Туннелирование с осциллирующим эффектом основных состояний квадратичного оператора на гиперболоиде

Математические заметки. 2024. Т. 116. № 6. С. 862–880.

Vybornyi E., Rumiantseva S.

In this paper, we consider the problem of constructing a semiclassical asymptotic estimate of the splitting between a pair of close lower-lying energy levels for a quadratic operator defined on the irreducible representation of the Lie algebra su(1,1). In Darboux coordinates on the hyperboloid the Hamiltonian defines the landscape of a symmetric double well. It is known that the asymptotics of tunnel splitting for the upper energy levels of this class of operators not only exhibit the typical exponential decay observed in double wells but also exhibit rapid oscillations. We demonstrate that such oscillatory behavior of tunneling persists even for the ground states of the system. The operator has a representation of the second-order differential operator in the space of holomorphic functions. The corresponding eigenfunctions are represented in terms of parabolic cylinder functions in the neighborhood of a multiple turning point  and in terms of standard WKB expansions. A theorem on the oscillatory tunnel effect for the operator's ground states is proven via the analyticity condition of the eigenfunctions within a unit radius circle. Additionally, it is shown that the tunnel asymptotics for the upper energy levels differ from those for the ground state by a factor of √π/e

Keywords: semiclassical approximation, квазиклассическое приближение, complex WKB method, комплексный метод ВКБ, tunnel splitting, parabolic cylinder functions, функции параболического цилиндра, туннельное расщепление

Publication based on the results of: Analytical and numerical methods for constructing solutions to nonlinear mathematical models (2024)