Tuesday, February 19, 2013

1302.4074 (Mouez Dimassi et al.)

Trace asymptotics formula for the Schrödinger operators with constant
magnetic fields

Mouez Dimassi, Anh Tuan Duong
In this paper, we consider the 2D- Schr\"odinger operator with constant magnetic field $H(V)=(D_x-By)^2+D_y^2+V_h(x,y)$, where $V$ tends to zero at infinity and $h$ is a small positive parameter. We will be concerned with two cases: the semi-classical limit regime $V_h(x,y)=V(h x,h y)$, and the large coupling constant limit case $V_h(x,y)=h^{-\delta} V(x,y)$. We obtain a complete asymptotic expansion in powers of $h^2$ of ${\rm tr}(\Phi(H(V),h))$, where $\Phi(\cdot,h)\in C^\infty_0(\mathbb R;\mathbb R)$. We also give a Weyl type asymptotics formula with optimal remainder estimate of the counting function of eigenvalues of $H(V)$.
View original: http://arxiv.org/abs/1302.4074

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