HIGHE RORDER EIGENVALUES FOR NON-LOCAL SCHRODINGER OPERATORS

Two-sided estimates for higher order eigenvalues are presented for a class of non-local Schrodinger operators by using the jump rate and the growth of the potential. For instance, let L be the generator of a Levy process with Levy measure v(dz) := rho(-z) dz such that rho(z) = rho(-z) and c(1) vertical bar z vertical bar(-(d+alpha 1)) <= rho(z) <= c(2) vertical bar z vertical bar(-(d+alpha 2)), vertical bar z vertical bar <= kappa for some constants kappa, c(1), c(2) > 0 and alpha(1), alpha(2) is an element of(0, 2); and let c(3) vertical bar x vertical bar(theta 1) <= V (x) <= c(1) vertical bar x vertical bar(theta 2) for some constants theta(1), theta(2), c(3), c(1) > 0 and large vertical bar x vertical bar. Then the eigenvalues lambda(1) <= lambda(2) center dot center dot center dot lambda(n) <= center dot center dot center dot of -L + V satisfies the following two-side estimate: there exists a constant C > 1 such that Cn theta(2)alpha(2/)/d(theta(2)+alpha(2)) >= lambda(n) >= C(-1)n theta(1)alpha(1)/d(theta(1)+alpha(1)), n >= 1. When alpha(1) is variable, a better lower bound estimate is derived.