Librations with large periods in tunneling: Efficient calculation and applications to trigonal dimers

In studying tunnel asymptotics for lower energy levels of the Schrödinger operator (such as the energy splitting in a symmetric double well or the width of a spectral band in a periodic problem), there naturally arise librations, i.e., periodic solutions of a classical system with inverted potential, which reach the boundary of the domain of possible motions twice during the period. In the limit, they give double-asymptotic solutions with two symmetric unstable equilibria (instantons). The tunnel asymptotics can be written in two ways: either in terms of the action on the instanton and the linearized dynamics in its neighborhood or in terms of a certain libration, called a tunnel libration. The second way is more constructive, since when used in numerical calculations, it reduces to two operations: finding a libration with a given energy and calculating the Floquet coefficients for a given libration. To apply this approach in practice, we propose to find librations with a given energy by using a numerical variational method that generalizes the ideas of the nudged elastic band method. As an application, we find the asymptotics for the widths of the lower spectral bands and gaps, expressed in terms of tunnel libration in a four-dimensional system describing the dimer in a trigonal-symmetric field, which was proposed by M. I. Katsnelson.