00-405 A.A. Balinsky, W.D. Evans and Roger T. Lewis

On the number of negative eigenvalues of Schr\"{o}dinger operators with an Aharonov-Bohm magnetic field (167K, "Postscript") Oct 17, 00

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Abstract. It is proved that for $V_+ = \max(V,0)$ in the subspace $ L^1 ( \mathbb{R}^+ , \ L^{\infty}(\mathbb{S}^1), \ rdr)$ of $L^1 (\mathbb{R}^2)$, there is a Cwikel-Lieb-Rosenblum type inequality for the number of negative eigenvalues of the operator $\biggl( \frac{1}{i} \vec{\nabla} + \vec{A} \biggr)^2 - V$ in $L^2 (\mathbb{R}^2)$ when $\vec{A}$ is an Aharonov-Bohm magnetic potential with non-integer flux. It is shown that $ L^1 ( \mathbb{R}^+ , \ L^{\infty}(\mathbb{S}^1), \ rdr)$ can not be replaced by $L^1 (\mathbb{R}^2)$ in the inequality.

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