The effects of 6nal state pion-pion interactions on the spectrum of E+ -+ 3m decay is studied by dispersion relation methods. In the approximations adopted we are led to a set of linear integral equations for the amplitudes of the, E -+ 3~decay. The kernels in these equations depend on the pion-pion S-wave scattering amplitudes. An approximate solution for these equations is obtained by iteration and the departures from a purely statistical spectrum for the decay are related to pion-pion S-wave scattering. The latter in turn is assumed to be well represented with a scattering length structure. The E+~3 2' spectrum then is parametrized by two quantities, the 7 = 0 and T=2 pion-pion S-wave scattering lengths, uo and a2. Such experimental results as presently exist indicate that a&ao is positive and that roughly a2ao =0.7, in units of the pion Compton wavelength.

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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.

In this paper we consider non-relativistic quantum mechanics on a space with an additional internal compact dimension, i.e. R 3 ⊗S 1 instead of R 3 . More specifically we study potential scattering for this case and the analyticity properties of the forward scattering amplitude, T nn (K), where K 2 is the total energy and the integer n denotes the internal excitation of the incoming particle. The surprising result is that the analyticity properties which are true in R 3 do not hold in R 3 ⊗ S 1 . For example, T nn (K), is not analytic in K for ImK > 0, for n such that (|n|/R) > µ, where R is the radius of S 1 , and µ −1 is the exponential range of the potential, V (r, φ) = O(e −µr ) for large r. We show by explicit counterexample that T nn (K) for n = 0, can have singularities on the physical energy sheet. We also discuss the motivation for our work, and briefly the lesson it teaches us. 1

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.