$\cN$-FOLD SUPERSYMMETRY IN QUANTUM MECHANICAL MATRIX MODELS

Abstract

We formulate N -fold supersymmetry in quantum mechanical matrix models. As an example, we construct general two-by-two Hermitian matrix 2-fold supersymmetric quantum mechanical systems. We find that there are two inequivalent such systems, both of which are characterized by two arbitrary scalar functions, and one of which does not reduce to the scalar system. The obtained systems are all weakly quasi-solvable.

Two-step shape invariance in the framework of $$\mathcal{N}$$ -fold supersymmetry

“…Although we have restricted our investigation to ordinary scalar Schrödinger operators, the concepts of ordinary, two-step, and multi-step SI are applicable to much wider systems. In fact, N -fold SUSY was successfully formulated also for Schrödinger operators with position-dependent mass [35], matrix ones [41], and ones with reflection operators [42]. Hence, we would be able to make systematic studies on various SI in these systems with the framework of N -fold SUSY, as have been done in this work.…”

Spectral design for matrix Hamiltonians: different methods of constructing of a matrix intertwining operator

“…which follows from (16), (20) and (21). It is easy to see that intertwining (20) and (22) to (18) and, consequently, to (19). The latter is obviously equivalent to independence of U 0 (x) and U (x) from x.…”

“…As well the corresponding polynomial supersymmetry algebra of the second order is constructed and different applications of the obtained results are examined in [21]. The author of [22] proposes to study a supersymmetry generated by two n × n matrix non-Hermitian, in general, Hamiltonians H + and H − and two n × n matrix differential operators Q + N and Q − N of the same order N with constant coefficients proportional to the identity matrix at (d/dx) N that intertwine H + and H − in the opposite directions and such that the products Q + N Q − N and Q − N Q + N are the same polynomials with matrix coefficients of H + and H − respectively. Moreover, the operators Q + N and Q − N are supposed to be related one to another by some unnatural operation which is not, in general, neither transposition nor Hermitian conjugation.…”

Minimal realizations of supersymmetry for matrix Hamiltonians

“…The matrix models with supersymmetry appear in Quantum Mechanics in several areas: in particular, for spectral design of potentials describing multichannel scattering and the motion of spin particles in external fields. The different cases of such models are considered in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and their systematic study was undertaken in [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] (see also the recent reviews [34,35]). In [17] intertwining of matrix Hermitian Hamiltonians by n × n first-order and 2 × 2 second-order matrix differential operators was investigated and the corresponding supersymmetric algebras were constructed.…”