Non-additive quantum mechanics for a position-dependent mass system: Dirac delta and quasi-periodic potentials

Abstract

Motivated by non-extensive statistical mechanics, in this work we consider a deformed Schrödinger equation (DSE) for position-dependent mass (PDM) systems, whose deformed plane-wave solutions allow to characterise a non-periodic lattice. We obtain a deformed version of the Bloch theorem and we illustrate the formalism presented with two examples of the literature: the Dirac and the Kronig-Penney potentials. We found that the Kronig-Penney potential offers a modelling for a lattice with defects expressed by a non-periodicity of the potential within the underlying non-extensive mathematical structure, which is evidenced by the displacement of the gaps with respect to the non-deformed case. The eigenfunctions, the reduced energy bands scheme and the density of states are affected by the deformation.