Georgia Institute of Technology College of Sciences

The Cwikel-Lieb-Rozenblum (CLR) inequality is a semi-classical estimate on the number of bound states for Schrödinger operators. In this talk I will give a brief overview of the CLR inequality and present a substantial refinement of Cwikel’s original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our new proof highlights a natural but overlooked connection of the CLR inequality with bounds for maximal Fourier multipliers from harmonic analysis and leads to a variational problem that can be reformulated in terms of a variant of Hadamard’s three-lines lemma. The solution of this variational problem relies on some interesting complex analysis techniques. (Based on joint work with T. Carvalho-Corso, D. Hundertmark, P. Kunstmann, S. Vugalter)