@article{5862535339d543668ab46cab5a9a952d,

title = "On the spectrum of the Dirichlet Laplacian in a narrow strip",

abstract = "We consider the Dirichlet Laplacian Δ in a family of bounded domains {-a < x < b, 0 < y < εh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ.",

author = "Leonid Friedlander and Michael Solomyak",

note = "Funding Information: Acknowledgments. The bulk of the work was done when the first author was the Weston Visiting Professor in the Weizmann Institute of Science in Rehovot, Israel. He thanks the Institute for its hospitality. The work was finished when both authors visited the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK. We acknowledge the hospitality of the Newton Institute. The first author was partially supported by NSF grant DMS 0648786. We are also grateful to V. Maz{\textquoteright}ya and S. Nazarov for their bibliographical advice and to A. Sobolev for discussions.",

year = "2009",

month = mar,

doi = "10.1007/s11856-009-0032-y",

language = "English (US)",

volume = "170",

pages = "337--354",

journal = "Israel Journal of Mathematics",

issn = "0021-2172",

publisher = "Springer New York",

number = "1",

}