The finite difference algorithm for higher order supersymmetry

Abstract

The higher order supersymmetric partners of the Schroedinger's Hamiltonians can be explicitly constructed by iterating a simple finite difference equation corresponding to the Baecklund transformation. The method can completely replace the Crum determinants. Its limiting, differential case offers some new operational advantages.Comment: LaTeX, 12 pages, 3 figures. To appear in Phys. Lett.

Group Theoretical Approach to the Intertwined Hamiltonians

“…This idea has been kept also in subsequent works [5,19], and in all of these articles the algorithm has been shown to be of use for obtaining new exactly solvable Hamiltonians. Moreover, the proof of Theorem 4.1 given recently by Mielnik, Nieto and Rosas-Ortiz, alternative to that which has been given here, still relies on the idea of iteration of the intertwining technique, see [28,Sec. 2] for details.…”

“…Theorem 4.1 (Finite difference Bäcklund algorithm [4,28,34,35]). Let w k (x), w l (x) be two solutions of the Riccati equations…”

“…In Section 4 we recall the action of a group on the set of Riccati equations described in [27]. Using this technique we will prove the finite difference theorem [4] in a way alternative to [28]. In Section 5 we will determine the elements of the transformation group preserving the subset of Riccati equations arising from the set of Schrödinger equations after applying the reduction process outlined in Section 3.…”

Supersymmetric quantum mechanics and Painlevé equations

“…2 (x) we have An important result that will be proven next is that the solution of the Riccati equation (2.11b) for α 2 can be algebraically determined using the solutions of the initial Riccati equation (2.5b) for the factorization energies 1 and 2 [16][17][18][19][20]. To do that, first let us take the two solutions of the initial Riccati equation…”

“…These equations immediately lead to the higher-order SUSY QM [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. In this treatment, the standard SUSY algebra with two generators…”