 07128 Barry Simon
 Critical LiebThirring bounds for onedimensional Schrodinger operators and Jacobi matrices with regular ground states
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May 24, 07

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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
\[
\sum_{\substack{E<E_0 \\ E\in\sigma(H)}}\, (E_0E)^{1/2} \leq C\int \abs{V(x)}\, dx
\]
for $H=H_0 +V$. We prove a similar result for Jacobi matrices above or below their spectrum.
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