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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 wereanalyzed. A number of mentions was maid with respect to various inconsistenciesand faulty statements omnipresent in the literature devoted to so-calledfractional quantum mechanics spectral problems. Presently, we give a decisivecomputer-assisted proof, for an exemplary finite and ultimately infinite Cauchywell problem, that spectral solutions proposed so far were plainly wrong. As aconstructive input, we provide an explicit spectral solution of the finiteCauchy well. The infinite well emerges as a limiting case in a sequence ofdeepening finite wells. The employed numerical methodology (algorithm based onthe Strang splitting method) has been tested for an exemplary Cauchy oscillatorproblem, whose analytic solution is available. An impact of the inherentspatial nonlocality of motion generators upon computer-assisted outcomes(potentially defective, in view of various cutoffs), i.e. detailed eigenvaluesand shapes of eigenfunctions, has been analyzed.

Solving fractional Schroedinger-type spectral problems: Cauchy oscillator and Cauchy well