Asymptotic behavior of the spectrum of combination scattering at Stokes phonons

A. I. Aptekarev   ; M. A. Lapik   ; Yu. N. Orlov

Teoretičeskaâ i matematičeskaâ fizika, Tome 193 (2017) no. 1, pp. 84-103

Résumé

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.

Détail

Keywords: creation operator, annihilation operator, polynomial quantum Hamiltonian, combination scattering, asymptotics of a discrete orthogonal polynomial, equilibrium measure in an external field.

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