CLR type bounds for a magnetic Schrödinger operator in two dimensions

The Cwikel-Lieb-Rosenblum (CLR) inequality provides an upper-bound on the number of negative eigenvalues of a Schrödinger operator $-\Delta-V$ in $L^2(\mathbb{R}^d)$, $V\geq 0$, for dimension $d\geq 3$ in terms of the $L^{d/2}(\mathbb{R}^d)$-norm of its potential. In dimensions one and two there is an absence of such bounds since any non-trivial potential $V$ produces at least one negative eigenvalue. In this talk I will consider the case of the two dimensional Aharonov-Bohm magnetic Schrödinger operator. I will show that the addition of this magnetic field lifts the energy enough for us to find CLR type bounds. The results discussed are from joint works with Rupert L. Frank, Ari Laptev and Lukas Schimmer.