Geodesic
Asymptotic Behavior of Eigenvalues of the Laplace Operator in Infinite Cylinders Perturbed by Transverse Extensions
Matematičeskie zametki, Tome 81 (2007) no. 3, pp. 328-334

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We present sufficient conditions for the existence of an eigenvalue of the Laplace operator with zero Dirichlet conditions in a weakly perturbed infinite cylinder in the case of localized perturbations which are extensions along the transverse coordinates with coefficients depending on the longitudinal coordinate. If such an eigenvalue exists, then, for this eigenvalue, we obtain an asymptotic formula with respect to a small parameter characterizing the values of extensions.
Keywords: Laplace operator, eigenvalue, asymptotics, small parameter, infinite cylinder, localized perturbations. 
@article{MZM_2007_81_3_a1,
     author = {V. V. Grushin},
     title = {Asymptotic {Behavior} of {Eigenvalues} of the {Laplace} {Operator} in {Infinite} {Cylinders} {Perturbed} by {Transverse} {Extensions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {328--334},
     publisher = {mathdoc},
     volume = {81},
     number = {3},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2007_81_3_a1/}
}
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V. V. Grushin. Asymptotic Behavior of Eigenvalues of the Laplace Operator in Infinite Cylinders Perturbed by Transverse Extensions. Matematičeskie zametki, Tome 81 (2007) no. 3, pp. 328-334. http://geodesic.mathdoc.fr/item/MZM_2007_81_3_a1/