Dynamical Resonances and SSF Singularities for a Magnetic Schrödinger Operator

Astaburuaga, María Angélica, et al. "Dynamical Resonances and SSF Singularities for a Magnetic Schrödinger Operator." Serdica Mathematical Journal 34.1 (2008): 179-218. <http://eudml.org/doc/281472>.

@article{Astaburuaga2008,
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 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.},
author = {Astaburuaga, María Angélica, Briet, Philippe, Bruneau, Vincent, Fernández, Claudio, Raikov, Georgi},
journal = {Serdica Mathematical Journal},
keywords = {Magnetic Schrödinger Operators; Resonances; Mourre Estimates; Spectral Shift Function; Magnetic Schrödinger operators; resonances; Mourre estimates; spectral shift function},
language = {eng},
number = {1},
pages = {179-218},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Dynamical Resonances and SSF Singularities for a Magnetic Schrödinger Operator},
url = {http://eudml.org/doc/281472},
volume = {34},
year = {2008},
}

TY - JOUR
AU - Astaburuaga, María Angélica
AU - Briet, Philippe
AU - Bruneau, Vincent
AU - Fernández, Claudio
AU - Raikov, Georgi
TI - Dynamical Resonances and SSF Singularities for a Magnetic Schrödinger Operator
JO - Serdica Mathematical Journal
PY - 2008
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 34
IS - 1
SP - 179
EP - 218
AB - 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.
LA - eng
KW - Magnetic Schrödinger Operators; Resonances; Mourre Estimates; Spectral Shift Function; Magnetic Schrödinger operators; resonances; Mourre estimates; spectral shift function
UR - http://eudml.org/doc/281472
ER -