No abstract

No abstract

No abstract

In this paper we study the number of bound states for potentials in one and two spatial dimensions. We first show that in addition to the well-known fact that an arbitrarily weak attractive potential has a bound state, it is easy to construct examples where weak potentials have an infinite number of bound states. These examples have potentials which decrease at infinity faster than expected. Using somewhat stronger conditions, we derive explicit bounds on the number of bound states in one dimension, using known results for the three-dimensional zero angular momentum. A change of variables which allows us to go from the one-dimensional case to that of two dimensions results in a bound for the zero angular momentum case. Finally, we obtain a bound on the total number of bound states in two dimensions, first for the radial case and then, under stronger conditions, for the non-central case.

For any relativistic quantum field theory in 2ϩ1 dimensions, with no zero mass particles, and satisfying the standard axioms, we establish a remarkable low-energy theorem. The S-wave phase shift, ␦ 0 (k), k being the c.m. momentum, vanishes as eitherThe constant c is universal and c ϭ/2. This result follows only from the rigorously established analyticity and unitarity properties for 2-particle scattering. This kind of universality was first noted in non-relativistic potential scattering, albeit with an incomplete proof which missed, among other things, an exceptional class of potentials where ␦ 0 (k) is O(k 2 ) near kϭ0. We treat the potential scattering case with full generality and rigor, and explicitly define the exceptional class. Finally, we look at perturbation theory in 3 4 and study its relation to our non-perturbative result. The remarkable fact here is that in n-th order the perturbative amplitude diverges like (ln k) n as k→0, while the full amplitude vanishes as (lnk) Ϫ1 . We show how these two facts can be reconciled.

No abstract

LPTHE Orsay 94/08CERN-TH.7158/94 momentum bound states in two dimensions are discussed.condition that rz V(r) has a single extremum. Consequences for zero angularWe also investigate the situation for -% $ Z < 0 and give a bound under the angular momentum.is possible to obtain a bound on the total number of bound states for arbitrary The question is now to know whether this bound is optimal. In Reed and

For pt.I see ibid., vol.5, p.257 (1988). The authors generalize the inverse problem in the coupling constant to the case of two potentials, one of them being known. Identifying this last potential with energy (constant potential) they obtain the solution of the inverse problem at non-zero energy. This generalizes the previous results obtained for the zero energy case. The interval in which they consider the Schrodinger equation may be finite or infinite, and the potential may be singular at the origin. Several soluble examples are given.