Geodesic
Bôcher Contractions of Conformally Superintegrable Laplace Equations
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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The explicit solvability of quantum superintegrable systems is due to symmetry, but the symmetry is often “hidden”. The symmetry generators of $2$nd order superintegrable systems in $2$ dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems on constant curvature spaces are related by geometric limits, induced by generalized Inönü–Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. However, the limits have no satisfactory Lie algebra contraction interpretations for underlying spaces with $1$- or $0$-dimensional Lie algebras. We show that these systems can be best understood by transforming them to Laplace conformally superintegrable systems, with flat space conformal symmetry group ${\rm SO}(4,{\mathbb C})$, and using ideas introduced in the 1894 thesis of Bôcher to study separable solutions of the wave equation in terms of roots of quadratic forms. We show that Bôcher's prescription for coalescing roots of these forms induces contractions of the conformal algebra $\mathfrak{so}(4,{\mathbb C})$ to itself and yields a mechanism for classifying all Helmholtz superintegrable systems and their limits. In the paper [Acta Polytechnica, to appear, arXiv:1510.09067], we announced our main findings. This paper provides the proofs and more details.
Keywords: conformal superintegrability; contractions; Laplace equations. 
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     title = {B\^ocher {Contractions} of {Conformally} {Superintegrable} {Laplace} {Equations}},
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}
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Ernest G. Kalnins; Willard Miller Jr.; Eyal Subag. Bôcher Contractions of Conformally Superintegrable Laplace Equations. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a37/

[1] Bôcher M., Ueber die Reihenentwickelungen der Potentialtheorie, B. G. Teubner, Leipzig, 1894

[2] Bromwich T. J. I., Quadratic forms and their classification by means of invariant-factors, Cambridge University Press, Cambridge, 1906 | Zbl

[3] Capel J. J., Kress J. M., “Invariant classification of second-order conformally flat superintegrable systems”, J. Phys. A: Math. Theor., 47 (2014), 495202, 33 pp., arXiv: 1406.3136 | DOI | MR | Zbl

[4] Capel J. J., Kress J. M., Post S., “Invariant classification and limits of maximally superintegrable systems in 3D”, SIGMA, 11 (2015), 038, 17 pp., arXiv: 1501.06601 | DOI | MR | Zbl

[5] Daskaloyannis C., Tanoudis Y., “Quantum superintegrable systems with quadratic integrals on a two dimensional manifold”, J. Math. Phys., 48 (2007), 072108, 22 pp., arXiv: math-ph/0607058 | DOI | MR | Zbl

[6] Evans N. W., “Super-integrability of the Winternitz system”, Phys. Lett. A, 147 (1990), 483–486 | DOI | MR

[7] Fordy A. P., “Quantum super-integrable systems as exactly solvable models”, SIGMA, 3 (2007), 025, 10 pp., arXiv: math-ph/0702048 | DOI | MR | Zbl

[8] Heinonen R., Kalnins E. G., Miller W. Jr., Subag E., “Structure relations and Darboux contractions for 2D 2nd order superintegrable systems”, SIGMA, 11 (2015), 043, 33 pp., arXiv: 1502.00128 | DOI | MR | Zbl

[9] Inönü E., Wigner E. P., “On the contraction of groups and their representations”, Proc. Nat. Acad. Sci. USA, 39 (1953), 510–524 | DOI | MR | Zbl

[10] Izmest'ev A. A., Pogosyan G. S., Sissakian A. N., Winternitz P., “Contractions of Lie algebras and separation of variables”, J. Phys. A: Math. Gen., 29 (1996), 5949–5962 | DOI | MR | Zbl

[11] Izmest'ev A. A., Pogosyan G. S., Sissakian A. N., Winternitz P., “Contractions of Lie algebras and the separation of variables: interbase expansions”, J. Phys. A: Math. Gen., 34 (2001), 521–554 | DOI | MR | Zbl

[12] Kalnins E. G., Kress J. M., Miller W. Jr., “Second-order superintegrable systems in conformally flat spaces. I: Two-dimensional classical structure theory”, J. Math. Phys., 46 (2005), 053509, 28 pp. | DOI | MR | Zbl

[13] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. II: The classical two-dimensional Stäckel transform”, J. Math. Phys., 46 (2005), 053510, 15 pp. | DOI | MR | Zbl

[14] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. III: Three-dimensional classical structure theory”, J. Math. Phys., 46 (2005), 103507, 28 pp. | DOI | MR | Zbl

[15] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. IV: The classical 3D Stäckel transform and 3D classification theory”, J. Math. Phys., 47 (2006), 043514, 26 pp. | DOI | MR | Zbl

[16] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. V: Two- and three-dimensional quantum systems”, J. Math. Phys., 47 (2006), 093501, 25 pp. | DOI | MR | Zbl

[17] Kalnins E. G., Kress J. M., Miller W. Jr., “Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties”, J. Phys. A: Math. Theor., 40 (2007), 3399–3411, arXiv: 0708.3044 | DOI | MR | Zbl

