Position-Dependent Mass Systems: Classical and Quantum Pictures

Superintegrable quantum mechanical systems with position dependent masses invariant with respect to three parametric Lie groups

“…In the following we will use the rotations and the inverse transformation (29) for optimisation of calculation.…”

“…In other words, the determining equations ( 16) and ( 36) are reduced to the five decoupled subsystems corresponding to the Killing tensors which are n-order homogeneous polynomials with n = 0, 1, 2, 3, 4, and arbitrary functions ϕ 1 , ϕ 2 , ..., ϕ 9 are reduced to constants. Moreover, since Hamiltonians (2) with arbitrary elements (6) are invariant with respect to the inverse transformation (29) we can restrict ourselves to the polynomials of order n < 3, since symmetries with n=3 and n=4 appears to be equivalent to ones with n = 1 and n = 0 correspondingly.…”

“…We also omit the subindices for λ a and λ ab . Notice than any second order symmetry corresponding to n = 0 and n = 1 is accompanied by the addition symmetry generated by the changes of variables (29).…”

“…The same is true for the Schrödinger equation with position dependent mass whose symmetries are studied much less. The latter equation is requested in many branches of modern theoretical physics, whose list can be found, e.g., in [29,30].…”

Superintegrable and Scale-Invariant Quantum Systems with Position-Dependent Mass

“…Moreover, such generalizations include the systems more generic than the standard Schrödinger equations, namely, Schrödinger equations with position dependent mass (PDM). The latter equations are requested in many branches of modern theoretical physics, whose list can be found, e.g., in [19,20,21]. Their symmetries are studied much less than those ones for the standard SE.…”