ИСТИНА

It is well-known that in the study of exponentially small “tunneling” effects in quantum mechanics, special trajectories of classical systems – instantons – play an important role. In the simplest problem of this kind – the ground state splitting of the Schr¨odinger operator with a symmetric “double well” potential – the instanton is a doubly asymptotic trajectory of a classical system with an “inverted” potential V (x) connecting its peaks. Various asymptotic formulas relating the tunneling splitting with the instanton action were derived with or without accurate justification by many authors (e.g. L.D. Landau, E.M. Lifshits, B. Helffer, J. Sj¨ostrand, B. Simon, E. Harrell, V.P. Maslov). In a series of papers by S.Yu. Dobrokhotov, A.Yu. Anikin, and co-authors, it was shown that the tunneling splitting can be much more efficiently calculated in terms of the action on the “libration” with a large period giving an instanton as a limit, rather than the instanton itself. By libration we mean a periodic solution whose velocity vanishes twice in a period. The existence of a family of librations near the instanton follows from the results of V.V. Kozlov and S.V. Bolotin [2]. The question of calculating the asymptotics for the tunneling splitting is essentially reduced to seeking some specific libration, called the “tunneling libration”, and calculating the value of the action on it, called the“tunneling action”. In the case of two degrees of freedom, a tunneling libration can be found by the primitive “shooting method”. The efficient calculation of tunneling libration for a larger number of degrees of freedom has not been previously considered. The goal of the talk is to show that tunneling libration can be efficiently calculated by using a modification of the variational method developed in the papers by I.A. Nosikov, M.V. Klimenko et al. Our main example with four degrees of freedom is connected with the quantum problem of rotating dimers. We also consider a simpler two-dimensional example, in which we compare the calculations of the tunneling libration and action by the shooting method and the variational method.