Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
16
The categories of this paper are: stat
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn