In this paper we extend our previous results on the bosonic Liouville theory, to the supersymmetric case. As in the bosonic case, we find that the quantization of the N=1 theory is limited to the region D≤1. We compute the exact critical exponents and the analogue of the Hausdorff dimension of super random surfaces. Our procedure is manifestly covariant and our results hold for the surface of arbitrary topology. We also examine the N=2, O(2) string theory and find that it appears to be well-defined for all D.

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In this paper we compute exactly, using the scaling properties of the Liouville theory, the Hausdorff dimension of the continuous random surfaces of Polyakov for D≤1. We find that for D<1, the mean square size of the surface grows as a logarithm of the area of the surface as well as the area of the surface raised to a power, the power being minus the string susceptibility. For D=1 the behavior changes, as expected, because the model undergoes a phase transition. In that case we find that the mean square size of the surface behaves as a combination of terms that grow as a logarithm of the area as well as its square, in qualitative agreement with the results of numerical experiments in discrete models.

A three-body scattering process in the presence of Coulomb interaction can be decomposed formally into a two-body single channel, a two-body multichannel and a genuine three-body scattering. The corresponding integral equations are coupled Lippmann-Schwinger and Faddeev-Merkuriev integral equations. We solve them by applying the Coulomb-Sturmian separable expansion method. We present elastic scattering and reaction cross sections of the e + + H system both below and above the H(n = 2) threshold. We found excellent agreements with previous calculations in most cases. PACS number(s): 34.10.+x, 34.85.+x, 21.45.+v, 03.65.Nk, 02.30.Rz, 02.60.Nm The three-body Coulomb scattering problem is one of the most challenging long-standing problems of nonrelativistic quantum mechanics. The source of the difficulties is related to the long-range character of the Coulomb potential. In the standard scattering theory it is supposed that the particles move freely asymptotically. That is not the case if Coulombic interactions are involved. As a result the fundamental equations of the three-body problems, the Faddeev-equations, become illbehaved if they are applied for Coulomb potentials in a straightforward manner.The first, and formally exact, approach was proposed by Noble [1]. His formulation was designed for solving the nuclear three-body Coulomb problem, where all Coulomb interactions are repulsive. The interactions were split into short-range and long-range Coulomb-like parts and the long-range parts were formally included in the "free" Green's operator. Therefore the corresponding Faddeev-Noble equations become mathematically wellbehaved and in the absence of Coulomb interaction they fall back to the standard equations. However, the associated Green's operator is not known. This formalism, as presented at that time, was not suitable for practical calculations.In Noble's approach the separation of the Coulomb-like potential into short-range and long-range parts were carried out in the two-body configuration space. Merkuriev extended the idea of Noble by performing the splitting in the three-body configuration space. This was a crucial development since it made possible to treat attractive Coulomb interactions on an equal footing as repulsive ones. This theory has been developed using integral equations with connected (compact) kernels and transformed into configuration-space differential equations with asymptotic boundary conditions [2]. In practical calculations, so far only the latter version of the theory has been considered. The primary reason is that the more complicated structure of the Green's operators in the kernels of the Faddeev-Merkuriev integral equations has not yet allowed any direct solution. However, use of integral equations is a very appealing approach since no boundary conditions are required.Recently, one of us has developed a novel method for treating the three-body problem with repulsive Coulomb interactions in three-potential picture [3]. In this approach a three-body Coulomb scattering process can ...

We describe techniques we have developed to analyze the physics of topological solitons in a model-independent way. Our central result is to show that topological solitons in generic field theories exhibit Bogomol’nyi bounds and Bogomol’nyi equations. Our methods turn the derivation of these Bogomol’nyi relationships into algebraic calculations and do not depend on the particular equations of motion. We present a discussion of the O(3) nonlinear σ-model as an example of our techniques.