The Wigner-Heisenberg Algebra in Quantum Mechanics

Nonlinear supersymmetric quantum mechanics: concepts and realizations

“…(0) (x; E (0) ) ∝ (−∂ + (∂W )(x)) (1) (x; E (0) ). (10) It is clear that one obtains a one-to-one correspondence between solutions of ( 8) and ( 9), but up to zero modes of operators (∓∂ + (∂W )(x)).…”

“…The concept of supersymmetric quantum mechanics (SUSY QM) embodies an algebraic form of transformations of a (complete or partial) spectral equivalence between different dynamical systems and in this sense it gives the algebraic tools for the spectral design of new quantum systems from a given set with controllable energy spectra. At present, there are a number of reviews [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] devoted to development and various applications, mostly of the linear SUSY QM. The very isospectral transformations realizing SUSY represent the Darboux (Darboux-Moutard) transformations [19][20][21][22][23][24][25][26][27][28][29], which, in the theory of ordinary differential equations, have been known about for a long time.…”