  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.
