Resonances and Spectral Shift Function near the Landau levels
| dc.creator | Bony, J. F. | |
| dc.creator | Bruneau, V. | |
| dc.creator | Raikov, G. | |
| dc.date | 2006-03-31 | |
| dc.date | 2006-06-27 | |
| dc.date.accessioned | 2026-07-07T07:07:25Z | |
| dc.date.available | 2026-07-07T07:07:25Z | |
| dc.description | We consider the 3D Schrödinger operator $H = H_0 + V$ where $H_0 = (-i\nabla - A)^2$, $A$ is a magnetic potential generating a constant magnetic field of strength $b>0$, and $V$ is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of $H$ admits a meromorphic extension from the upper half-plane to an appropriate complex manifold ${\mathcal M}$, and define the resonances of $H$ as the poles of this meromorphic extension. We study their distribution near any fixed Landau level $2bq$, $q \in {\mathbb N}$. First, we obtain a sharp upper bound of the number of resonances in a vicinity of $2bq$. Moreover, under appropriate hypotheses, we establish corresponding lower bounds which imply the existence of an infinite number of resonances, or the absence of resonances in certain sectors adjoining $2bq$. Finally, we deduce a representation of the derivative of the spectral shift function (SSF) for the operator pair $(H,H_0)$ as a sum of a harmonic measure related to the resonances, and the imaginary part of a holomorphic function. This representation justifies the Breit-Wigner approximation, implies a trace formula, and provides information on the singularities of the SSF at the Landau levels. | |
| dc.description | 32 pages, 4 figures. Revised version with more precise notation concerning subsets of the Riemann surface | |
| dc.identifier | https://arxiv.org/abs/math/0603731 | |
| dc.identifier | http://arxiv.org/abs/math/0603731 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/110381 | |
| dc.subject | Spectral Theory | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35P25; 35J10; 47F05; 81Q10 | |
| dc.title | Resonances and Spectral Shift Function near the Landau levels | |
| dc.type | text |