### 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.

Original language | English (US) |
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Pages (from-to) | 337-354 |

Number of pages | 18 |

Journal | Israel Journal of Mathematics |

Volume | 170 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2009 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*170*(1), 337-354. https://doi.org/10.1007/s11856-009-0032-y

**On the spectrum of the Dirichlet Laplacian in a narrow strip.** / Friedlander, Leonid; Solomyak, Michael.

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 170, no. 1, pp. 337-354. https://doi.org/10.1007/s11856-009-0032-y

}

TY - JOUR

T1 - On the spectrum of the Dirichlet Laplacian in a narrow strip

AU - Friedlander, Leonid

AU - Solomyak, Michael

PY - 2009/3

Y1 - 2009/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=65549169108&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=65549169108&partnerID=8YFLogxK

U2 - 10.1007/s11856-009-0032-y

DO - 10.1007/s11856-009-0032-y

M3 - Article

AN - SCOPUS:65549169108

VL - 170

SP - 337

EP - 354

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -