Geodesic
Asymptotic behaviour of quasi-eigenvalues of the Laplace operator in the case of the outside of a circular disc
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 19 (1979) no. 5, pp. 1217-1227

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An analisys of the exact solution is used to obtain the asymptotic behaviour as $|K| \to \infty$ of the quasi-eigenvalues $K$, closest to the $Im~K = 0$ axis, of the Laplace operator in the case of the outside of a circular disc (it is assumed that the Neumann boundary condition holds on the disc itself). A geometrical interpretation is given for the asymptotic expressions for the quasi-eigenfunctions of the Laplace operator, in terms of geometrical optics. 
@article{ZVMMF_1979_19_5_a12,
     author = {N. S. Grigor'ev},
     title = {Asymptotic behaviour of~quasi-eigenvalues of~the {Laplace} operator in~the case of~the outside of~a~circular disc},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1217--1227},
     publisher = {mathdoc},
     volume = {19},
     number = {5},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_1979_19_5_a12/}
}
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N. S. Grigor'ev. Asymptotic behaviour of quasi-eigenvalues of the Laplace operator in the case of the outside of a circular disc. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 19 (1979) no. 5, pp. 1217-1227. http://geodesic.mathdoc.fr/item/ZVMMF_1979_19_5_a12/