Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields

@article{Raikov1999,
abstract = {We consider the Pauli operator $\{\bf H\}(\mu ):= \big (\sum _\{j=1\}^m \sigma _j \big (-i\{\partial \over \partial x_j\} - \mu A_j\big ) \big )^2 + V$ selfadjoint in $L^2(\{\Bbb R\}^m; \{\Bbb C\}^2)$, $m=2,3$. Here $\sigma _j$, $j=1, \ldots , m$, are the Pauli matrices, $A: = (A_1, \ldots , A_m)$ is the magnetic potential, $\mu > 0$ is the coupling constant, and $V$ is the electric potential which decays at infinity. We suppose that the magnetic field generated by $A$ satisfies some regularity conditions; in particular, its norm is lower-bounded by a positive constant, and, in the case $m=3$, its direction is constant. We investigate the asymptotic behaviour as $\mu \rightarrow \infty $ of the number of the eigenvalues of $\{\bf H\}(\mu )$ smaller than $\lambda $, the parameter $\lambda < 0$ being fixed. Furthermore, if $m=2$, we study the asymptotics as $\mu \rightarrow \infty $ of the number of the eigenvalues of $\{\bf H\}(\mu )$ situated on the interval $(\lambda _1, \lambda _2)$ with $0 < \lambda _1 < \lambda _2$.},
author = {Raikov, Georgi D.},
journal = {Annales de l'institut Fourier},
keywords = {eigenvalue asymptotics; quantum mechanical equations; Pauli operators; strong magnetic fields},
language = {eng},
number = {5},
pages = {1603-1636},
publisher = {Association des Annales de l'Institut Fourier},
title = {Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields},
url = {http://eudml.org/doc/75395},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Raikov, Georgi D.
TI - Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 5
SP - 1603
EP - 1636
AB - We consider the Pauli operator ${\bf H}(\mu ):= \big (\sum _{j=1}^m \sigma _j \big (-i{\partial \over \partial x_j} - \mu A_j\big ) \big )^2 + V$ selfadjoint in $L^2({\Bbb R}^m; {\Bbb C}^2)$, $m=2,3$. Here $\sigma _j$, $j=1, \ldots , m$, are the Pauli matrices, $A: = (A_1, \ldots , A_m)$ is the magnetic potential, $\mu > 0$ is the coupling constant, and $V$ is the electric potential which decays at infinity. We suppose that the magnetic field generated by $A$ satisfies some regularity conditions; in particular, its norm is lower-bounded by a positive constant, and, in the case $m=3$, its direction is constant. We investigate the asymptotic behaviour as $\mu \rightarrow \infty $ of the number of the eigenvalues of ${\bf H}(\mu )$ smaller than $\lambda $, the parameter $\lambda < 0$ being fixed. Furthermore, if $m=2$, we study the asymptotics as $\mu \rightarrow \infty $ of the number of the eigenvalues of ${\bf H}(\mu )$ situated on the interval $(\lambda _1, \lambda _2)$ with $0 < \lambda _1 < \lambda _2$.
LA - eng
KW - eigenvalue asymptotics; quantum mechanical equations; Pauli operators; strong magnetic fields
UR - http://eudml.org/doc/75395
ER -