Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 104, 16 pages      arXiv:0805.0166      https://doi.org/10.3842/SIGMA.2009.104
Contribution to the Proceedings of the 5-th Microconference Analytic and Algebraic Methods V

Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations

Ryu Sasaki a, Wen-Li Yang b, c and Yao-Zhong Zhang c
a) Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
b) Institute of Modern Physics, Northwest University, Xian 710069, P.R. China
c) School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

Received September 20, 2009, in final form November 10, 2009; Published online November 18, 2009

Abstract
Bethe ansatz formulation is presented for several explicit examples of quasi exactly solvable difference equations of one degree of freedom which are introduced recently by one of the present authors. These equations are deformation of the well-known exactly solvable difference equations of the Meixner-Pollaczek, continuous Hahn, continuous dual Hahn, Wilson and Askey-Wilson polynomials. Up to an overall factor of the so-called pseudo ground state wavefunction, the eigenfunctions within the exactly solvable subspace are given by polynomials whose roots are solutions of the associated Bethe ansatz equations. The corresponding eigenvalues are expressed in terms of these roots.

Key words: Bethe ansatz solution; quasi-exactly solvable models.

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References

  1. Infeld L., Hull T.E., The factorization method, Rev. Modern Phys. 23 (1951), 21-68.
    Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, hep-th/9405029.
  2. Odake S., Sasaki R., Shape invariant potentials in "discrete quantum mechanics", J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 507-521, hep-th/0410102.
  3. Odake S., Sasaki R., Equilibrium positions, shape invariance and Askey-Wilson polynomials, J. Math. Phys. 46 (2005), 063513, 10 pages, hep-th/0410109.
    Odake S., Sasaki R., Calogero-Sutherland-Moser systems, Ruijsenaars-Schneider-van Diejen systems and orthogonal polynomials, Prog. Theoret. Phys. 114 (2005), 1245-1260, hep-th/0512155.
    Odake S., Sasaki R., Equilibrium positions and eigenfunctions of shape invariant ("discrete") quantum mechanics, Rokko Lectures in Mathematics (Kobe University) 18 (2005), 85-110, hep-th/0505070.
  4. Odake S., Sasaki R., Unified theory of annihilation-creation operators for solvable ("discrete") quantum mechanics, J. Math. Phys. 47 (2006), 102102, 33 pages, quant-ph/0605215.
    Odake S., Sasaki R., Exact solution in the Heisenberg picture and annihilation-creation operators, Phys. Lett. B 641 (2006), 112-117, quant-ph/0605221.
  5. Odake S., Sasaki R., Exact Heisenberg operator solutions for multi-particle quantum mechanics, J. Math. Phys. 48 (2007), 082106, 12 pages, arXiv:0706.0768.
  6. Odake S., Sasaki R., Exactly solvable 'discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states, Prog. Theoret. Phys. 119 (2008), 663-700, arXiv:0802.1075.
  7. Odake S., Sasaki R., Orthogonal polynomials from Hermitian matrices, J. Math. Phys. 49 (2008), 053503, 43 pages, arXiv:0712.4106.
  8. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and Its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  9. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, math.CA/9602214.
  10. Sasaki R., Quasi exactly solvable difference equations, J. Math. Phys. 48 (2007), 122104, 11 pages, arXiv:0708.0702.
  11. Sasaki R., New quasi exactly solvable difference equation, J. Nonlinear Math. Phys. 15 (2008), suppl. 3, 373-384, arXiv:0712.2616.
