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42
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Boundstate asymptotic estimate for windowcoupled Dirichlet strips and layers
, 1997
"... We consider the discrete spectrum of the Dirichlet Laplacian on a manifold consisting of two adjacent parallel straight strips or planar layers coupled by a finite number N of windows in the common boundary. If the windows are small enough, there is just one isolated eigenvalue. We find upper and lo ..."
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Cited by 15 (4 self)
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We consider the discrete spectrum of the Dirichlet Laplacian on a manifold consisting of two adjacent parallel straight strips or planar layers coupled by a finite number N of windows in the common boundary. If the windows are small enough, there is just one isolated eigenvalue. We find upper and lower asymptotic bounds on the gap between the eigenvalue and the essential spectrum in the planar case, as well as for N = 1 in three dimensions. Based on these results, we formulate a conjecture on the weak–coupling asymptotic behaviour of such bound states. 1
Tater: Point interactions in a strip
, 1996
"... We study the behavior of a quantum particle confined to a hard–wall strip of a constant width in which there is a finite number N of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein’s formula, we analyze its spectral and scattering properties. The bo ..."
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We study the behavior of a quantum particle confined to a hard–wall strip of a constant width in which there is a finite number N of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein’s formula, we analyze its spectral and scattering properties. The bound state–problem is analogous to that of point interactions in the plane: since a two–dimensional point interaction is never repulsive, there are m discrete eigenvalues, 1 ≤ m ≤ N, the lowest of which is nondegenerate. On the other hand, due to the presence of the boundary the point interactions give rise to infinite series of resonances; if the coupling is weak they approach the thresholds of higher transverse modes. We derive also spectral and scattering properties for point perturbations in several related models: a cylindrical surface, both of a finite and infinite heigth, threaded by a magnetic flux, and a straight strip which supports a potential independent of the transverse coordinate. As for strips with an infinite number of point perturbations, we restrict ourselves to the situation when the latter are arranged periodically; we show that in distinction to the case of a point–perturbation array in the plane, the spectrum may exhibit any finite number of gaps. Finally, we study numerically conductance fluctuations in case of random point perturbations.
Embedded Trapped Modes for Obstacles in TwoDimensional Waveguides
 Quarterly J. Mechanics Applied Mathematics
, 2000
"... In this pape we inve153; the ee53; of e be33 trappe mo de forsymme70 obstacle whichare place onthe ce tre425 of a twodime41234 acoustic wave47741 Mo are sought whichare antisymme3 aboutthe ce tre344 ofthe channe but which have fre14;371 thatare above the first cuto# for antisymme37 wave propa ..."
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Cited by 12 (6 self)
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In this pape we inve153; the ee53; of e be33 trappe mo de forsymme70 obstacle whichare place onthe ce tre425 of a twodime41234 acoustic wave47741 Mo are sought whichare antisymme3 aboutthe ce tre344 ofthe channe but which have fre14;371 thatare above the first cuto# for antisymme37 wave propagation downthe guide Inthe te3722;35 of spe;34 the4 this mes; thatthe ee; value associate withthe trappe mode is e be34 inthe continuous spe;47 ofthe re1 ant ope1525 A nume701; proce;1 base on a boundary inte353 te hnique is dee4 e to se51 h fore be174 trappe mo for bodie ofge347 shape In addition two approximate solutions for trappe mo deare found;the first is for longplate onthe ce tre15 ofthe channe andthe se23 is forsle321 bodie whichare pe517;224 ofplate pe1 e14; tothe guide walls. It is found thate be35 trappe mo de do notet;1 for arbitrary symmeary bodie but if anobstacle isde32 by twoge315;1 parame1 the branche of trappe mo de may be obtaine by varying both ofthe parame;1 simu...
A Framework for Network
 Protocol Software,” Proceedings OOPSLA’95, ACM SIGPLAN Notices
, 1995
"... Existence of eigenvalues of a linear operator pencil in a curved waveguide — localized shelf waves on a curved coast ..."
