Analysis with weak trace ideals and the number of bound states of Schrödinger operators

Abstract:

We discuss interpolation theory for the operator ideals $I_p^w$ defined on a separable Hilbert space as those operators A whose singular values ${\mu _n}(A)$ obey ${\mu _n} \leqslant c{n^{ - 1/p}}$ for some c. As an application we consider the functional $N(V) = \dim$ (spectral projection on $( - \infty ,0)$ for $- \Delta + V$) on functions V on ${{\mathbf {R}}^n},n \geqslant 3$. We prove that for any $\epsilon > 0:N(V) \leqslant C_\epsilon (\left \| V \right \|_{n/2 + \epsilon } + \left \| V \right \|_{n/2 - \epsilon })^{n/2}$ where ${\left \| \cdot \right \|_p}$ is an ${L^p}$ norm and that ${\lim \nolimits _{\lambda \to \infty }}N(\lambda V)/{\lambda ^{n/2}} = {(2\pi )^{ - n}}{\tau _n}\smallint |{V_ - }(x){|^{n/2}}{d^n}x$ for any $V \in {L^{n/2 - }} \cap {L^{n/2 + }}$. Here ${V_ - }$ is the negative part of V and ${\tau _n}$ is the volume of the unit ball in ${{\mathbf {R}}^n}$.