  This paper is a direct offspring of Ref. [J. Math. Phys. 54, 072103, (2013)]
where basic tenets of the nonlocally induced random and quantum dynamics were
analyzed. A number of mentions was maid with respect to various inconsistencies
and faulty statements omnipresent in the literature devoted to so-called
fractional quantum mechanics spectral problems. Presently, we give a decisive
computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy
well problem, that spectral solutions proposed so far were plainly wrong. As a
constructive input, we provide an explicit spectral solution of the finite
Cauchy well. The infinite well emerges as a limiting case in a sequence of
deepening finite wells. The employed numerical methodology (algorithm based on
the Strang splitting method) has been tested for an exemplary Cauchy oscillator
problem, whose analytic solution is available. An impact of the inherent
spatial nonlocality of motion generators upon computer-assisted outcomes
(potentially defective, in view of various cutoffs), i.e. detailed eigenvalues
and shapes of eigenfunctions, has been analyzed.
