970 research outputs found
Magnetic WKB Constructions
This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB
expansions for the eigenfunctions were only established in presence of a
non-zero electric potential. Here we tackle the pure magnetic case. Thanks to
Feynman-Hellmann type formulas and coherent states decomposition, we develop
here a magnetic Born-Oppenheimer theory. Exploiting the multiple scales of the
problem, we are led to solve an effective eikonal equation in pure magnetic
cases and to obtain WKB expansions. We also investigate explicit examples for
which we can improve our general theorem: global WKB expansions, quasi-optimal
estimates of Agmon and upper bound of the tunelling effect (in symmetric
cases). We also apply our strategy to get more accurate descriptions of the
eigenvalues and eigenfunctions in a wide range of situations analyzed in the
last two decades
Quantum Hall Effect on the Hyperbolic Plane
In this paper, we study both the continuous model and the discrete model of
the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is
identified as a geometric invariant associated to an imprimitivity algebra of
observables. We define a twisted analogue of the Kasparov map, which enables us
to use the pairing between -theory and cyclic cohomology theory, to identify
this geometric invariant with a topological index, thereby proving the
integrality of the Hall conductivity in this case.Comment: AMS-LaTeX, 28 page
Quantization of edge currents for continuous magnetic operators
For a magnetic Hamiltonian on a half-plane given as the sum of the Landau
operator with Dirichlet boundary conditions and a random potential, a
quantization theorem for the edge currents is proven. This shows that the
concept of edge channels also makes sense in presence of disorder. Moreover,
Gaussian bounds on the heat kernel and its covariant derivatives are obtained
An adaptive finite element method for solving 3D electromagnetic volume integral equation with applications in microwave thermometry
An adaptive finite element method (AFEM) for the numerical solution of an electromagnetic volume integral equation (VIE) is presented. To solve the model VIE, the problem is formulated as an optimal control problem for minimization of Tikhonov\u27s regularization functional. A posteriori error estimates in the obtained finite element reconstruction and in the underlying Tikhonov\u27s functional are derived. Based on these estimates, adaptive finite element algorithms are formulated and numerically tested on the problem of microwave hyperthermia in cancer treatment. In this problem, the temperature change of a target in the computational domain results in the change of its dielectric properties. Numerical examples of monitoring this change show robust and qualitative three-dimensional reconstructions of the target using the proposed adaptive algorithms
A posteriori error estimates for the Electric Field Integral Equation on polyhedra
We present a residual-based a posteriori error estimate for the Electric
Field Integral Equation (EFIE) on a bounded polyhedron. The EFIE is a
variational equation formulated in a negative order Sobolev space on the
surface of the polyhedron. We express the estimate in terms of
square-integrable and thus computable quantities and derive global lower and
upper bounds (up to oscillation terms).Comment: Submitted to Mathematics of Computatio
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