Schneider, R., Tachikawa, Y., & Yamaura, T. "On hydrogen-like bound states in N = 4 super Yang-Mills." Journal of High Energy Physics 2017.9 (2017): 1-11. / https://arxiv.org/abs/1706.07587
相対論的水素原子（Klein-Gordon版）においては束縛状態のso(4)対称性が破れるが，それを復活させる話．

■水素原子のSUSY拡張

• Kirchberg, A., Länge, J. D., Pisani, P. A. G., & Wipf, A. (2003). Algebraic solution of the supersymmetric hydrogen atom in d dimensions. Annals of Physics, 303(2), 359-388. / https://arxiv.org/abs/hep-th/0208228

• Wipf, A., Kirchberg, A., & Länge, D. (2006). Algebraic Solution of the Supersymmetric Hydrogen Atom. Bulg. J. Phys, 33, 206-216. / https://arxiv.org/abs/hep-th/0511231

■一次元水素原子

• Loudon, Rodney. "One-dimensional hydrogen atom." American journal of physics 27.9 (1959): 649-655.
一次元水素に関する研究のさきがけ．

• Loudon, Rodney. "One-dimensional hydrogen atom." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472.2185 (2016): 20150534.

• Pérez-Obiol, Axel, and Taksu Cheon. "One dimensional hydrogen problem with general connection condition at the origin and non-Rydberg spectra." Physica E: Low-dimensional Systems and Nanostructures 119 (2020): 114013. / https://arxiv.org/abs/1911.12966

■水素原子コヒーレント状態

• Klauder, John R. "Coherent states for the hydrogen atom." Journal of Physics A: Mathematical and General 29.12 (1996): L293. / https://arxiv.org/abs/quant-ph/9511033

■Kustaanheimo-Stiefel変換

• Hestenes, David, and Pertti Lounesto. "Geometry of spinor regularization." Celestial mechanics 30.2 (1983): 171-179.

• Kibler, M., A. Ronveaux, and T. Negadi. "On the hydrogen‐oscillator connection: Passage formulas between wave functions." Journal of mathematical physics 27.6 (1986): 1541-1548.

• Kibler, M., & Winternitz, P. (1988). Lie algebras under constraints and non-bijective canonical transformations. Journal of Physics A: Mathematical and General, 21(8), 1787.

• Lambert, D., & Kibler, M. (1988). An algebraic and geometric approach to non-bijective quadratic transformations. Journal of Physics A: Mathematical and General, 21(2), 307.

• Schumacher, P. W. (1987). Results of true-anomaly regularization in orbital mechanics (Doctoral dissertation, Virginia Polytechnic Institute and State University).

• Cordani, Bruno. "On the generalisation of the Kustaanheimo-Stiefel transformation." Journal of Physics A: Mathematical and General 22.13 (1989): 2441.

• Le Van Hoang and L.I. Komarov. "Theory of the generalized Kustaanheimo-Stiefel transformation." Physics Letters A 177.2 (1993): 121-124

• Waldvogel, J. (2008). Quaternions for regularizing celestial mechanics: the right way. Celestial Mechanics and Dynamical Astronomy, 102(1), 149-162.

• Saha, P. (2009). Interpreting the Kustaanheimo–Stiefel transform in gravitational dynamics. Monthly Notices of the Royal Astronomical Society, 400(1), 228-231.

• Marquette, Ian. "Generalized five-dimensional Kepler system, Yang-Coulomb monopole, and Hurwitz transformation." Journal of mathematical physics 53.2 (2012): 022103. / https://arxiv.org/abs/1109.4672

• Vicens, J. R. (2016). Regularization in Astrodynamics: applications to relative motion, low-thrust missions, and orbit propagation (Doctoral dissertation, Universidad Politécnica de Madrid).

• Lavrenov, A. (2019). Generalized KS transformations, $ ND $ singular oscillator and generalized MICZ-Kepler system. arXiv preprint arXiv:1908.03572.

