Abstract
For a class of polynomial quantum Hamiltonians used in models of combination scattering in quantum optics, we obtain the asymptotic behavior of the spectrum for large occupation numbers in the secondary quantization representation. Hamiltonians of this class can be diagonalized using a special system of polynomials determined by recurrence relations with coefficients depending on a parameter (occupation number). For this system of polynomials, we determine the asymptotic behavior a discrete measure with respect to which they are orthogonal. The obtained limit measures are interpreted as equilibrium measures in extremum problems for a logarithmic potential in an external field and with constraints on the measure. We illustrate the general case with an exactly solvable example where the Hamiltonian can be diagonalized by the canonical Bogoliubov transformation and the special orthogonal polynomials degenerate into the Krawtchouk classical discrete polynomials.
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References
Ja. Perina, Quantum Statistics of Nonlinear Optics, Reidel, Dordrecht (1984).
V. V. Vedenyapin and Yu. N. Orlov, “Conservation laws for polynomial Hamiltonians and for discrete models of the Boltzmann equation,” Theor. Math. Phys., 121, 1516–1523 (1999).
N. N. Bogolyubov and N. N. Bogolyubov Jr., Introduction to the Quantum Statistical Mechanics [in Russian], Nauka, Moscow (1984).
V. G. Bagrov and B. F. Samsonov, “Darboux transformation, factorization, supersymmetry in one-dimensional quantum mechanics,” Theor. Math. Phys., 104, 1051–1060 (1995).
B. F. Samsonov and J. Negro, “Darboux transformations of the Jaynes–Cummings Hamiltonian,” J. Phys. A: Math. Gen., 37, 10115–10127 (2004).
S. B. Leble, “Dressing method in matter + radiation quantum models,” Theor. Math. Phys., 152, 977–990 (2007).
E. A. Rakhmanov, “Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable,” Sb. Math., 187, 1213–1228 (1996).
A. Borodin, “Riemann–Hilbert problem and discrete Bessel kernel,” Int. Math. Res. Notices, 2000, 467–494 (2000)
arXiv:math/9912093v2 (1999).
J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin, and P. D. Miller, Discrete orthogonal polynomials: Asymptotics and application (Ann. Math. Stud., Vol. 164), Princeton Univ. Press, Princeton, N. J. (2007).
D. Dai and R. Wong, “Global asymptotics of Krawtchouk polynomials–a Riemann–Hilbert approach,” Chin. Ann. Math. Ser. B, 28, 1–34 (2007).
A. I. Aptekarev and D. N. Tulyakov, “Asymptotics of Meixner polynomials and Christoffel–Darboux kernels,” Trans. Moscow Math. Soc., 73, 67–106 (2012).
G. Szegö, Orthogonal Polynomials (AMS Colloq. Publ., Vol. 23), Amer. Math. Soc., Providence, R. I. (1959).
A. B. J. Kuijlaars and W. Van Assche, “The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients,” J. Approx. Theory, 99, 167–197 (1999).
A. I. Aptekarev and W. Van Assche, “Asymptotic of discrete orthogonal polynomials and the continuum limit of the Toda lattice,” J. Phys. A: Math. Gen., 34, 10627–10639 (2001).
A. I. Aptekarev, A. Branquinho, and F. Marcell´an, “Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation,” J. Comput. Appl. Math., 78, 139–160 (1997).
A. I. Aptekarev, J. S. Geronimo, and W. Van Assche, “Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients,” J. Approx. Theory, 150, 214–238 (2008).
A. I. Aptekarev and D. N. Tulyakov, “The leading term of the Plancherel–Rotach asymptotic formula for solutions of recurrence relations,” Sb. Math., 205, 1696–1719 (2014).
A. I. Aptekarev, V. Kalyagin, López G. Lagomasino, and I. A. Rocha, “On the limit behavior of recurrence coefficients for multiple orthogonal polynomials,” J. Approx. Theory, 139, 346–370 (2006).
A. I. Aptekarev, G. López Lagomasino, and I. Alvarez Rocha, “Ratio asymptotics of Hermite–Padé polynomials for Nikishin systems,” Sb. Math., 196, 1089–1107 (2005).
R. S. Varga, Matrix Iterative Analysis (Springer Ser. Comput. Math., Vol. 27), Springer, Berlin (2000).
E. B. Saff and V. Totik, Logarithmic Potentials with External Fields (Grundlehren Math. Wiss., Vol. 316), Springer, Berlin (1997).
P. D. Dragnev and E. B. Saff, “Constrained energy problems with applications to orthogonal polynomials of a discrete variable,” J. Anal. Math., 72, 223–259 (1997).
A. B. J. Kuijlaars and E. A. Rakhmanov, “Zero distributions for discrete orthogonal polynomials,” J. Comput. Appl. Math., 99, 255–274 (1998).
P. Deift and K. T.-R. McLaughlin, A Continuum Limit of the Toda Lattice (Mem. AMS, Vol. 131, No. 624), Amer. Math. Soc., Providence, R. I. (1998).
V. S. Buyarov and E. A. Rakhmanov, “Families of equilibrium measures in an external field on the real axis,” Sb. Math., 190, 791–802 (1999).
B. Beckermann, “On a conjecture of E. A. Rakhmanov,” Construct. Approx., 16, 427–448 (2000).
M. Krawtchouk, “Sur une généralisation des polynômes d’Hermite,” C. R. Acad. Sci. Paris, 189, 620–622 (1929).
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This research was supported by a grant from the Russian Science Foundation (Project No. 14-21-00025).
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Aptekarev, A.I., Lapik, M.A. & Orlov, Y.N. Asymptotic behavior of the spectrum of combination scattering at Stokes phonons. Theor Math Phys 193, 1480–1497 (2017). https://doi.org/10.1134/S0040577917100063
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DOI: https://doi.org/10.1134/S0040577917100063