## Abstract

We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function N_{L}(E), the number of bound states of the operator L = Δ+V in ℝ^{d} below -E. Here V is a bounded potential behaving asymptotically like P(ω)r^{-2} where P is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator Δ_{S} ^{d-1} +P on the sphere S^{d-1} has negative eigenvalues -μ1, ⋯ ,-μ_{n} less than -(d-2)^{2}/4, we prove that N_{L}(E) may be estimated as N_{L}(E) =log(E^{-1})/ 2πn∑i=1 √μi - (d - 2)^{2}/4 + O(1). Thus, in particular, if there are no such negative eigenvalues, then L has a finite discrete spectrum. Moreover, under some additional assumptions including the fact that d = 3 and that there is exactly one eigenvalue -μ1 less than -1/4, with all others > -1/4, we show that the negative spectrum is asymptotic to a geometric progression with ratio exp(-2π/ √μ1 - 1/4 ).

Original language | English |
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Pages (from-to) | 4145-4167 |

Number of pages | 23 |

Journal | Transactions of the American Mathematical Society |

Volume | 360 |

Issue number | 8 |

DOIs | |

Publication status | Published - Aug 2008 |