• Bars, I., & Rosner, J. L. (2020). Duality between hydrogen atom and oscillator systems via hidden SO (d, 2) symmetry and 2T-physics. Journal of Physics A: Mathematical and Theoretical, 53(23), 234001. / https://arxiv.org/abs/2001.08818

• Inomata, A., & Junker, G. (2021). Power law duality in classical and quantum mechanics. Symmetry, 13(3), 409.

• Dereli, T., Nounahon, P., & Popov, T. (2022, February). A remarkable dynamical symmetry of the Landau problem. In Journal of Physics: Conference Series (Vol. 2191, No. 1, p. 012009). IOP Publishing.

■水素原子と無限次元リー代数

• Daboul, Jamil, Peter Slodowy, and Claudia Daboul. "The hydrogen algebra as centerless twisted Kac-Moody algebra." Physics Letters B 317.3 (1993): 321-328.
水素原子の力学的対称性は束縛状態ならばリー代数so(4)，散乱状態ならばso(3,1)というリー代数で説明される．これらのリー代数はエネルギー同一の状態がなす部分空間に制限することによって得られるものである．この制限を取り払うと有限リー代数の範疇では収まらず，無限次元リー代数が出てくるというな話．

• Daboul, Claudia, and Jamil Daboul. "From hydrogen atom to generalized Dynkin diagrams." Physics Letters B 425.1-2 (1998): 135-144.
上記論文を一般D次元に拡張してそれぞれに対する無限次元リー代数の性質を同定したという話．

• Daboul, Jamil, and Pavel Winternitz. "Time-dependent realization of the infinite-dimensional hydrogen algebra." Physics Letters A 282.3 (2001): 163-168.

• Daboul, J., and P. Winternitz. "A superintegrable time-dependent system with kac-moody symmetry." Physics of Atomic Nuclei 65.6 (2002): 1000-1007.

• Daboul, Jamil. "Contraction of broken symmetries via Kac-Moody formalism." Journal of mathematical physics 47.8 (2006): 083507. / https://arxiv.org/abs/math-ph/0608008

■線形化可能問題

• Leach, P. G. L. (2003). Equivalence classes of second-order ordinary differential equations with only a three-dimensional Lie algebra of point symmetries and linearisation. Journal of mathematical analysis and applications, 284(1), 31-48.

• Gubbiotti, G., & Nucci, M. C. (2017). Are all classical superintegrable systems in two-dimensional space linearizable?. Journal of Mathematical Physics, 58(1), 012902. / https://arxiv.org/abs/1602.00705

• Nucci, M. C., & Campoamor-Stursberg, R. (2021). Maximally superintegrable systems in flat three-dimensional space are linearizable. Journal of Mathematical Physics, 62(1), 012702.

• Nucci, M. C., & Campoamor-Stursberg, R. (2022). Minimally superintegrable systems in flat three-dimensional space are also linearizable. Journal of Mathematical Physics, 63(12), 123510.

■水素原子の球面版

• Higgs, Peter W. "Dynamical symmetries in a spherical geometry. I." Journal of Physics A: Mathematical and General 12.3 (1979): 309.
水素原子のシュレディンガー方程式の球面版を作るとその力学的対称性を表す代数としてHiggs代数という有限次元W代数の一種が出てくる．リー代数の範疇には収まらない．

• De Boer, Jan, Frederique Harmsze, and Tjark Tjin. "Non-linear finite W-symmetries and applications in elementary systems." Physics Reports 272.4 (1996): 139-214. / https://arxiv.org/abs/hep-th/9503161

• Delisle-Doray, L., Hussin, V., Kuru, Ş., & Negro, J. (2019). Classical ladder functions for Rosen–Morse and curved Kepler–Coulomb systems. Annals of Physics, 405, 69-82. / https://arxiv.org/abs/1812.11582
水素原子の動径方向微分方程式を一次元量子力学の問題とみなしたときに，そのポテンシャルを周期化するとRosen-Morseポテンシャルを得る．また水素原子のシュレディンガー方程式の球面版を作るとRosen-Morseポテンシャルに似たポテンシャルを得る．これらは可解ポテンシャルであり，水素原子同様にSUSY QMの方法を使って解くことができる．

• Ohya, Satoshi. "Algebraic Description of Shape Invariance Revisited." arXiv preprint arXiv:1711.02422 (2017).

