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Higher order eigenvalues for non-local Schrödinger operators

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

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


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