Asymptotic Behaviors of the Eigenvalues of Schrödinger Operator with Critical Potential

Abstract

We study the asymptotic behaviors of the discrete eigenvalue of Schrödinger operator $P(\lambda )=P_0+\lambda V$ with $P_0=-\mathrm{\Delta }+q\left(\theta \right)/r^2.$ We obtain the leading terms of discrete eigenvalues of $P(\lambda )$ when the eigenvalues tend to 0. In particular, we obtain the asymptotic behaviors of eigenvalues when ${(P_0-\alpha )}^{-1}$ has singularity at $\alpha =0$.