Manifestly Supersymmetric Lax Integrable Hierarchies

A systematic method of constructing manifestly supersymmetric $1+1$-dimensional KP Lax hierarchies is presented. Closed expressions for the Lax operators in terms of superfield eigenfunctions are obtained. All hierarchy equations being eigenfunction equations are shown to be automatically invariant under the (extended) supersymmetry. The supersymmetric Lax models existing in the literature are found to be contained (up to a gauge equivalence) in our formalism.Comment: LaTeX, 10 pg

Coordinate Realizations of Deformed Lie Algebras with Three Generators

Differential realizations in coordinate space for deformed Lie algebras with three generators are obtained using bosonic creation and annihilation operators satisfying Heisenberg commutation relations. The unified treatment presented here contains as special cases all previously given coordinate realizations of $so(2,1),so(3)$ and their deformations. Applications to physical problems involving eigenvalue determination in nonrelativistic quantum mechanics are discussed.Comment: 11 pages, 0 figure

New Solvable Singular Potentials

We obtain three new solvable, real, shape invariant potentials starting from the harmonic oscillator, P\"oschl-Teller I and P\"oschl-Teller II potentials on the half-axis and extending their domain to the full line, while taking special care to regularize the inverse square singularity at the origin. The regularization procedure gives rise to a delta-function behavior at the origin. Our new systems possess underlying non-linear potential algebras, which can also be used to determine their spectra analytically.Comment: 19 pages, 4 figure

Zero Curvature Formalism for Supersymmetric Integrable Hierarchies in Superspace

We generalize the Drinfeld-Sokolov formalism of bosonic integrable hierarchies to superspace, in a way which systematically leads to the zero curvature formulation for the supersymmetric integrable systems starting from the Lax equation in superspace. We use the method of symmetric space as well as the non-Abelian gauge technique to obtain the supersymmetric integrable hierarchies of the AKNS type from the zero curvature condition in superspace with the graded algebras, sl(n+1,n), providing the Hermitian symmetric space structure.Comment: LaTeX, 9 pg

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

Supersymmetric Quantum Mechanics and Solvable Models

We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of h-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on h

Exact Solutions of the Schroedinger Equation: Connection between Supersymmetric Quantum Mechanics and Spectrum Generating Algebras

Using supersymmetric quantum mechanics, one can obtain analytic expressions for the eigenvalues and eigenfunctions for all nonrelativistic shape invariant Hamiltonians. These Hamiltonians also possess spectrum generating algebras and are hence solvable by an independent, group theoretical method. In this paper, we demonstrate the equivalence of the two methods of solution, and review related progress in this field

Negaton and Positon solutions of the soliton equation with self-consistent sources

The KdV equation with self-consistent sources (KdVES) is used as a model to illustrate the method. A generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for $N$-times repeated GBDT are presented. This GBDT provides non-auto-B\"{a}cklund transformation between two KdV equations with different degrees of sources and enable us to construct more general solutions with $N$ arbitrary $t$-dependent functions. By taking the special $t$-function, we obtain multisoliton, multipositon, multinegaton, multisoliton-positon, multinegaton-positon and multisoliton-negaton solutions of KdVES. Some properties of these solutions are discussed.Comment: 13 pages, Latex, no figues, to be published in J. Phys. A: Math. Ge

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