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Cited by 10 (2 self)
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Existence of eigenvalues of a linear operator pencil in a curved waveguide — localized shelf waves on a curved coast
Edge Resonance in an Elastic SemiStrip
 Quarterly J. Mech. Appl. Math
, 1998
"... . We study the elasticity operator in a semistrip subject to free boundary conditions. In the case of zero Poisson ratio we prove the existence of a positive eigenvalue embedded in the essential spectrum. Physically, the eigenvalue corresponds to a "trapped mode", that is, a harmonic osci ..."
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. We study the elasticity operator in a semistrip subject to free boundary conditions. In the case of zero Poisson ratio we prove the existence of a positive eigenvalue embedded in the essential spectrum. Physically, the eigenvalue corresponds to a "trapped mode", that is, a harmonic oscillation of the semistrip localized near the edge. This effect, known in mechanics as the "edge resonance", has been extensively studied numerically and experimentally. Our result provides a mathematical justification. 1. Introduction In this paper we study a particular selfadjoint operator A with spectrum oe(A) = [0; +1) and prove the existence of a positive eigenvalue e . Our proof is based on the use of certain symmetries of A which allow us to identify suitable invariant subspaces. The crucial step is to choose an invariant subspace in such a way that the restriction of A to this subspace has essential spectrum separated from zero. Then the existence of an eigenvalue below this essential spectr...
Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors., in "Inverse Problems
"... using time reversal mirrors ..."
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The Branch Structure of Embedded Trapped Modes in TwoDimensional Waveguides
"... In this pape we inve[ the ee[ of branche of e be trappe mo inthe vicinity of symme[ obstacle whichare place onthe ce tre of a twodime [ acoustic wave Mo deare sought whichare antisymme aboutthe ce tre ofthe channe and which have fre[ thatare above the first cuto# for antisymme wave propagatio ..."
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Cited by 5 (4 self)
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In this pape we inve[ the ee[ of branche of e be trappe mo inthe vicinity of symme[ obstacle whichare place onthe ce tre of a twodime [ acoustic wave Mo deare sought whichare antisymme aboutthe ce tre ofthe channe and which have fre[ thatare above the first cuto# for antisymme wave propagation downthe guide Inpre work [1], a proce[ for finding such mode wasde e e and it was shown nume[ that a branch of trappe mo dee1 for ane[ which starts from a flatplate onthe ce tre1 ofthe guide and te1[ with a flatplate pe e[ tothe guide walls. In this work we show thatfurthe branche of such mode e for botheh[ andre1[ blocks, e, h of which starts with aplate ofdi#e tle onthe ce tre ofthe guide Approximations tothe trappe mode wave numbe forre1[ blocksare de e from a twote matche ec[[ e[e andthe are compare to the re1 fromthe nume[ sche de1 e in [1].The transition from trappe mode to standingwave which occurs atone ee of e[ h ofthe branche is inve[ inde11 Ke ords:e be trappe mode bran...
Existence of eigenvalues of a linear operator pencil in a curved waveguide – localized shelf waves on a curved coast
 SIAM J. Math. Anal
"... Abstract. The study of the possibility of existence of the nonpropagating, trapped continental shelf waves (CSWs) along curved coasts reduces mathematically to a spectral problem for a selfadjoint operator pencil in a curved strip. Using the methods developed in the setting of the waveguide trapped ..."
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Abstract. The study of the possibility of existence of the nonpropagating, trapped continental shelf waves (CSWs) along curved coasts reduces mathematically to a spectral problem for a selfadjoint operator pencil in a curved strip. Using the methods developed in the setting of the waveguide trapped mode problem, we show that such CSWs exist for a wide class of coast curvature and depth profiles. Key words. continental shelf waves; curved coasts; trapped modes; essential spectrum; operator pencil