1,721,007 research outputs found
Spectrum of the magnetic Schroedinger operator in a waveguide with combined boundary conditions
No full textHomogenization of the planar waveguide with frequently alternating boundary conditions
No full textWe consider the Laplacian in a planar strip with a Dirichlet boundary condition on the upper boundary and with a frequent alternation boundary condition on the lower boundary. The alternation is introduced by the periodic partition of the boundary into small segments on which Dirichlet and Neumann conditions are imposed in turns. We show that under certain conditions the homogenized operator is the Dirichlet Laplacian and prove the uniform resolvent convergence. The spectrum of the perturbed operator consists of its essential part only and has a band structure. We construct the leading terms of the asymptotic expansions for the first band functions. We also construct the complete asymptotic expansion for the bottom of the spectrum
Planar waveguide with “twisted” boundary conditions: Small width
No full textWe consider a planar waveguide with "twisted" boundary conditions. By twisting we mean a special combination of Dirichlet and Neumann boundary conditions. Assuming that the width of the waveguide goes to zero, we identify the effective (limiting) operator as the width of the waveguide tends to zero, establishes the uniform resolvent convergence in various possible operator norms, and gives the estimates for the rates of convergence. We show that studying the resolvent convergence can be treated as a certain threshold effect and we present an elegant technique which justifies such point of view. (C) 2012 American Institute of Physics. [doi:10.1063/1.3681895
On a waveguide with an infinite number of small windows
No full textWe consider a waveguide modeled by the Laplacian in a straight planar strip with the Dirichlet condition on the upper boundary, while on the lower one we impose periodically alternating boundary conditions with a small period. We study the case when the homogenization leads us to the Neumann boundary condition on the lower boundary. We establish the uniform resolvent convergence and provide the estimates for the rate of convergence. We construct the two-terms asymptotics for the first band functions of the perturbed operator and also the complete two-parametric asymptotic expansion for the bottom of its spectrum. (C) 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved
Planar waveguide with “twisted” boundary conditions: Discrete spectrum
No full textWe consider a planar waveguide with combined Dirichlet and Neumann conditions imposed in a "twisted" way. We study the discrete spectrum and describe it dependence on the configuration of the boundary conditions. In particular, we show that in certain cases the model can have discrete eigenvalues emerging from the threshold of the essential spectrum. We give a criterium for their existence and construct them as convergent holomorphic series. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3670875
Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows
No full textWe consider a planar waveguide modeled by the Laplacian in a straight infinite strip with
the Dirichlet boundary condition on the upper boundary and with frequently alternating
boundary conditions (Dirichlet and Neumann) on the lower boundary. The homogenized
operator is the Laplacian subject to the Dirichlet boundary condition on the upper boundary
and to the Dirichlet or Neumann condition on the lower one. We prove the uniform
resolvent convergence for the perturbed operator in both cases and obtain the estimates
for the rate of convergence. Moreover, we construct the leading terms of the asymptotic
expansions for the first band functions and the complete asymptotic expansion for the
bottom of the spectrum
On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition
No full textWe consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet-Bloch decomposition, the two terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term
Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics
No full textWe consider a magnetic Schrodinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for the first band functions, as well as the complete asymptotics expansion (up to an exponentially small term) for the bottom of the band spectrum. RI Cardone, Giuseppe/I-2998-2012 OI Cardone, Giuseppe/0000-0002-5050-890
Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve.
Full text linkWe consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm-resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm-resolvent convergence, we prove the convergence of the spectrum
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