■水素原子におけるcomplete symmetry group

• Krause, J. "On the complete symmetry group of the classical Kepler system." Journal of Mathematical Physics 35.11 (1994): 5734-5748.

• Nucci, M. C., and P. G. L. Leach. "The harmony in the Kepler and related problems." Journal of Mathematical Physics 42.2 (2001): 746-764. / https://arxiv.org/abs/nlin/0011011

■水素原子と量子情報

• Toranzo, I. V., A. Martínez-Finkelshtein, and J. S. Dehesa. "Heisenberg-like uncertainty measures for D-dimensional hydrogenic systems at large D." Journal of Mathematical Physics 57.8 (2016): 082109. / https://arxiv.org/abs/1609.01113

■水素原子のCalogero拡張

• Hakobyan, Tigran, and Armen Nersessian. "Runge-Lenz vector in the Calogero-Coulomb problem." Physical Review A 92.2 (2015): 022111. / https://arxiv.org/abs/1504.00760v2

• Feigin, Misha, and Tigran Hakobyan. "Algebra of Dunkl Laplace-Runge-Lenz vector." arXiv preprint arXiv:1907.06706 (2019).

■水素原子外場中

• Le, Dai-Nam, Ngoc-Tram D. Hoang, and Van-Hoang Le. "Exact analytical solutions of a two-dimensional hydrogen atom in a constant magnetic field." Journal of Mathematical Physics 58.4 (2017): 042102.

• Chatar, W., et al. "The hydrogen atom in the van der Waals potential combined by magnetic and electric fields, Painlevé analysis, and integrability." Journal of Mathematical Physics 60.6 (2019): 062702.

• Liberman, Mikhail A. "Hydrogen atom in a magnetic field as an exactly solvable system without dynamical symmetries?." Physics Letters A (2022): 128250. / https://arxiv.org/abs/2203.02730
一様磁場中の水素原子においては超可積分性はなくなり隠れた対称性もなくなるが，それにもかかわらずある種の解析解が得られる

■Bertrandの定理

• Santos, F. C., V. Soares, and A. C. Tort. "An English translation of Bertrand’s theorem." Lat. Am. J. Phys. Educ 5.4 (2011): 694./ https://arxiv.org/abs/0704.2396

• Albouy, A., & Zhao, L. (2022). Darboux inversions of the Kepler problem. Regular and Chaotic Dynamics, 27(3), 253-280. / https://arxiv.org/abs/2201.00808

■一般化LRLベクトル

• Bacry, Henri, Henri Ruegg, and Jean-Marie Souriau. "Dynamical groups and spherical potentials in classical mechanics." Communications in Mathematical Physics 3.5 (1966): 323-333.

• Mukunda, N. "Dynamical symmetries and classical mechanics." Physical Review 155.5 (1967): 1383.
任意のn次元系においてso(n+1)及びsu(n)の代数構造は見出せるという話．すなわち水素原子や等方調和振動子等のMaximum Superintegrable systemに限らない．

• Fradkin, D. M. "Existence of the dynamic symmetries O 4 and SU 3 for all classical central potential problems." Progress of Theoretical Physics 37.5 (1967): 798-812.

• Peres, Asher. "A classical constant of motion with discontinuities." Journal of Physics A: Mathematical and General 12.10 (1979): 1711.

• Holas, A., and N. H. March. "A generalisation of the Runge-Lenz constant of classical motion in a central potential." Journal of Physics A: Mathematical and General 23.5 (1990): 735.

• Pillay, T.* &Leach, PGL. "Generalized Laplace-Runge-Lenz vectors and nonlocal symmetries." South African journal of science 95.9 (1999): 403-407.

