Weak type estimates for singular values and the number of bound states of Schrödinger operators


If $A$ if an operator defined by the kernel $f(x-y)g(y)$ with $g\in L^p(\mathbf{R}^n)$ and $f\in \mathrm{Weak}\, L^{p’} (\mathbf{R}^*)$ where $1/p+1/p’ = 1$ and $2\lt p\lt \infty$, then it is shown that $A$ is a compact operator on $L^2(\mathbf{R}^n)$ and that its singular values $\mu_1,\mu_2,\cdots$ satisfy the estimate $\mathrm{sup}_{k\geqq 1} k^{1,p}\mu_k\leqq C_p \Vert f\Vert_{L^{p’},\infty}\Vert g\Vert_{L^{p^\ast}}$. One application of this is the following result: Let $n\geqq 3$ and let $N(V) = \mathrm{dim}$ (spectral projection on $(-\infty,0]$ for $-\Delta+V$) where $V$ is any function in $L^{n/2}(\mathbf{R}^n)$, then $N(V)\leqq \mathrm{const.} \int \vert V_{-}(x)\vert^{n/2}d^nx$, where $V_{-}$ is the negative part of $V$. Furthermore
\mathrm{lim}_{\lambda\to\infty} N(\lambda V)/\lambda^{n/2} = (2\pi)^{-n}\tau_n \int \vert V_{-}(x)^{n/2}d^n x,
where $\tau_n$ is the volume of the unit ball of $\mathrm{R}^n$.


Michael Cwikel