Spectral shift function in strong magnetic fields
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- by V. Bruneau, A. Pushnitski and G. Raikov
- St. Petersburg Math. J. 16 (2005), 181-209
- DOI: https://doi.org/10.1090/S1061-0022-04-00847-7
- Published electronically: December 17, 2004
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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 $\lambda$.References
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Bibliographic Information
- V. Bruneau
- Affiliation: Mathématiques Appliquées de Bordeaux, Université Bordeaux I, 351 Cours de la Libération, 33405 Talence, France
- MR Author ID: 607313
- 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
- MR Author ID: 144060
- Email: graykov@uchile.cl
- 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).
- © Copyright 2004 American Mathematical Society
- Journal: St. Petersburg Math. J. 16 (2005), 181-209
- MSC (2000): Primary 35J10
- DOI: https://doi.org/10.1090/S1061-0022-04-00847-7
- MathSciNet review: 2069004
Dedicated: Dedicated to Professor Mikhail Birman on the occasion of his 75th birthday