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Spectral shift function in strong magnetic fields


Authors: V. Bruneau, A. Pushnitski and G. Raikov
Original publication: Algebra i Analiz, tom 16 (2004), nomer 1.
Journal: St. Petersburg Math. J. 16 (2005), 181-209
MSC (2000): Primary 35J10
Published electronically: December 17, 2004
MathSciNet review: 2069004
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Abstract: The three-dimensional Schrödinger operator $H$ with constant magnetic field of strength $b>0$ is considered under the assumption that the electric potential $V \in L^1({{\mathbb R}^3})$ admits certain power-like estimates at infinity. The asymptotic behavior as $b \rightarrow \infty$ of the spectral shift function $\xi(E;H,H_0)$ is studied for the pair of operators $(H,H_0)$ at the energies $E = {\mathcal{E}} b + \lambda$, ${\mathcal{E}}>0$ and $\lambda \in{\mathbb R}$ being fixed. Two asymptotic regimes are distinguished. In the first regime, called asymptotics far from the Landau levels, we pick ${\mathcal{E}}/2 \not \in {\mathbb Z}_+$ and $\lambda \in {\mathbb R}$; then the main term is always of order $\sqrt{b}$, and is independent of $\lambda$. In the second asymptotic regime, called asymptotics near a Landau level, we choose ${\mathcal{E}}= 2 q_0$, $q_0 \in {\mathbb Z}_+$, and $\lambda \neq 0$; in this case the leading term of the SSF could be of order $b$ or $\sqrt{b}$ for different $\lambda$.


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Additional Information

V. Bruneau
Affiliation: Mathématiques Appliquées de Bordeaux, Université Bordeaux I, 351 Cours de la Libération, 33405 Talence, France
Email: vbruneau@math.u-bordeaux.fr

A. Pushnitski
Affiliation: Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom
Email: A.B.Pushnitski@lboro.ac.uk

G. Raikov
Affiliation: Departamento de Matemáticas, Universidad de Chile, Las Palmeras 3425, Casilla 653, Santiago, Chile
Email: graykov@uchile.cl

DOI: http://dx.doi.org/10.1090/S1061-0022-04-00847-7
Keywords: Schr\"odinger operator, spectral shift function, asymptotics
Received by editor(s): October 27, 2003
Published electronically: December 17, 2004
Additional Notes: V. Bruneau and G. Raikov were supported by the Chilean Science Foundation Fondecyt (grants nos. 1020737 and 7020737). A. Pushnitski was supported by the EPSRC (grant no. GR/R53210/01).
Dedicated: Dedicated to Professor Mikhail Birman on the occasion of his 75th birthday
Article copyright: © Copyright 2004 American Mathematical Society