Exponentially small splitting of eigenvalues of the Laplace operator corresponding to two natural oscillations of ”jumping ball” type. (Russian) Zbl 0602.58051

Tabanov, M. B.

Probl. Mat. Fiz. 11, 261-269 (1986).

Asymptotic expressions are derived for the exponential splitting of eigenvalues of the Laplace operator in the two-dimensional region under the condition that in the region natural oscillations arise which correspond to two invariant Lagrange tori of the corresponding classical billiard and have in the quasiclassical approximation double eigenvalues. It is assumed that a substantial part of the boundary of the region differs slightly from the sections of parallel lines. It is shown that the exponentially small splitting arises because of the ”tunneling” interaction of oscillations through the connecting zone.

Reviewer: O.Dumbrajs

MSC:

58J50	Spectral problems; spectral geometry; scattering theory on manifolds
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

Asymptotic expressions; exponential splitting of eigenvalues of the Laplace operator; natural oscillations; invariant Lagrange tori