[18] Kalnins E. G., Kress J. M., Miller W. Jr., Post S., “Laplace-type equations as conformal superintegrable systems”, Adv. in Appl. Math., 46 (2011), 396–416, arXiv: 0908.4316 | DOI | MR | Zbl

[19] Kalnins E. G., Kress J. M., Miller W. Jr., Winternitz P., “Superintegrable systems in Darboux spaces”, J. Math. Phys., 44 (2003), 5811–5848, arXiv: math-ph/0307039 | DOI | MR | Zbl

[20] Kalnins E. G., Kress J. M., Pogosyan G. S., Miller W. Jr., “Completeness of superintegrability in two-dimensional constant-curvature spaces”, J. Phys. A: Math. Gen., 34 (2001), 4705–4720, arXiv: math-ph/0102006 | DOI | MR | Zbl

[21] Kalnins E. G., Miller W. Jr., “Quadratic algebra contractions and second-order superintegrable systems”, Anal. Appl. (Singap.), 12 (2014), 583–612, arXiv: 1401.0830 | DOI | MR | Zbl

[22] Kalnins E. G., Miller W. Jr., Post S., “Wilson polynomials and the generic superintegrable system on the 2-sphere”, J. Phys. A: Math. Theor., 40 (2007), 11525–11538 | DOI | MR | Zbl

[23] Kalnins E. G., Miller W. Jr., Post S., “Models for quadratic algebras associated with second order superintegrable systems in 2D”, SIGMA, 4 (2008), 008, 21 pp., arXiv: 0801.2848 | DOI | MR | Zbl

[24] Kalnins E. G., Miller W. Jr., Post S., “Coupling constant metamorphosis and $N$th-order symmetries in classical and quantum mechanics”, J. Phys. A: Math. Theor., 43 (2010), 035202, 20 pp., arXiv: 0908.4393 | DOI | MR | Zbl

[25] Kalnins E. G., Miller W. Jr., Post S., “Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials”, SIGMA, 9 (2013), 057, 28 pp., arXiv: 1212.4766 | DOI | MR | Zbl

[26] Kalnins E. G., Miller W. Jr., Reid G. J., “Separation of variables for complex Riemannian spaces of constant curvature. I: Orthogonal separable coordinates for ${\rm S}_{nC}$ and ${\rm E}_{nC}$”, Proc. Roy. Soc. London Ser. A, 394 (1984), 183–206 | DOI | MR | Zbl

[27] Kalnins E. G., Miller W. Jr., Subag E., Bôcher contractions of conformally superintegrable Laplace equations: detailed computations, arXiv: 1601.02876 | MR

[28] Kalnins E. G., Miller W. Jr., Subag E., “Laplace equations, conformal superintegrability and Bôcher contractions”, Acta Polytechnica (to appear) , arXiv: 1510.09067

[29] Koenigs G., “Sur les géodésiques a intégrales quadratiques”: Darboux G., Lecons sur la théorie générale des surfaces et les applications geométriques du calcul infinitesimal, v. 4, Chelsea, New York, 1972, 368–404

[30] Kress J. M., “Equivalence of superintegrable systems in two dimensions”, Phys. Atomic Nuclei, 70 (2007), 560–566 | DOI

[31] Miller W. Jr., Li Q., “Wilson polynomials/functions and intertwining operators for the generic quantum superintegrable system on the 2-sphere”, J. Phys. Conf. Ser., 597 (2015), 012059, 11 pp., arXiv: 1411.2112 | DOI

[32] Miller W. Jr., Post S., Winternitz P., “Classical and quantum superintegrability with applications”, J. Phys. A: Math. Theor., 46 (2013), 423001, 97 pp., arXiv: 1309.2694 | DOI | MR | Zbl

[33] Nesterenko M., Popovych R., “Contractions of low-dimensional Lie algebras”, J. Math. Phys., 47 (2006), 123515, 45 pp., arXiv: math-ph/0608018 | DOI | MR | Zbl

[34] NIST digital library of mathematical functions, http://dlmf.nist.gov/

[35] Post S., “Models of quadratic algebras generated by superintegrable systems in 2D”, SIGMA, 7 (2011), 036, 20 pp., arXiv: 1104.0734 | DOI | MR | Zbl

[36] Tempesta P., Turbiner A. V., Winternitz P., “Exact solvability of superintegrable systems”, J. Math. Phys., 42 (2001), 4248–4257, arXiv: hep-th/0011209 | DOI | MR | Zbl

[37] Tempesta P., Winternitz P., Harnad J., Miller W., Pogosyan G., Rodriguez M. (eds), Superintegrability in classical and quantum systems, CRM Proceedings and Lecture Notes, 37, Amer. Math. Soc., Providence, RI, 2004 | MR | Zbl

[38] Weimar-Woods E., “The three-dimensional real Lie algebras and their contractions”, J. Math. Phys., 32 (1991), 2028–2033 | DOI | MR | Zbl