Dynamical resonances and SSF singularities for a magnetic Schroedinger operator

Details Dynamical resonances and SSF singularities for a magnetic Schroedinger operator Dynamical resonances and SSF singularities for a magnetic Schroedinger operator

2007-10-02

arxiv.org/abs/0710.0502v4

Serdica Math. J. 34 (2008) 179-218

Mathematics

Spectral Theory

29 pages, published version

Scientific paper

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 $\varkappa$ 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 $H$.

Affiliated with

Also associated with

No associations

Rating

Dynamical resonances and SSF singularities for a magnetic Schroedinger operator does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Dynamical resonances and SSF singularities for a magnetic Schroedinger operator, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Dynamical resonances and SSF singularities for a magnetic Schroedinger operator will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-7186