Higher Order Eigenvalues for Non-Local Schr\"odinger Operators

Two-sided estimates for higher order eigenvalues are presented for a class of non-local Schr"odinger operators by using the jump rate and the growth of the potential. For instance, let $L$ be the generator of a L'evy process with L'evy measure $\nu (dz):=\rho (z)dz$ such that $\rho (z)=\rho (-z)$ and $$c_1\mathrm{\mid }z{\mathrm{\mid }}^{-(d+{\alpha }_1)}\le \rho (z)\le c_2\mathrm{\mid }z{\mathrm{\mid }}^{-(d+{\alpha }_2)},\mathrm{\mid }z\mathrm{\mid }\le \kappa$$ for some constants $\kappa ,c_1,c_2>0$ and ${\alpha }_1,{\alpha }_2\in (0,2),$ and let $c_3\mathrm{\mid }x{\mathrm{\mid }}^{{\theta }_1}\le V(x)\le c_4\mathrm{\mid }x{\mathrm{\mid }}^{{\theta }_2}$ for some constants ${\theta }_1,{\theta }_2,c_3,c_4>0$ and large $\mathrm{\mid }x\mathrm{\mid }$. Then the eigenvalues ${\lambda }_1\le {\lambda }_2\le \cdots {\lambda }_n\le \cdots$ of $-L+V$ satisfies the following two-side estimate: for any $p>1$, there exists a constant $C>1$ such that $$C{n}^{\frac{{\theta }_2{\alpha }_2}{d({\theta }_2+{\alpha }_2)}}\ge {\lambda }_n\ge C^{-1}{n}^{\frac{{\theta }_1{\alpha }_1}{d({\theta }_1+{\alpha }_1)}},n\ge 1.$$ When ${\alpha }_1$ is variable, a better lower bound estimate is derived.