  12. Ushveridze A.G., Quasi-exactly solvable models in quantum mechanics, Institute of Physics Publishing, Bristol, 1994.
    Morozov A.Y., Perelomov A.M., Roslyi A.A., Shifman M.A., Turbiner A.V., Quasi-exactly-solvable quantal problems: one-dimensional analog of rational conformal field theories, Internat. J. Modern Phys. A 5 (1990), 803-832.
  13. Turbiner A.V., Quasi-exactly-solvable problems and sl(2) algebra, Comm. Math. Phys. 118 (1988), 467-474.
  14. Andrianov A.A., Ioffe M.V., Spiridonov V.P., Higher-derivative supersymmetry and the Witten index, Phys. Lett. A 174 (1993), 273-279, hep-th/9303005.
    Bagrov V.G., Samsonov B.F., Darboux transformation, factorization and supersymmetry in one-dimensional quantum mechanics, Theoret. and Math. Phys. 104 (1995), 1051-1060.
    Klishevich S.M., Plyushchay M.S., Supersymmetry of parafermions, Modern Phys. Lett. A 14 (1999), 2739-2752, hep-th/9905149.
    Aoyama H., Kikuchi H., Okouchi I., Sato M., Wada S., Valley views: instantons, large order behaviors, and supersymmetry, Nuclear Phys. B 553 (1999), 644-710, hep-th/9808034.
    Aoyama H., Sato M., Tanaka T., General forms of a N-fold supersymmetric family, Phys. Lett. B 503 (2001), 423-429, quant-ph/0012065.
  15. Sasaki R., Takasaki K., Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry, J. Phys. A: Math. Gen. 34 (2001), 9533-9553, Corrigendum, J. Phys. A: Math. Gen. 34 (2001), 10335, hep-th/0109008.
  16. Odake S., Sasaki R., Multi-particle quasi exactly solvable difference equations, J. Math. Phys. 48 (2007), 122105, 8 pages, arXiv:0708.0716.
  17. Wiegmann P.B., Zabrodin A.V., Bethe-ansatz for Bloch electron in magnetic field, Phys. Rev. Lett. 72 (1994), 1890-1893.
    Wiegmann P.B., Zabrodin A.V., Algebraization of difference eigenvalue equations related to Uq(sl2), Nuclear Phys. B 451 (1995), 699-724, cond-mat/9501129.
  18. Felder G., Varchenko A., Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(sl2), Nuclear Phys. B 480 (1996), 485-503, q-alg/9605024.
  19. Hou B.Y., Sasaki R., Yang W.-L., Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(sln) and its applications, Nuclear Phys. B 663 (2003), 467-486, hep-th/0303077.
    Hou B.Y., Sasaki R., Yang W.-L., Eigenvalues of Ruijsenaars-Schneider model associated with An–1 root system in Bethe ansatz formalism, J. Math. Phys. 45 (2004), 559-575, hep-th/0309194.
  20. Manojlovic N., Nagy Z., Construction of the Bethe state for the Eτ,η(so(3)) elliptic quantum group, SIGMA 3 (2007), 004, 10 pages, math.QA/0612086.
    Manojlovic N., Nagy Z., Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(A2(2)), J. Math. Phys. 48 (2007), 123515, 11 pages, arXiv:0704.3032.
  21. Degasperis A., Ruijsenaars S.N.M., Newton-equivalent Hamiltonians for the harmonic oscillator, Ann. Physics 293 (2001), 92-109.
  22. Turbiner A.V., Quantum mechanics: problems intermediate between exactly solvable and completely unsolvable, Soviet Phys. JETP 67 (1988), 230-236.
    Gonzárez-López A., Kamran N., Olver P., Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators, Comm. Math. Phys. 153 (1993), 117-146.
  23. Smirnov Y., Turbiner A., Lie algebraic discretization of differential equations, Modern Phys. Lett. A 10 (1995), 1795-1802, funct-an/9501001.
    Chrissomalakos C., Turbiner A., Canonical commutation relation preserving maps, J. Phys. A: Math. Gen. 34 (2001), 10475-10485, math-ph/0104004.
  24. Odake S., Sasaki R., Unified theory of exactly and quasi-exactly solvable 'discrete' quantum mechanics. I. Formalism, arXiv:0903.2604.


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