The efimov effect of three-body schrödinger operators: Asymptotics for the number of negative eigenvalues

Abstract

The Efimov effect is one of the most remarkable results in the spectral theory for three-body Schrödinger operators. Roughly speaking, the effect will be explained as follows: If all three two-body subsystems have no negative eigenvalues and if at least two of these two-body subsystems have resonance states at zero energy, then the three-body system under consideration has an infinite number of negative eigenvalues accumulating at zero. This remarkable spectral property was first discovered by Efimov [1] and the problem has been discussed in several physical journals. For related references, see, for example, the book [3]. The mathematically rigorous proof of the result has been given by the works [4, 8, 9]. The aim of the present work is to study the asymptotic distribution of these negative eigenvalues below zero (bottom of essential spectrum). Denote by N(E), E > 0, the number of negative eigenvalues less than – E. Then the main result obtained here is, somewhat loosely stating, that N(E) behaves like | log E | as E → 0. We first formulate precisely the main theorem and then make a brief comment on the recent related result obtained by Sobolev [7].