• Leach, Peter GL, and George P. Flessas. "Generalisations of the Laplace–Runge–Lenz vector." Journal of Nonlinear Mathematical Physics 10.3 (2003): 340-423.

• Ben-Ya'acov, Uri. "Laplace–Runge–Lenz symmetry in general rotationally symmetric systems." Journal of mathematical physics 51.12 (2010): 122902. / https://arxiv.org/abs/1005.1817

• Grandati, Yves, Alain Bérard, and Hervé Mohrbach. "On Peres approach to Fradkin-Bacry-Ruegg-Souriau’s perihelion vector." Open Physics 9.1 (2011): 88-95.

• Anco, Stephen C., Tyler Meadows, and Vincent Pascuzzi. "Some new aspects of first integrals and symmetries for central force dynamics." Journal of Mathematical Physics 57.6 (2016): 062901. / https://arxiv.org/abs/1508.07258

• Chen, Zhe, Ian Marquette, and Yao-Zhong Zhang. "Extended Laplace–Runge–Lentz vectors, new family of superintegrable systems and quadratic algebras." Annals of Physics 402 (2019): 78-90. / https://arxiv.org/abs/1809.08401

• Bouarroudj, S., & Konstein, S. E. (2020). SO (4)-symmetry of mechanical systems with 3 degrees of freedom. Journal of Nonlinear Mathematical Physics, 27(1), 162-169. / https://arxiv.org/abs/1812.07458
"Not all 3d systems have o(4) symmetry"と言っおり，Mukundaの主張への反例を挙げている．

■MIC-Z Kepler系

• McIntosh, Harold V., and Arturo Cisneros. "Degeneracy in the presence of a magnetic monopole." Journal of Mathematical Physics 11.3 (1970): 896-916.

• Phan, Ngoc-Hung, and Van-Hoang Le. "Generalized Runge-Lenz vector and a hidden symmetry of the nine-dimensional MICZ-Kepler problem." Journal of mathematical physics 53.8 (2012): 082103.

■Position-dependent effective mass

• Quesne, Christiane, and V. M. Tkachuk. "Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem." Journal of Physics A: Mathematical and General 37.14 (2004): 4267.　/ https://arxiv.org/abs/math-ph/0403047

■未分類

• LE, D., & Le, V. H. (2021). REVIEW PAPER NORMED DIVISION ALGEBRAS APPLICATION TO THE MONOPOLE PHYSICS. Communications in Physics, 31(3), 235-257.

• Frauenfelder, U., & Weber, J. (2019). The fine structure of Weber’s hydrogen atom: Bohr–Sommerfeld approach. Zeitschrift für angewandte Mathematik und Physik, 70(4), 1-12. / https://arxiv.org/abs/1902.09612

• Albouy, A. (2019). Lambert’s Theorem: Geometry or Dynamics?. Celestial Mechanics and Dynamical Astronomy, 131(9), 1-30. / https://arxiv.org/abs/1711.03049

• Houri, Tsuyoshi, Makoto Sakamoto, and Kentaro Tatsumi. "Spectral intertwining relations in exactly solvable quantum-mechanical systems." Progress of Theoretical and Experimental Physics 2017.6 (2017). / https://arxiv.org/abs/1701.04307

• Zhang, P. M., et al. "“Kepler Harmonies” and conformal symmetries." Physics Letters B 792 (2019): 324-328. / https://arxiv.org/abs/1903.01436

• Subag, Eyal M. "Symmetries of the hydrogen atom and algebraic families." Journal of Mathematical Physics 59.7 (2018): 071702. / https://arxiv.org/abs/1710.07744
よくわかっていないが，水素原子の対称性についてHarish-Chandra modulesという概念を使ってうまい解釈をしているらしい．

• Meng, Guowu. "A generalization of the Kepler problem." Physics of Atomic Nuclei 71.5 (2008): 946-950. / https://arxiv.org/abs/math-ph/0509002

• 岩井敏洋, & 片山登揚. (1994). Kepler 型対称性をもつ自由度 3 のハミルトン系: 非線形可積分の例 (非線型可積分系の研究の現状と展望). 数理解析研究所講究録, 868, 198-210.

