# Dynamical Resonances and SSF Singularities for a Magnetic Schrödinger Operator

Astaburuaga, María Angélica; Briet, Philippe; Bruneau, Vincent; Fernández, Claudio; Raikov, Georgi

Serdica Mathematical Journal (2008)

- Volume: 34, Issue: 1, page 179-218
- ISSN: 1310-6600

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topAstaburuaga, 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 -

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