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 $\nu(d z):= \rho(z) d z$ such that $\rho(z)=\rho(-z)$ and $$c_1 |z|^{-(d+\alpha_1)}\le \rho(z)\le c_2|z|^{-(d+\alpha_2)},\ \ |z|\le \kappa$$ for some constants $\kappa, c_1,c_2>0$ and $\alpha_1,\alpha_2\in (0,2),$ and let $c_3|x|^{\theta_1} \le V(x)\le c_4|x|^{\theta_2}$ for some constants $\theta_1,\theta_2, c_3,c_4>0$ and large $|x|$. 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.