• 岩井敏洋. (1994). SU (2, 2) に付随する力学系 (力学系理論における幾何と解析). 数理解析研究所講究録, 863, 121-131.

• Chanda, Sumanto. "A Study of Geometry, Physics and Integrability of Geodesics on Curved Spaces.", D-thesis at Department of Physics University of Calcutta (2018).

• Chanda, Sumanto, Gary William Gibbons, and Partha Guha. "Jacobi–Maupertuis metric and Kepler equation." International Journal of Geometric Methods in Modern Physics 14.07 (2017): 1730002. / https://arxiv.org/abs/1612.07395

• Keane, Aidan J., Richard K. Barrett, and John FL Simmons. "The classical Kepler problem and geodesic motion on spaces of constant curvature." Journal of Mathematical Physics 41.12 (2000): 8108-8116. / https://arxiv.org/abs/1411.5355

• Houri, Tsuyoshi. "Liouville integrability of Hamiltonian systems and spacetime symmetry." Seminar, Kobe U., April 22, 2015

• Cariglia, Marco. "Conformal triality of the Kepler problem." Journal of Geometry and Physics 106 (2016): 205-209. / https://arxiv.org/abs/1508.03408

• Rosas-Ortiz, Oscar, and Rodrigo Munoz. "Non-Hermitian SUSY hydrogen-like Hamiltonians with real spectra." Journal of Physics A: Mathematical and General 36.31 (2003): 8497. / https://arxiv.org/abs/quant-ph/0302190v2
よくわかっていないが，水素原子のnon-Hermitian版を考えている．SUSY QMの方法が使えるらしい．

• Meng, Guowu. "Euclidean Jordan algebras, hidden actions, and J-Kepler problems." Journal of mathematical physics 52.11 (2011): 112104. / https://arxiv.org/abs/0911.2977v3
よくわかっていないが，Jordan代数を使ってKepler問題を拡張すると面白いらしい．

• Inzunza, Luis, and Mikhail S. Plyushchay. "Hidden superconformal symmetry: Where does it come from?." Physical Review D 97.4 (2018): 045002. / https://arxiv.org/abs/1711.00616

• Kibler, Maurice, and Tidjani Négadi. "On the q-analogue of the hydrogen atom." Journal of Physics A: Mathematical and General 24.22 (1991): 5283. / https://arxiv.org/abs/quant-ph/0408151v1
q類似水素原子．

• Gadella, M., Negro, J., Nieto, L. M., Pronko, G. P., & Santander, M. (2011). Spectrum generating algebras for the free motion in S 3. Journal of mathematical physics, 52(6), 063509. / https://arxiv.org/abs/1007.1423
超球面S^3上におけるスペクトル生成代数so(4,2)について

• Tempesta, P., & Tondo, G. (2012). Generalized Lenard chains, separation of variables, and superintegrability. Physical Review E, 85(4), 046602. / https://arxiv.org/abs/1205.6937

• Dereziński, Jan, and Przemysław Majewski. "From conformal group to symmetries of hypergeometric type equations." SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 12 (2016): 108.

• Moser, Jürgen. "Regularization of Kepler's problem and the averaging method on a manifold." Communications on pure and applied mathematics 23.4 (1970): 609-636.

• Milnor, John. "On the geometry of the Kepler problem." The American Mathematical Monthly 90.6 (1983): 353-365.

• Nambu, Yoichiro. "Infinite-component wave equations with hydrogenlike mass spectra." Physical Review 160.5 (1967): 1171.

• Zhang, S. (1999). An extended hydrogen atom with Yangian (sl (2)). Journal of Physics A: Mathematical and General, 32(6), 1083.

• Zhang, Rui-Bin. "Orthosymplectic Lie superalgebras in superspace analogues of quantum Kepler problems." Communications in mathematical physics 280.2 (2008): 545-562. / https://arxiv.org/abs/0709.0565

• Iwai, Toshihiro. "Quantization of the conformal Kepler problem and its application to the hydrogen atom." Journal of Mathematical Physics 23.6 (1982): 1093-1099.

