Superintegrable systems with position dependent mass

Abstract

First order integrals of motion for Schr\"odinger equations with position dependent masses are classified. Seventeen classes of such equations with non-equivalent symmetries are specified. They include integrable, superintegrable and maximally superintegrable systems. Among them is a system invariant with respect to the Lie algebra of Lorentz group and a system whose integrals of motion form algebra so(4). Three of the obtained systems are solved exactly.Comment: Two unnecessary items are deleted from Table

Semiconductor Quantum Wells with BenDaniel-Duke Boundary Conditions and Janus Nanorods

“…1 in [1]). Recent papers give explicit methods to obtain explicit solutions of the Schrödinger equation with PDM, for various forms of this dependence and for several classes of potentials [2][3][4].…”

The Maximal "Kinematical" Invariance Group for an Arbitrary Potential Revised

“…Analogously, considering non-equivalent three dimensional subalgebras D, P 3 , L 3 , D, P 1 , P 2 , L 1 , L 2 , L 3 , L 3 , P 1 , P 2 (A4) P 1 , P 2 , P 3 , L 3 + P 3 , P 1 , P 2 , D + µL 3 , P 1 , P 2 , µ > 0 (A5) we obtain potentials presented in Items 11-14 and recover once more the cases given in Items 7,8,10. Notice that now we have also a simple algebra so(3) realized by L 1 , L 2 and L 3 .…”

“…Recently we extend the Boyer classification to the case of QM systems with position dependent mass [10], [11], [12]. It was a good opportunity to revise the classical paper [3] bearing in mind that some results of this paper appear as particular cases of our analysis.…”

Superintegrable relativistic systems in scalar background fields

“…[14,15,16,17]. This gives an explanation for why the symmetries of the non-relativistic system (18) are given in terms of 3-d conformal Killing vectors [15]; the non-relativistic system is the limit of our relativistic system, the dynamics of which is equivalent to that in a nontrivial metric (3), the isometry group of which is the 4-d conformal Killing group.…”

“…1. Before moving on, we consider again the non-relativistic limit, in which the example above should reduce to case 10 in table 2 of [15]. In that table three integrals of motion are identified, belonging to the non-relativistic limit of the conformal group (10), and corresponding in our notation to p 1 , p 2 and L z = xp 2 − yp 1 .…”