We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of di
screte eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.
We study the survival probability associated with a semi-classical matrix Shrodinger operator that models the predissociation of a general molecule in the Born-Oppenheimer approximation. We show that it is given by its usual time-dependent exponentia
l contribution, up to a reminder term that is exponentially small (in the semiclassical parameter) with arbitrarily large rate of decay. The result applies in any dimension, and in presence of a number of resonances that may tend to infinity as the semiclassical parameter tends to 0.
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 havin
g a nonzero component on the longitudinal direction of the system. MSC-2010 number: 35B34,35P25, 81Q10, 82D77.
We study the resonance phenomena for time periodic perturbations of a Hamiltonian $H$ on the Hilbert space $L^2(mathbb R ^d)$. Here, resonances are characterized in terms of time behavior of the survival probability. Our approach uses the Floquet-How
land formalism combined with the results of L. Cattaneo, J.M. Graf and W. Hunziker on resonances for time independent perturbations.
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 contin
uous 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 $varkappa$ 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 cite{cgh}. 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 $H$.
Consider a charged, perfect quantum gas, in the effective mass approximation, and in the grand-canonical ensemble. We prove in this paper that the generalized magnetic susceptibilities admit the thermodynamic limit for all admissible fugacities, unif
ormly on compacts included in the analyticity domain of the grand-canonical pressure. The problem and the proof strategy were outlined in cite{3}. In cite{4} we proved in detail the pointwise thermodynamic limit near $z=0$. The present paper is the last one of this series, and contains the proof of the uniform bounds on compacts needed in order to apply Vitalis Convergence Theorem.
We consider a 2D Schroedinger operator H0 with constant magnetic field, on a strip of finite width. The spectrum of H0 is absolutely continuous, and contains a discrete set of thresholds. We perturb H0 by an electric potential V which decays in a sui
table sense at infinity, and study the spectral properties of the perturbed operator H = H0 + V . First, we establish a Mourre estimate, and as a corollary prove that the singular continuous spectrum of H is empty, and any compact subset of the complement of the threshold set may contain at most a finite set of eigenvalues of H, each of them having a finite multiplicity. Next, we introduce the Krein spectral shift function (SSF) for the operator pair (H,H0). We show that this SSF is bounded on any compact subset of the complement of the threshold set, and is continuous away from the threshold set and the eigenvalues of H. The main results of the article concern the asymptotic behaviour of the SSF at the thresholds, which is described in terms of the SSF for a pair of effective Hamiltonians.
