On the number of bound states for Schrdinger operators with operator-valued potentials.pdf

ON THE NUMBER OF BOUND STATES FOR SCHRO?DINGER OPERATORS WITH OPERATOR-VALUED POTENTIALS DIRK HUNDERTMARK Abstract. Cwikel’s bound is extended to an operator-valued setting. One application of this result is a semi-classical bound for the number of negative bound states for Schro?dinger operators with operator-valued potentials. We recover Cwikel’s bound for the Lieb–Thirring constant L0,3 which is far worse than the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimension d ≥ 3) for the quotient L0,d/L cl 0,d, where L cl 0,d is the so-called classical constant. This gives some improvement in large dimensions. 1. Introduction The Lieb-Thirring inequalities bound certainmoments of the negative eigen- values of a one-particle Schro?dinger operator by the corresponding classical phase space moment. More precisely, for “nice enough” potentials one has trL2(Rd)(??+ V )γ? ≤ Cγ,d (2π)d ∫∫ RdRd dξdx (ξ2 + V (x))γ?. (1) Here and in the following, (x)? = 1 2 (|x| ? x) is the negative part of a real number or a self-adjoint operator. Doing the ξ integration explicitly with the help of scaling the above inequality is equivalent to its more often used form trL2(Rd)(??+ V )γ? ≤ Lγ,d ∫ Rd dx V (x) γ+d/2 ? , (2) where the Lieb-Thirring constant Lγ,d is given by Lγ,d = Cγ,dL cl γ,d with the classical Lieb-Thirring constant Lclγ,d = 1 (2π)d ∫ Rd dp(1? p2)γ+. (3) This integral is, of course, explicitly given by a quotient of Gamma functions, but we will have no need for this. The Lieb-Thirring inequalities are valid as soon as the potential V is in Lγ+d/2(Rd). Department of Mathematics 253–37, California Institute of Technology, Pasadena, CA 91125, U.S.A.; E-mail: dirkh@caltech.edu. 2000 Mathematics subject classification. Primary: 35P15, 47B10; Secondary: 81Q10, 47L20. Key words: CLR estimate, weak type estimates, singular values. c?2000 by the author. Reproduction of this article, in its entirety, by any means is permitted for non-commer