Geodesic
On the Dirichlet problem for the Helmholtz equation on the plane with boundary conditions on an almost closed curve
Sbornik. Mathematics, Tome 191 (2000) no. 6, pp. 821-848

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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. 
@article{SM_2000_191_6_a2,
     author = {R. R. Gadyl'shin},
     title = {On the {Dirichlet} problem for the {Helmholtz} equation on the plane with boundary conditions on an almost closed curve},
     journal = {Sbornik. Mathematics},
     pages = {821--848},
     publisher = {mathdoc},
     volume = {191},
     number = {6},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_6_a2/}
}
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R. R. Gadyl'shin. On the Dirichlet problem for the Helmholtz equation on the plane with boundary conditions on an almost closed curve. Sbornik. Mathematics, Tome 191 (2000) no. 6, pp. 821-848. http://geodesic.mathdoc.fr/item/SM_2000_191_6_a2/