Lieb-Thirring inequality for the 2D Pauli operator

By the Aharonov-Casher theorem, the Pauli operator $P$ has no zero eigenvalue when the normalized magnetic flux $\alpha$ satisfies $\mathrm{\mid }\alpha \mathrm{\mid }<1$, but it does have a zero energy resonance. We prove that in this case a Lieb-Thirring inequality for the $\gamma$-th moment of the eigenvalues of $P+V$ is valid under the optimal restrictions $\gamma \ge \mathrm{\mid }\alpha \mathrm{\mid }$ and $\gamma >0$. Besides the usual semiclassical integral, the right side of our inequality involves an integral where the zero energy resonance state appears explicitly. Our inequality improves earlier works that were restricted to moments of order $\gamma \ge 1$.