Constantin Rasinariu

New solvable singular potentials

We obtain three new solvable, real, shape-invariant potentials starting from the harmonic oscillator, Poschl-Teller I and Poschl-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 behaviour at the origin. Our new systems possess underlying nonlinear potential algebras, which can also be used to determine their spectra analytically.

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 ~-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on ~.

Exactly solvable systems and the quantum Hamilton¿Jacobi formalism

Abstract We connect quantum Hamilton–Jacobi theory with supersymmetric quantum mechanics (SUSYQM). We show that the shape invariance, which is an integrability condition of SUSYQM, translates into fractional linear relations among the quantum momentum functions.

Exactly solvable problems of quantum mechanics and their spectrum generating algebras: A review

In this review, we summarize the progress that has been made in connecting supersymmetry and spectrum generating algebras through the property of shape invariance. This monograph is designed to be used by our fellow researchers, by other interested physicists, and by students at the graduate and even undergraduate levels who would like a brief introduction to the field.

Baxter T-Q equation for shape invariant potentials. The finite-gap potentials case

The Darboux transformation applied recurrently on a Schrodinger operator generates what is called a 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.

Generation of a novel exactly solvable potential

Abstract 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 Plancks constant ħ is complete. Additionally, a set of “extended” superpotentials has been identified, each containing a conventional superpotential as a kernel and additional ħ -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.

Shape invariance in phase space

Shape invariance is a powerful solvability condition, that allows for complete knowledge of the energy spectrum, and eigenfunctions of a system. After a short introduction into the deformation quantization formalism, this paper explores the implications of the supersymmetric quantum mechanics and shape invariance techniques to the phase space formalism. We show that shape invariance induces a new set of relations between the Wigner functions of the system, that allows for their direct calculation, once we know one of them. The simple harmonic oscillator and the Morse potential are solved as examples.

The supersymmetric WKB formalism is not exact for all additive shape invariant potentials

Following the verification of the conjecture made by Comtet, Bandrauk and Campbell that the supersymmetry-inspired semiclassical method known as SWKB is exact for the conventional additive shape invariant potentials, it was widely believed that SWKB yields exact results for all additive shape invariant potentials. In this paper we present a concrete example of an additive shape invariant potential for which the SWKB method fails to produce exact results.

The motion of two identical masses connected by an ideal string symmetrically placed over a corner

We introduce a novel, two-mass system that slides up an inclined plane while its center of mass moves down. The system consists of two identical masses connected by an ideal string symmetrically placed over a corner-shaped support. This system is similar to a double-cone that rolls up an inclined set of V-shaped rails. We find the double-cones motion easy to demonstrate but difficult to analyze. Our example here is more straightforward to follow, and the experimental observations are in good agreement with the theoretical predictions.