Algebraic Shape Invariant Models

Motivated by the shape invariance condition in supersymmetric quantum mechanics, we develop an algebraic framework for shape invariant Hamiltonians with a general change of parameters. This approach involves nonlinear generalizations of Lie algebras. Our work extends previous results showing the equivalence of shape invariant potentials involving translational change of parameters with standard $SO(2,1)$ potential algebra for Natanzon type potentials.Comment: 8 pages, 2 figure

Interpreting data from polarized proton beams

The reactions pp --> pp and pp --> [Delta]++n with polarized beam and/or polarized target are currently under investigation at the Argonne ZGS. We discuss how to interpret various measured quantities in terms of amplitudes whose behavior is familiar (as functions of s, t). For pp total cross sections and elastic scattering, Argonne measurements will yield Im [phiv]2 (s,t = 0) and the rather complicated combination 2|[phiv]5|2 + Re ([phiv]1[phiv]2* - [phiv]3[phiv]4*), where [phiv]i (i = 1, ... 5) are conventional s-channel helicity amplitudes. The forward direction (t = 0) is of special interest. We find that for both pp --> pp and pp --> [Delta]++n, polarized beam -- polarized target experiments plus the rather general (testable) assumption that amplitudes with the same s-channel helicity flip quantum numbers are proportional, are sufficient to fully determine all non-vanishing amplitudes at t = 0. Numerical estimates of some observables, based on calculations in a specific model, are also given.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/22295/1/0000736.pd

An improvement of Gribov's reggeon calculus

We derive an expression for Regge cuts and the associated enhancement of Regge poles, following Gribov's derivation of the reggeon calculus, but refraining from making an approximation made by Gribov. We show that Gribov's loop integrand should be multiplied by . This factor is identically unity for the Regge cut discontinuity, but is different from unity for enhanced singularities.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/22146/1/0000575.pd

Unitarity bounds on diffraction dissociation

Using s-channel unitarity and the standard picture that diffraction dissociation and elastic scattering are the shadow of non-difractive particle production, we derive rigorous upper bounds for the diffraction dissociation cross section. The bounds are valid at each impact parameter, and are derived for an arbitrary number N of difractive channels. Our results are a generalization of previously derived bounds for the special simple case of N = 2 channels.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/21751/1/0000145.pd

Baxter T-Q Equation for Shape Invariant Potentials. The Finite-Gap Potentials Case

The Darboux transformation applied recurrently on a Schroedinger operator generates what is called a {\em dressing chain}, or from a different point of view, a set of supersymmetric shape invariant potentials. The finite-gap potential theory is a special case of the chain. For the finite-gap case, the equations of the chain can be expressed as a time evolution of a Hamiltonian system. We apply Sklyanin's method of separation of variables to the chain. We show that the classical equation of the separation of variables is the Baxter T-Q relation after quantization.Comment: 25 pages, no figures Extended section 10, one reference added. Version accepted for publication in Jurnal of Mathematical Physic

New Shape Invariant Potentials in Supersymmetric Quantum Mechanics

Quantum mechanical potentials satisfying the property of shape invariance are well known to be algebraically solvable. Using a scaling ansatz for the change of parameters, we obtain a large class of new shape invariant potentials which are reflectionless and possess an infinite number of bound states. They can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for energy eigenvalues, eigenfunctions and transmission coefficients are given. Included in our potentials as a special case is the self-similar potential recently discussed by Shabat and Spiridonov.Comment: 8pages, Te

Obtaining real parts of scattering amplitudes directly from cross section data using derivative analyticity relations

We show that one can obtain real parts of scattering amplitudes by knowing the imaginary parts at only nearby energies. This is accomplished by re-casting the dispersion integral into an equivalent form which we will calla "derivative analyticity relation". Predictions are given for forward amplitudes where [sigma]T is measured: pp, . We deduce the real part of the elastic pp amplitude away from the forward direction at ISR energies.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/22369/1/0000816.pd

Slowly Rotating Homogeneous Stars and the Heun Equation

The scheme developed by Hartle for describing slowly rotating bodies in 1967 was applied to the simple model of constant density by Chandrasekhar and Miller in 1974. The pivotal equation one has to solve turns out to be one of Heun's equations. After a brief discussion of this equation and the chances of finding a closed form solution, a quickly converging series solution of it is presented. A comparison with numerical solutions of the full Einstein equations allows one to truncate the series at an order appropriate to the slow rotation approximation. The truncated solution is then used to provide explicit expressions for the metric.Comment: 16 pages, uses document class iopart, v2: minor correction

Exactly solvable models of supersymmetric quantum mechanics and connection to spectrum generating algebra

For nonrelativistic Hamiltonians which are shape invariant, analytic expressions for the eigenvalues and eigenvectors can be derived using the well known method of supersymmetric quantum mechanics. Most of these Hamiltonians also possess spectrum generating algebras and are hence solvable by an independent group theoretic method. In this paper, we demonstrate the equivalence of the two methods of solution by developing an algebraic framework for shape invariant Hamiltonians with a general change of parameters, which involves nonlinear extensions of Lie algebras.Comment: 12 pages, 2 figure