Orateur
Description
Consider the Landau Hamiltonian $H=-\frac{\partial^2}{\partial x_1^2} + \left( - i \frac{\partial}{\partial x_2} - Bx_1 \right)^2$ in Landau gauge acting on $L^2(\mathbb R^2)$. Let $1_{\{B\}}(H)$ be the spectral projection onto the (infinitely degenerate) eigenspace corresponding to the eigenvalue $\lambda = B$. Leschke, Sobolev and Spitzer established an asymptotic expansion of the form [ \operatorname{tr}h(1_{[-l,l]^2} 1_{{B }}(H) 1_{[-l,l]^2}) = \alpha l^2 + \beta l + o(l) ] as $l\to \infty$, for fairly general functions $h$ with $h(0) = 0$. Our main result is a corresponding asymptotic expansion when replacing the projecton $1_{\{B\}}(H)$ by a subprojection $P$ onto a subspace incorporating only ``half'' the eigenfunctions (in a sense that will be made precise). We will see that in this case the subleading behavior of the asymptotic expansion features an additional logarithmic enhancement term $\log l$.
