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

TL;DR

This paper derives two-sided estimates for higher order eigenvalues of non-local Schr"odinger operators, linking eigenvalues to jump rates and potential growth, with applications to Lévy processes.

Contribution

It provides new bounds for eigenvalues of non-local Schr"odinger operators considering variable jump measures and potential growth.

Findings

Eigenvalues are bounded between explicit power functions of n.

Bounds depend on jump measure decay rates and potential growth.

Improved bounds are obtained when the jump measure parameter varies.

Abstract

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.