On the eigenvalues of a “dumb-bell with a thin handle”

We consider the Neumann boundary-value problem of finding the small-parameter asymptotics of the eigenvalues and eigenfunctions for the Laplace operator in a singularly perturbed domain consisting of two bounded domains joined by a thin “handle”. The small parameter is the diameter of the cross-section of the handle. We show that as the small parameter tends to zero these eigenvalues converge either to the eigenvalues corresponding to the domains joined or to the eigenvalues of the Dirichlet problem for the Sturm–Liouville operator on the segment to which the thin handle contracts. The main results of this paper are the complete power small-parameter asymptotics of the eigenvalues and the corresponding eigenfunctions and explicit formulae for the first terms of the asymptotics. We consider critical cases generated by the choice of the place where the thin “handle” is joined to the domains, as well as by the multiplicity of the eigenvalues corresponding to the domains joined.