Higher Order Eigenvalues for Non-Local Schrödinger Operators

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 $\nu (dz):=\rho (z)dz$ 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.

21 pages

References in corpus (3)

• Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of Lévy processes
• Schrodinger Operators with Purely Discrete Spectrum
• A geometric interpretation of the transition density of a symmetric Lévy Process