Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields

Raikov, Georgi D.. "Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields." Annales de l'institut Fourier 49.5 (1999): 1603-1636. <http://eudml.org/doc/75395>.

@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, $\mu &gt; 0$ 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 $\lambda &lt; 0$ 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 $0 &lt; \lambda _1 &lt; \lambda _2$.},
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, $\mu &gt; 0$ 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 $\lambda &lt; 0$ 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 $0 &lt; \lambda _1 &lt; \lambda _2$.
LA - eng
KW - eigenvalue asymptotics; quantum mechanical equations; Pauli operators; strong magnetic fields
UR - http://eudml.org/doc/75395
ER -