Quantum mechanics with spatial non-local effects and position-dependent mass

Abstract

A new higher order Schrödinger equation characterized by a position-dependent mass is introduced based on long-range spatial kernel effects and von Roos arguments. The extended Schrödinger equation depends on the sign of the moments Mk, k = 0, 1, 2, ⋯ and a stabilized quantum dynamic is realized for M2 > 0 and M4 > 0. We have discussed its implications in several quantum mechanical systems where more than a few were raised, mainly the emergence of the quantum Pais-Uhlenbeck and the relativistic quantum harmonic oscillators besides the complex periodic potential characterized by a PT symmetry.