A.V. Sobolev
Discrete Spectrum Asymptotics for the Schroedinger Operator with a Singular Potential and a Magnetic Field
The paper is published: Rev. Math. Phys. 8, 6 (1996) 861-903

MSC:

35P20 Asymptotic distribution of eigenvalues and eigenfunctions for PDO

35J10 Schrodinger operator, See also {35Pxx}

Abstract: Object of the study is the operator $H = H_0(h, \mu) + V$ in
$L^2(\Rd),\ d\ge 2$, where $H_0(h, \mu)$
is the Schr\"odinger operator with
a magnetic field of intensity $\mu\ge 0$ and the Planck constant
$h\in (0, h_0\rbr$. The
electric (real-valued) potential $V = V(x)$ is assumed to be
asymptotically homogeneous of order $-\b,\ \b \ge 0$ as $x\ra 0$. One
obtains asymptotic formulae with remainder estimates as $h\ra 0,\
\mu h\le C$ for the trace $\cm_s = \tr \{\psi g_s(H)\}$ where
$\psi\in\co(\Rd),\ g(\l) = \l_-^s, s\in\lbr 0, 1\rbr$. Due to the
condition $\mu h\le C$ the leading term of $\cm_s$ does not
depend on $\mu$. It depends on the relation between the parameters
$d, s$ and $\b$. There are five regions, in which either leading
terms or remainder estimates have different form.
In one of these regions $\cm_s$ admits a
two-term asymptotics. In this case, for an asymptotically Coulomb
potential the second term coincides with the well known Scott
correction term.

Keywords: trace asymptotics, singular potential, semiclassical asymptotics