2,801 research outputs found
Separation of variables in perturbed cylinders
We study the Laplace operator subject to Dirichlet boundary conditions in a
two-dimensional domain that is one-to-one mapped onto a cylinder (rectangle or
infinite strip). As a result of this transformation the original eigenvalue
problem is reduced to an equivalent problem for an operator with variable
coefficients. Taking advantage of the simple geometry we separate variables by
means of the Fourier decomposition method. The ODE system obtained in this way
is then solved numerically yielding the eigenvalues of the operator. The same
approach allows us to find complex resonances arising in some non-compact
domains. We discuss numerical examples related to quantum waveguide problems.Comment: LaTeX 2e, 18 pages, 6 figure
Uniform resolvent estimates and absence of eigenvalues for Lamé operators with complex potentials
We consider the -order perturbed Lamé operator .
It is well known that if one considers the free case, namely the spectrum of is purely continuous and coincides with the non-negative semi-axis.
The first purpose of the paper is to show that, at least in part, this spectral property is preserved in the perturbed setting. Precisely, developing a suitable multipliers technique, we will prove the absence of point spectrum for Lamé operator with potentials which satisfy a variational inequality with suitable small constant. We stress that our result also covers complex-valued perturbation terms. Moreover the techniques used to prove the absence of eigenvalues enable us to provide uniform resolvent estimates for the perturbed operator under the same assumptions about
Adaptive BEM with optimal convergence rates for the Helmholtz equation
We analyze an adaptive boundary element method for the weakly-singular and
hypersingular integral equations for the 2D and 3D Helmholtz problem. The
proposed adaptive algorithm is steered by a residual error estimator and does
not rely on any a priori information that the underlying meshes are
sufficiently fine. We prove convergence of the error estimator with optimal
algebraic rates, independently of the (coarse) initial mesh. As a technical
contribution, we prove certain local inverse-type estimates for the boundary
integral operators associated with the Helmholtz equation
A Bloch wave numerical scheme for scattering problems in periodic wave-guides
We present a new numerical scheme to solve the Helmholtz equation in a
wave-guide. We consider a medium that is bounded in the -direction,
unbounded in the -direction and -periodic for large ,
allowing different media on the left and on the right. We suggest a new
numerical method that is based on a truncation of the domain and the use of
Bloch wave ansatz functions in radiation boxes. We prove the existence and a
stability estimate for the infinite dimensional version of the proposed
problem. The scheme is tested on several interfaces of homogeneous and periodic
media and it is used to investigate the effect of negative refraction at the
interface of a photonic crystal with a positive effective refractive index.Comment: 25 pages, 10 figure
Reconstruction of dielectric constants of multi-layered optical fibers using propagation constants measurements
We present new method for the numerical reconstruction of the variable
refractive index of multi-layered circular weakly guiding dielectric waveguides
using the measurements of the propagation constants of their eigenwaves. Our
numerical examples show stable reconstruction of the dielectric permittivity
function for random noise level using these measurements
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