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23
CurvatureInduced Bound States In Quantum Waveguides In Two And Three Dimensions
 Math. Phys
, 1995
"... Dirichlet Laplacian on curved tubes of a constant cross section in two and three dimensions is investigated. It is shown that if the tube is nonstraight and its curvature vanishes asymptotically, there is always a bound state below the bottom of the essential spectrum. An upper bound to the number ..."
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Cited by 110 (12 self)
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Dirichlet Laplacian on curved tubes of a constant cross section in two and three dimensions is investigated. It is shown that if the tube is nonstraight and its curvature vanishes asymptotically, there is always a bound state below the bottom of the essential spectrum. An upper bound to the number of these bound states in thin tubes is derived. Furthermore, if the tube is only slightly bent, there is just one bound state; we derive its behaviour with respect to the bending angle. Finally, perturbation theory of these eigenvalues in any thin tube with respect to the tube radius is constructed and some open questions are formulated. October 1994 CPT94/P.3023 anonymous ftp or gopher: cpt.univmrs.fr Unit'e Propre de Recherche 7061 1 and PHYMAT, Universit'e de Toulon et du Var, 83130 Lagarde, France duclos@naxos.unice.fr 2 Nuclear Physics Institute, AS CR, 25068 Rez near Prague and Doppler Institute, Czech Technical University, Brehov'a 7, 11519 Prague, Czech Republic exner@uj...
Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case
 Journal of Physics A: Mathematical and General
"... Abstract. We consider a family of open sets Mε which shrinks with respect to an appropriate parameter ε to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of Mε converges to the spectrum of the (differential) ..."
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Cited by 27 (7 self)
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Abstract. We consider a family of open sets Mε which shrinks with respect to an appropriate parameter ε to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of Mε converges to the spectrum of the (differential) Laplacian on the graph with Dirichlet boundary conditions at the vertices, i.e., a graph operator without coupling between different edges. The smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood. The lower bound is given by the first transversal mode of the edge neighbourhood. We also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a Laplacian acting on the edge with an additional potential coming from the curvature. 1.
Asymptotic Estimates for Bound States in Quantum Waveguides Coupled Laterally Through a Narrow Window
, 1995
"... . Consider the Laplacian in a straight planar strip of width d , with the Neumann boundary condition at a segment of length 2a of one of the boundaries, and Dirichlet otherwise. For small enough a this operator has a single eigenvalue ffl(a) ; we show that there are positive c 1 ; c 2 such that \G ..."
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Cited by 27 (6 self)
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. Consider the Laplacian in a straight planar strip of width d , with the Neumann boundary condition at a segment of length 2a of one of the boundaries, and Dirichlet otherwise. For small enough a this operator has a single eigenvalue ffl(a) ; we show that there are positive c 1 ; c 2 such that \Gammac 1 a 4 ffl(a) \Gamma (ß=d) 2 \Gammac 2 a 4 . An analogous conclusion holds for a pair of Dirichlet strips, of generally different widths, with a window of length 2a in the common boundary. 1 Introduction Recent progress in "mesoscopic" physics brought not only new physical effects but also some interesting spectral problems. One of them concerns the existence of bound states which appear if a Dirichlet tube of a constant cross section is locally deformed, e.g., bent or protruded, or coupled to another tube  see [BGRS, ES, DE, SRW] and references therein. In this paper we are concerned with another system of this type, which consists of a pair of parallel Dirichlet strips cou...
Spectra of Graph Neighborhoods and Scattering
"... Let (Gε)ε>0 be a family of ’εthin’ Riemannian manifolds modeled on a finite metric graph G, for example, the εneighborhood of an embedding of G in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the LaplaceBeltrami operator on Gε as ε → 0, for ..."
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Cited by 14 (3 self)
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Let (Gε)ε>0 be a family of ’εthin’ Riemannian manifolds modeled on a finite metric graph G, for example, the εneighborhood of an embedding of G in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the LaplaceBeltrami operator on Gε as ε → 0, for
Exponential bounds on curvatureinduced resonances in a twodimensional Dirichlet tube
 Helv. Phys. Acta
, 1998
"... Abstract. We consider curvature–induced resonances in a planar two–dimensional Dirichlet tube of a width d. It is shown that the distances of the corresponding resonance poles from the real axis are exponentially small as d → 0+, provided the curvature of the strip axis satisfies certain analyticity ..."
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Cited by 12 (1 self)
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Abstract. We consider curvature–induced resonances in a planar two–dimensional Dirichlet tube of a width d. It is shown that the distances of the corresponding resonance poles from the real axis are exponentially small as d → 0+, provided the curvature of the strip axis satisfies certain analyticity and decay requirements. 1
Computed eigenmodes of planar regions
 IN RECENT ADVANCES IN DIFFERENTIAL EQUATIONS AND MATHEMATICAL PHYSICS, VOLUME 412 OF CONTEMP. MATH
, 2006
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Thin tubes in mathematical physics, global analysis and spectral geometry
 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS
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Solving the Helmholtz equation for membranes of arbitrary shape, sent to Journal of Physics A
, 2008
"... Abstract. I calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on Little Sinc Functions. The matrix representation of the PDE obtained using this method is explicit and it does not require the calculation of integrals. To illustrate the virtues of thi ..."
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Cited by 2 (1 self)
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Abstract. I calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on Little Sinc Functions. The matrix representation of the PDE obtained using this method is explicit and it does not require the calculation of integrals. To illustrate the virtues of this approach, I have considered a large number of examples, part of them taken from the literature, and part of them new. When possible, I have tested the accuracy of these results by comparing them with the exact results (when available) or with results from the literature. In particular, in the case of the Lshaped membrane, the first example discussed in the paper, I show that it is possible to extrapolate the results obtained with different grid sizes to obtain higly precise results. Finally, I also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results.