Asymptotics of the eigenvalues of boundary value problems for the Laplace operator in a three-dimensional domain with a thin closed tube

We construct and justify asymptotic representations for the eigenvalues and eigenfunctions of boundary value problems for the Laplace operator in a three-dimensional domain $ \Omega (\varepsilon )=\Omega \setminus \overline {\Gamma }_\varepsilon $ with a thin singular set $ \Gamma _\varepsilon $ lying in the $ c\varepsilon $-neighborhood of a simple smooth closed contour $ \Gamma $. We consider the Dirichlet problem, a mixed boundary value problem with the Neumann conditions on $ \partial \Gamma _\varepsilon $, and also a spectral problem with lumped masses on $ \Gamma _\varepsilon $. The asymptotic representations are of diverse character: we find an asymptotic series in powers of the parameter $ \vert{\ln \varepsilon }\vert^{-1}$ or $ \varepsilon $. The most comprehensive and complicated analysis is presented for the lumped mass problem; namely, we sum the series in powers of $ \vert{\ln \varepsilon }\vert^{-1}$ and obtain an asymptotic expansion with the leading term holomorphically depending on $ \vert{\ln \varepsilon }\vert^{-1}$ and with the remainder $ O(\varepsilon ^\delta )$, $ \delta \in (0,1)$. The main role in asymptotic formulas is played by solutions of the Dirichlet problem in $ \Omega \setminus \Gamma $ with logarithmic singularities distributed along the contour $ \Gamma $.