Advanced Search

We consider the operator $-d^2/d{r}^2-V$ in $L_2\left({ℝ}_{+}\right)$ with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound $\mathrm{tr}{(-d^2/d{r}^2-V)}_{-}^{\gamma }\le C_{\gamma ,\alpha }{\int }_{{ℝ}_{+}}{(V\left(r\right)-1/\left(4r^2\right))}_{+}^{\gamma +(1+\alpha )/2}{r}^{\alpha }dr$ for any $\alpha \in [0,1)$ and $\gamma \ge (1-\alpha )/2$. This includes a Lieb-Thirring inequality in the critical endpoint case.