## Semiclassical analysis of SchrÃ¶dinger operators with coulomb-like singular potentials

J. Math. Sci. Univ. Tokyo
Vol. 1 (1994), No. 3, Page 589--615.

Nakano, Fumihiko
Semiclassical analysis of SchrÃ¶dinger operators with coulomb-like singular potentials
In this paper, we study the behavior of eigenvalues and eigenfunctions of SchrÃ¶dinger operators whose potentials have finitely many negative singularities. We prove that if potentials behave like $O(|x-p_i|^{-Ï})(0<Ï<2)$ near singular points $x=p_i$, then eigenvalues behave like $O(h^{-\frac{2Ï}{2-Ï}})$ when the Planck constant $h$ approaches to zero. Then we obtain the asymptotic expansion of the eigenvalues and eigenfunctions in $h$. We also study the splitting of the lowest eigenvalues and show that the asymptotic is estimated by a suitable Riemann metric called Agmon distance.