215 research outputs found
Semiclassical Resonances of Schr\"odinger operators as zeroes of regularized determinants
We prove that the resonances of long range perturbations of the
(semiclassical) Laplacian are the zeroes of natural perturbation determinants.
We more precisely obtain factorizations of these determinants of the form and give
semiclassical bounds on as well as a representation of
Koplienko's regularized spectral shift function. Here the index
depends on the decay rate at infinity of the perturbation.Comment: 37 pages, published versio
Convergence of a Vector Penalty Projection Scheme for the Navier-Stokes Equations with moving body
In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to
treat the displacement of a moving body in incompressible viscous flows in the
case where the interaction of the fluid on the body can be neglected. The
presence of the obstacle inside the computational domain is treated with a
penalization method introducing a parameter . We show the stability of
the scheme and that the pressure and velocity converge towards a limit when the
penalty parameter , which induces a small divergence and the time
step t tend to zero with a proportionality constraint =
t. Finally, when goes to 0, we show that the problem
admits a weak limit which is a weak solution of the Navier-Stokes equations
with no-sleep condition on the solid boundary. R{\'e}sum{\'e} Dans ce travail
nous analysons un sch{\'e}ma de projection vectorielle (voir [1]) pour traiter
le d{\'e}placement d'un corps solide dans un fluide visqueux incompressible
dans le cas o` u l'interaction du fluide sur le solide est n{\'e}gligeable. La
pr{\'e}sence de l'obstacle dans le domaine solide est mod{\'e}lis{\'e}e par une
m{\'e}thode de p{\'e}nalisation. Nous montrons la stabilit{\'e} du sch{\'e}ma
et la convergence des variables vitesse-pression vers une limite quand le param
etre qui assure une faible divergence et le pas de temps t
tendent vers 0 avec une contrainte de proportionalit{\'e} =
t. Finalement nous montrons que leprob{\`i} eme converge au
sens faible vers une solution des equations de Navier-Stokes avec une condition
aux limites de non glissement sur lafront{\`i} ere immerg{\'e}e quand le param
etre de p{\'e}nalisation tend vers 0
Discrete Spectrum of Quantum Hall Effect Hamiltonians I. Monotone Edge Potential
We consider the unperturbed operator ,
self-adjoint in . Here is a magnetic potential which generates a
constant magnetic field , and the edge potential is a non-decreasing
non constant bounded function depending only on the first coordinate
of . Then the spectrum of has a band structure and is
absolutely continuous; moreover, the assumption implies the existence of infinitely many spectral gaps for .
We consider the perturbed operators where the electric
potential is non-negative and decays at infinity. We
investigate the asymptotic distribution of the discrete spectrum of in
the spectral gaps of . We introduce an effective Hamiltonian which governs
the main asymptotic term; this Hamiltonian involves a pseudo-differential
operator with generalized anti-Wick symbol equal to . Further, we restrict
our attention on perturbations of compact support and constant sign. We
establish a geometric condition on the support of which guarantees the
finiteness of the eigenvalues of in any spectral gap of . In the
case where this condition is violated, we show that, generically, the
convergence of the infinite series of eigenvalues of (resp. ) to the
left (resp. right) edge of a given spectral gap, is Gaussian.Comment: 32 page
ON THE GROUND STATE ENERGY OF THE LAPLACIAN WITH A MAGNETIC FIELD CREATED BY A RECTILINEAR CURRENT
ABSTRACT. We consider the three-dimensional Laplacian with a magnetic field created by an infinite rectilinear current bearing a constant current. The spectrum of the associated Hamiltonian is the positive half-axis as the range of an infinity of band functions all decreasing toward 0. We make a precise asymptotics of the band functions near the ground state energy and we exhibit a semi-classical behavior. We perturb the Hamiltonian by an electric potential. Helped by the analysis of the band functions, we show that for slow decaying potential an infinite number of negative eigenvalues are created whereas only finite number of eigenvalues appears for fast decaying potential. The criterion about finiteness depends essentially on the decay rate of the potential with respect to the distance to the wire
Dynamical Resonances and SSF Singularities for a Magnetic Schrödinger Operator
We consider the Hamiltonian H of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator H has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H by appropriate scalar potentials V and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic field, and obtain an asymptotic expansion of the resonances as the coupling constant ϰ of the perturbation tends to zero. Further, under the assumption that the Fermi Golden Rule holds true, we deduce estimates for the time evolution of the resonance states with and without analyticity assumptions; in the second case we obtain these results as a corollary of suitable Mourre estimates and a recent article of Cattaneo, Graf and Hunziker [11]. Next, we describe sets of perturbations V for which the Fermi Golden Rule is valid at each embedded eigenvalue of H; these sets turn out to be dense in various suitable topologies. Finally, we assume that V decays fast enough at infinity and is of definite sign, introduce the Krein spectral shift function for the operator pair (H+V, H), and study its singularities at the energies which coincide with eigenvalues of infinite multiplicity of the unperturbed operator H
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