Квазиклассические асимптотики осциллирующего туннелирования для квадратичного гамильтониана на алгебре su(1,1)

Математические заметки. 2022. Т. 112. № 5. С. 665–681.

Vybornyi E., Rumiantseva S.

The paper considers the problem of constructing semiclassical asymptotics of tunnel splitting of the spectrum of an operator given on an irreducible representation of the Lie algebra su(1, 1). It is assumed that the operator is a quadratic function of the generators of the algebra. Coherent states and coherent unitary transformation are presented, which make it possible to reduce the problem to the consideration of a second-order differential operator in the space of holomorphic functions. In this paper, quasi-classical asymptotic spectral series and corresponding wave functions are constructed in the form of expansions over coherent states. Under certain system parameters, the minimum energy corresponds to a pair of non-degenerate equilibrium positions, and the discrete spectrum of the operator has an exponentially small tunnel splitting of levels. Asymptotic formulas for tunneling splitting of energies are proved in the paper by applying the complex WKB method. It is shown that, unlike the one-dimensional Schrodinger operator, tunnel splitting in this problem not only decreases exponentially, but also contains an oscillating multiplier, which can be interpreted as interference of tunneling over various instantons. It is also shown that at certain values of the parameters, there is a complete suppression of tunneling and a twofold degeneration of part of the spectrum levels, which is atypical for one-dimensional systems.

Keywords: квазиклассическое приближение, комплексный метод ВКБ

Publication based on the results of: Mathematical methods and modeling in the study of physical phenomena (2022)