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Asymptotic behavior of the spectrum of combination scattering at Stokes phonons. (English. Russian original) Zbl 1387.81398

Theor. Math. Phys. 193, No. 1, 1480-1497 (2017); translation from Teor. Mat. Fiz. 193, No. 1, 84-103 (2017).
Summary: 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.

MSC:

81V80 Quantum optics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81U10 \(n\)-body potential quantum scattering theory
82D20 Statistical mechanics of solids
81V70 Many-body theory; quantum Hall effect
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References:

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