Semiclassical analysis of Schrödinger operators with coulomb-like singular potentials
Vol. 1 (1994), No. 3, Page 589--615.
Nakano, Fumihiko
Semiclassical analysis of Schrödinger operators with coulomb-like singular potentials
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Abstract:
In this paper, we study the behavior of eigenvalues and eigenfunctions of Schrödinger 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.
Mathematics Subject Classification (1991): Primary 35Q40; Secondary 35P20
Mathematical Reviews Number: MR1322693
Received: 1994-01-20
