Geodesic
The splitting of Laplacian operator eigenvalue in the case of ellipse with cuts
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 19, Tome 179 (1989), pp. 173-178

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For the splitting of eigenvalues of the Laplacian operator in the case of Dirichlet boundary problem for ellipse with cuts the exponential asymptotics with the pre-exponent factor in a quasi-classical form is proved. The prove is based on the separation variable method. 
@article{ZNSL_1989_179_a19,
     author = {D. Ya. Terman},
     title = {The splitting of {Laplacian} operator eigenvalue in the case of ellipse with cuts},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {173--178},
     publisher = {mathdoc},
     volume = {179},
     year = {1989},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1989_179_a19/}
}
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D. Ya. Terman. The splitting of Laplacian operator eigenvalue in the case of ellipse with cuts. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 19, Tome 179 (1989), pp. 173-178. http://geodesic.mathdoc.fr/item/ZNSL_1989_179_a19/