98 research outputs found
Exponential decay and resonances in a driven system
We study the resonance phenomena for time periodic perturbations of a
Hamiltonian on the Hilbert space . Here, resonances are
characterized in terms of time behavior of the survival probability. Our
approach uses the Floquet-Howland formalism combined with the results of L.
Cattaneo, J.M. Graf and W. Hunziker on resonances for time independent
perturbations.Comment: 16 page
Stark resonances in 2-dimensional curved quantum waveguides
In this paper we study the influence of an electric field on a two
dimen-sional waveguide. We show that bound states that occur under a
geometrical deformation of the guide turn into resonances when we apply an
electric field of small intensity having a nonzero component on the
longitudinal direction of the system. MSC-2010 number: 35B34,35P25, 81Q10,
82D77
A rigorous approach to the magnetic response in disordered systems
This paper is a part of an ongoing study on the diamagnetic behavior of a
3-dimensional quantum gas of non-interacting charged particles subjected to an
external uniform magnetic field together with a random electric potential. We
prove the existence of an almost-sure non-random thermodynamic limit for the
grand-canonical pressure, magnetization and zero- field orbital magnetic
susceptibility. We also give an explicit formulation of these thermodynamic
limits. Our results cover a wide class of physically relevant random potentials
which model not only crystalline disordered solids, but also amorphous solids.Comment: 35 pages. Revised version. Accepted for publication in RM
Hardy inequalities in globally twisted waveguides
We establish various Hardy-type inequalities for the Dirichlet Laplacian in
perturbed periodically twisted tubes of non-circular cross-sections. We also
state conjectures about the existence of such inequalities in more general
regimes, which we support by heuristic and numerical arguments.Comment: 18 pages, 1 figur
A rigorous proof of the Landau-Peierls formula and much more
We present a rigorous mathematical treatment of the zero-field orbital
magnetic susceptibility of a non-interacting Bloch electron gas, at fixed
temperature and density, for both metals and semiconductors/insulators. In
particular, we obtain the Landau-Peierls formula in the low temperature and
density limit as conjectured by T. Kjeldaas and W. Kohn in 1957.Comment: 30 pages - Accepted for publication in A.H.
Stark resonances in a quantum waveguide with analytic curvature
We investigate the influence of an electric field on trapped modes arising in
a two-dimensional curved quantum waveguide i.e. bound states of
the corresponding Laplace operator . Here the
curvature of the guide is supposed to satisfy some assumptions of analyticity,
and decays as at infinity. We show
that under conditions on the electric field , has resonances near the discrete
eigenvalues of
Spectral optimisation of Dirac rectangles
We are concerned with the dependence of the lowest positive eigenvalue of the
Dirac operator on the geometry of rectangles, subject to infinite-mass boundary
conditions. We conjecture that the square is a global minimiser both under the
area or perimeter constraints. Contrary to well-known non-relativistic
analogues, we show that the present spectral problem does not admit explicit
solutions. We prove partial optimisation results based on a variational
reformulation and newly established lower and upper bounds to the Dirac
eigenvalue. We also propose an alternative approach based on symmetries of
rectangles and a non-convex minimisation problem; this implies a sufficient
condition formulated in terms of a symmetry of the minimiser which guarantees
the conjectured results.Comment: 11 pages; due to a gap in the proof in our previous version (see
Remark 1), we obtain just partial results, by an alternative approach;
version accepted for publication in Journal of Mathematical Physic
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