David Fernandez

Supersymmetric quantum mechanics and Painleve equations

In the first lecture we will make a brief overview of supersymmetric quantum mechanics (SUSY QM), as a tool for generating new exactly solvable potentials departing from a given initial one. We will illustrate the technique with the harmonic and radial oscillators. In the second lecture we will address the polynomial Heisenberg algebras (PHA), and the way the SUSY partners of the harmonic and radial oscillators realize the even and odd degree PHA respectively. We will explore as well the most general systems ruled by the second and third degree PHA, and the way these systems connect with the Painleve IV (PIV) and V (PV) equations respectively. Finally, in the third lecture we will discuss and algorithm for generating solutions of the PIV and PV equations departing from the harmonic and radial oscillators respectively.

References

J.M. Carballo, D.J. Fernandez, J. Negro, L.M. Nieto, Polynomial Heisenberg algebras, J. Phys. A: Math. Gen. 37 (2004) 10349-10362
D. Bermudez, D.J. Fernandez, Supersymmetric quantum mechanics and Painleve IV equation, SIGMA 7 (2011) 025
D. Bermudez, D.J. Fernandez, Supersymmetric quantum mechanics and Painleve equations, AIP Conf. Proc. 1575 (2014) 50-88
D. Bermudez, D.J. Fernandez, J. Negro, Solutions to the Painleve V equation through supersymmetric quantum mechanics, J. Phys. A: Math. Theor. 49 (2016) 335203

XXXIX Workshop on Geometric Methods in Physics	19.06–25.06.2022
XI School on Geometry and Physics	27.06–1.07.2022

Event sponsored by:
University of Bialystok