M. A. Astaburuaga, Ph. Briet, V. Bruneau, C. Fernandez, G. Raikov
Dynamical resonances and SSF singularities for a magnetic
Schroedinger operator
(427K, pdf)
ABSTRACT. 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 $H$.