• Mardoyan, L. G., A. N. Sissakian, and V. M. Ter-Antonyan. "Hidden symmetry of the Yang–Coulomb monopole." Modern Physics Letters A 14.19 (1999): 1303-1307. / https://arxiv.org/abs/hep-th/9803010

• Rowe, D. J., M. J. Carvalho, and J. Repka. "Dual pairing of symmetry and dynamical groups in physics." Reviews of Modern Physics 84.2 (2012): 711. / https://arxiv.org/abs/1207.0148

• Cariglia, Marco. "Hidden symmetries of dynamics in classical and quantum physics." Reviews of Modern Physics 86.4 (2014): 1283. / https://arxiv.org/abs/1411.1262

• Liang, Haozhao, Jie Meng, and Shan-Gui Zhou. "Hidden pseudospin and spin symmetries and their origins in atomic nuclei." Physics Reports 570 (2015): 1-84. / https://arxiv.org/abs/1411.6774

• Wheeler, James T. "Not-so-classical mechanics: unexpected symmetries of classical motion." Canadian journal of physics 83.2 (2005): 91-138. / https://arxiv.org/abs/physics/0511054

• Wheeler, James T. "Not so classical Mechanics." (2018).

■水素原子の essential spectrum

• Rejto, P. (1966). On the essential spectrum of the hydrogen energy and related operators. Pacific Journal of Mathematics, 19(1), 109-140.

■自由落下における対称性

• Iwai, Toshihiro, and See‐Gew Rew. "Classical and quantum symmetry groups of a free‐fall particle." Journal of mathematical physics 26.1 (1985): 55-61.

• Truong, Tuong Trong. "Dynamical Symmetries of the 2D Newtonian Free Fall Problem Revisited." Symmetry 14.1 (2021): 27.

■Darboux Transformation

• Darboux, Gaston. "On a proposition relative to linear equations." CR Acad. Sci. Paris 94.physics/9908003 (1882): 1456-59.

• Rosu, Haret C. "Short survey of Darboux transformations." arXiv preprint quant-ph/9809056 (1998).

■Calogero-Moser

• Etingof, Pavel. "Lectures on Calogero-Moser systems." arXiv preprint math/0606233 (2006).

• Polychronakos, Alexios P. "Feynman’s proof of the commutativity of the Calogero integrals of motion." Annals of Physics 403 (2019): 145-151. / https://arxiv.org/abs/1812.03843

■ディラック電子系とSUSY

• Ezawa, Motohiko. "Supersymmetry and unconventional quantum Hall effect in monolayer, bilayer and trilayer graphene." Physica E: Low-dimensional Systems and Nanostructures 40.2 (2007): 269-272.

• Lu, Chi-Ken, and Igor F. Herbut. "Supersymmetric Runge–Lenz–Pauli vector for Dirac vortex in topological insulators and graphene." Journal of Physics A: Mathematical and Theoretical 44.29 (2011): 295003. / https://arxiv.org/abs/1102.5163v2

■Sl(2)

• Lindblad, G., & Nagel, B. (1970). Continuous bases for unitary irreducible representations of $ SU (1, 1) $. In Annales de l'IHP Physique théorique (Vol. 13, No. 1, pp. 27-56).

• Kostant, Bertram. "On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of Sl (2, R)." Representation Theory of the American Mathematical Society 4.8 (2000): 181-224.

■Quantum Hall

• Ikeda, Kazuki. "Hofstadter’s butterfly and Langlands duality." Journal of Mathematical Physics 59.6 (2018): 061704. / https://arxiv.org/abs/1708.00436

■剛体球

• Hara, O., Gotō, T., Tsai, S. Y., & Yabuki, H. (1967). A Quantum Theory of the Rigid Body in Terms of a Two-Component Spinor. Progress of Theoretical Physics, 37(6), 1319-1329.

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