Barry Simon
Critical Lieb-Thirring bounds for one-dimensional Schrodinger operators and Jacobi matrices with regular ground states
(226K, pdf)

ABSTRACT.  Let $V_0$ be a potential so that $H_0 =-\f{d^2}{dx^2}+V_0$ has $\inf \sigma (H_0)=E_0$. Suppose there is a function $u$ so that $H_0u=E_0u$ and $0<c_1\leq u(x)\leq c_2$ for constants $c_1,c_2$. Then we prove there is a $C$ so that 
\[ 
\sum_{\substack{E<E_0 \\ E\in\sigma(H)}}\, (E_0-E)^{1/2} \leq C\int \abs{V(x)}\, dx 
\] 
for $H=H_0 +V$. We prove a similar result for Jacobi matrices above or below their spectrum.