# American Institute of Mathematical Sciences

January  2018, 17(1): 191-208. doi: 10.3934/cpaa.2018012

## Higher order eigenvalues for non-local Schrödinger operators

 1 Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom 2 Center for Applied Mathematics, Tianjin University, Tianjin 300072, China 3 Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom

* Corresponding author

Received  April 2017 Revised  July 2017 Published  September 2017

Fund Project: The second named author is supported by Supported in part by NNSFC (11431014,11626245,11626250)

Two-sided estimates for higher order eigenvalues are presented for a class of non-local Schrödinger operators by using the jump rate and the growth of the potential. For instance, let
 $L$
be the generator of a Lévy process with Lévy measure
 $ν(\text{d} z):= \rho(z)\text{d} z$
such that
 $\rho(z)=\rho(-z)$
and
 $c_1 |z|^{-(d+\alpha_1)}≤ \rho(z)≤ c_2|z|^{-(d+\alpha_2)},\ \ |z|≤ \kappa$
for some constants
 $\kappa, c_1,c_2>0$
and
 $\alpha_1,\alpha_2∈ (0,2),$
and let
 $c_3|x|^{θ_1} ≤ V(x)≤ c_4|x|^{θ_2}$
for some constants
 $θ_1,θ_2, c_3,c_4>0$
and large
 $|x|$
. Then the eigenvalues
 $\lambda_1≤ \lambda_2≤··· \lambda_n≤ ···$
of
 $-L+V$
satisfies the following two-side estimate: there exists a constant
 $C>1$
such that
 $C n^{\frac{θ_2\alpha_2}{d(θ_2+\alpha_2)}}≥ \lambda_n ≥ C^{-1} n^{\frac{θ_1\alpha_1}{d(θ_1+\alpha_1)}},\ \ n≥ 1.$
When
 $\alpha_1$
is variable, a better lower bound estimate is derived.
Citation: Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schrödinger operators. Communications on Pure & Applied Analysis, 2018, 17 (1) : 191-208. doi: 10.3934/cpaa.2018012
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