Tunneling with Oscillating Effect of Ground States of a Quadratic Operator on a Hyperboloid

In the paper, the problem of constructing the semiclassical asymptotics of the difference between a pair of close lower energy levels of a quadratic operator defined on an irreducible representation of the Lie algebra $$\mathrm{su}(1,1)$$ is considered. In the Darboux coordinates on a hyperboloid, the Hamiltonian defines the landscape of a symmetric double well. As is known, the asymptotics of the tunnel splitting of the upper energy levels for this class of operators is not only exponentially decaying, which is usual in the case in double wells, but also oscillates rapidly. In this paper, we show that this effect is preserved when considering the ground energy states. It is shown that, in the space of holomorphic functions, the operator takes the form of a second-order differential operator. The eigenfunctions corresponding to the energies under study in a neighborhood of a multiple turning point are expressed in terms of parabolic cylinder functions and WKB asymptotics. A theorem on the oscillating tunnel effect for the ground states of the operator is proved using the condition of analyticity of the eigenfunctions in the unit circle. It is also shown that the tunnel asymptotics for the upper energy levels differ from the asymptotics for the ground state by the factor of $$\sqrt{\pi/e}$$ .