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Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

Published online by Cambridge University Press:  30 May 2008

David Krejčiřík
David Krejčiřík*
Affiliation:
Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 250 68 Řež near Prague, Czech Republic; krejcirik@ujf.cas.cz
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Abstract

We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the curves tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the curvature radii of the Neumann boundary to the Dirichlet one is the biggest. We also show that the asymptotics can be obtained from a form of norm-resolvent convergence which takes into account the width-dependence of the domain of definition of the operators involved.

Keywords

Laplacian in tubesDirichlet and Neumann boundary conditionsdimension reductionnorm-resolvent convergencebinding effect of curvaturewaveguides
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 3 , July 2009 , pp. 555 - 568
Copyright
© EDP Sciences, SMAI, 2008

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References

D. Borisov and P. Freitas, Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains. Ann. Inst. H. Poincaré Anal. Non Linéaire (2008) doi: 10.1016/j.anihpc.2007.12.001.
Bouchitté, G., Mascarenhas, M.L. and Trabucho, L., On the curvature and torsion effects in one dimensional waveguides. ESAIM: COCV 13 (2007) 793–808. CrossRef
Carron, G., Exner, P. and Krejčiřík, D., Topologically nontrivial quantum layers. J. Math. Phys. 45 (2004) 774–784. CrossRef
E.B. Davies, Spectral theory and differential operators. Camb. Univ. Press, Cambridge (1995).
Dittrich, J. and Kříž, J., Curved planar quantum wires with Dirichlet and Neumann boundary conditions. J. Phys. A 35 (2002) L269–L275. CrossRef
Duclos, P. and Exner, P., Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1995) 73–102. CrossRef
Duclos, P., Exner, P. and Krejčiřík, D., Bound states in curved quantum layers. Commun. Math. Phys. 223 (2001) 13–28. CrossRef
Exner, P. and Šeba, P., Bound states in curved quantum waveguides. J. Math. Phys. 30 (1989) 2574–2580. CrossRef
Freitas, P., Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi. J. Funct. Anal. 251 (2007) 376–398. CrossRef
Freitas, P. and Krejčiřík, D., Instability results for the damped wave equation in unbounded domains. J. Diff. Eq. 211 (2005) 168–186. CrossRef
Freitas, P. and Krejčiřík, D., Waveguides with combined Dirichlet and Robin boundary conditions. Math. Phys. Anal. Geom. 9 (2006) 335–352. CrossRef
Freitas, P. and Krejčiřík, D., Location of the nodal set for thin curved tubes. Indiana Univ. Math. J. 57 (2008) 343–376. CrossRef
L. Friedlander and M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip, I. Israel J. Math. (to appear).
L. Friedlander and M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip, II. Amer. Math. Soc. (to appear).
Goldstone, J. and Jaffe, R.L., Bound states in twisting tubes. Phys. Rev. B 45 (1992) 14100–14107. CrossRef
D. Grieser, Thin tubes in mathematical physics, global analysis and spectral geometry, in Analysis on Graphs and its Applications (Cambridge, 2007), Proceedings of Symposia in Pure Mathematics, Amer. Math. Soc. (to appear).
Johnson, E.R., Levitin, M. and Parnovski, L., Existence of eigenvalues of a linear operator pencil in a curved waveguide – localized shelf waves on a curved coast. SIAM J. Math. Anal. 37 (2006) 1465–1481. CrossRef
L. Karp and M. Pinsky, First-order asymptotics of the principal eigenvalue of tubular neighborhoods, in Geometry of random motion (Ithaca, N.Y., 1987), Contemp. Math. 73, Amer. Math. Soc., Providence, RI (1988) 105–119.
Krejčiřík, D., Quantum strips on surfaces. J. Geom. Phys. 45 (2003) 203–217. CrossRef
Krejčiřík, D., Hardy inequalities in strips on ruled surfaces. J. Inequal. Appl. 2006 (2006) 46409. CrossRef
D. Krejčiřík and J. Kříž, On the spectrum of curved quantum waveguides. Publ. RIMS, Kyoto University 41 (2005) 757–791.
Ch. Lin, Z. Lu, Existence of bound states for layers built over hypersurfaces in $\mathbb{R}^{n+1}$ . J. Funct. Anal. 244 (2007) 1–25. CrossRef
Ch. Lin, Z. Lu, Quantum layers over surfaces ruled outside a compact set. J. Math. Phys. 48 (2007) 053522. CrossRef
Olendski, O. and Mikhailovska, L., Localized-mode evolution in a curved planar waveguide with combined Dirichlet and Neumann boundary conditions. Phys. Rev. E 67 (2003) 056625. CrossRef
M. Reed and B. Simon, Methods of modern mathematical physics, I. Functional analysis. Academic Press, New York (1972).
Schatzman, M., On the eigenvalues of the Laplace operator on a thin set with Neumann boundary conditions. Appl. Anal. 61 (1996) 293–306. CrossRef