Nonlocally-induced (quasirelativistic) bound states: Harmonic confinement and the finite well

Abstract

Nonlocal Hamiltonian-type operators, like e.g. fractional and quasirelativistic, seem to be instrumental for a conceptual broadening of current quantum paradigms. However physically relevant properties of related quantum systems have not yet received due (and scientifically undisputable) coverage in the literature. In the present paper we address Schr\"{o}dinger-type eigenvalue problems for $H=T+V$, where a kinetic term $T=T_m$ is a quasirelativistic energy operator $T_m = \sqrt{-\hbar ^2c^2 \Delta + m^2c^4} - mc^2$ of mass $m\in (0,\infty)$ particle. A potential $V$ we assume to refer to the harmonic confinement or finite well of an arbitrary depth. We analyze spectral solutions of the pertinent nonlocal quantum systems with a focus on their $m$-dependence. Extremal mass $m$ regimes for eigenvalues and eigenfunctions of $H$ are investigated: (i) $m\ll 1$ spectral affinity ("closeness") with the Cauchy-eigenvalue problem ($T_m \sim T_0=\hbar c |\nabla |$) and (ii) $m \gg 1$ spectral affinity with the nonrelativistic eigenvalue problem ($T_m \sim -\hbar ^2 \Delta /2m$). To this end we generalize to nonlocal operators an efficient computer-assisted method to solve Schr\"{o}dinger eigenvalue problems, widely used in quantum physics and quantum chemistry. A resultant spectrum-generating algorithm allows to carry out all computations directly in the configuration space of the nonlocal quantum system. This allows for a proper assessment of the spatial nonlocality impact on simulation outcomes. Although the nonlocality of $H$ might seem to stay in conflict with various numerics-enforced cutoffs, this potentially serious obstacle is kept under control and effectively tamed.Comment: 23 pages, 16 figure