Generation of a Novel Exactly Solvable Potential

We report a new shape invariant (SI) isospectral extension of the Morse potential. Previous investigations have shown that the list of "conventional" SI superpotentials that do not depend explicitly on Planck's constant $\hbar$ is complete. Additionally, a set of "extended" superpotentials has been identified, each containing a conventional superpotential as a kernel and additional $\hbar$-dependent terms. We use the partial differential equations satisfied by all SI superpotentials to find a SI extension of Morse with novel properties. It has the same eigenenergies as Morse but different asymptotic limits, and does not conform to the standard generating structure for isospectral deformations.Comment: 9 pages, 3 figure

Translational Shape Invariance and the Inherent Potential Algebra

For all quantum-mechanical potentials that are known to be exactly solvable, there are two different, and seemingly independent methods of solution. The first approach is the potential algebra of symmetry groups; the second is supersymmetric quantum mechanics, applied to shape-invariant potentials, which comprise the set of known exactly solvable potentials. Using the underlying algebraic structures of Natanzon potentials, of which the translational shape-invariant potentials are a special subset, we demonstrate the equivalence of the two methods of solution. In addition, we show that, while the algebra for the general Natanzon potential is so(2,2), the subgroup so(2,1) suffices for the shape invariant subset. Finally, we show that the known set of exactly solvable potentials in fact constitutes the full set of such potentials

Tragedy of the common canal

This paper uses laboratory experiments to investigate the effects of alternative solutions to a common-pool resource with a unidirectional flow. The focus is on the comparative economic efficiency of communications, bilateral “Coasian” bargaining, auctions and price-based allocations. All treatments improve allocative efficiency relative to a baseline environment. Communication and bilateral bargaining are not generally as effective as market allocations. An exogenously imposed, optimal fee results in the greatest efficiency gain, followed by auction allocations that determine the usage fee endogenously.externalities, experiments, auctions, Coasian bargaining, common pool resource

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

Shape Invariance in Supersymmetric Quantum Mechanics and its Application to Selected Special Functions of Modern Physics

We applied the methods of supersymmetric quantum mechanics to differential equations that generate well-known special functions of modern physics. This application provides new insight into these functions and generates recursion relations among them. Some of these recursion relations are apparently new (or forgotten), as they are not available in commonly used texts and handbooks. This method can be easily extended to explore other special functions of modern physics