Dynamical supersymmetries of the harmonic oscillator

On Schrödinger superalgebras

“…generalizing what Beckers et al [7] found for N = N + = 2. In most frequent cases, when all epsilon's are equal to 1 and N = N + , we denote our Schrödinger algebras simply by sch(n/N ).…”

“…In the LLM algebra, however, there are N = 2 'Q-type' and 'S-type' supercharges, but just N/2 = 1 'T-type' charge. The commutation relations ot the LLM-algebra are similar to, but still different from, those of [7] for n = 2.…”

“…But, in two space dimensions, the 'ordinary' N = 2 superconformal symmetry of the type of Beckers et al [7] has 18 generators: in particular, there are N = 2 of each of the 'Q-type', 'S-type' and 'T-type' charges. In the LLM algebra, however, there are N = 2 'Q-type' and 'S-type' supercharges, but just N/2 = 1 'T-type' charge.…”

On supersymmetries in nonrelativistic quantum mechanics

“…(3.9) and (3.26). Among them only 13 close and lead to the largest kinematical invariance superalgebra osp(2/2) sh(2/2) [14]. Associated with these comments are changes of variables easily determined from the formulas (2.9)-(2.11) and directly connected with other results [1,15] when, for example, a = ω and b = 0.…”

“…then all the potentials 14) lead to four symmetries generating the direct sum so(2, 1) ⊕ I. Such properties are of special interest for the following discussion of the supersymmetries of Eq.…”