On the number of bound states of the Schroedinger Hamiltonian--a review

Abstract

We consider a non-relativistic, time independent quantum mechanical system consisting of a finite number of particles interacting via a potential, V. A sufficient condition on V that the system have an infinite number of bound states is that the particles must cluster near the continuum limit into two spatially separated clusters, and the sum of the inter-cluster two-body potentials must decay no faster than the inverse square of the inter-cluster separation. This result is proven following the work of B. Simon and W. Hunziker by showing the system reduces to a variant of the two-body problem. Many bounds for the number of bound states N(V) of the two-body system are reviewed. Most depend on integrals of V. These bounds are used to derive conditions on V so that N(V) =. If we introduce a coupling parameter, s, so that H(s)-A + sV is the two-body Hamiltonian, then we find, following the work of B. Simon [18] that N(sV) grows as s^3/2.