Higher order eigenvalues for non-local Schrödinger operators

Niels Jacob; Feng-Yu Wang

doi:10.3934/cpaa.2018012

Article Contents

2018, Volume 17, Issue 1: 191-208. Doi: 10.3934/cpaa.2018012

This issue Previous Article Time decay in dual-phase-lag thermoelasticity: Critical case Next Article Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary

Higher order eigenvalues for non-local Schrödinger operators

Niels Jacob^1, and
Feng-Yu Wang^2,3, ,

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
* Corresponding author

Received: March 31, 2017

Revised: June 30, 2017

Published: December 2017

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

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Abstract

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.

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Mathematics Subject Classification: Primary:35P15, 47G30;Secondary:60G52.

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