On the Dirichlet problem for the Helmholtz equation on the plane with boundary conditions on an almost closed curve

In this article the two-dimensional Dirichlet boundary-value problem is considered for the Helmholtz operator with boundary conditions on an almost closed curve $\Gamma_\varepsilon $ where $\varepsilon\ll 1$ is the distance between the end-points of the curve. A complete asymptotic expression is constructed for a pole of the analytic continuation of the Green's function of this problem as the pole converges to a simple eigenfrequency of the limiting interior problem in the case when the corresponding eigenfunction of the limiting problem has a second-order zero at the centre of contraction of the gap. The influence of symmetry of the gap on the absolute value of the imaginary parts of the poles is investigated.