  An upgraded concept of solvability of Schr\"{o}dinger-type equations is
proposed. In a broader methodical context of non-perturbative quantum theory
the innovation involves potentials which are piece-wise analytic yielding
differential equations solvable in terms of special functions. In our
illustrative example, Whittaker functions are employed and a single point of
non-analyticity is admitted in the origin. In a symbolic-manipulation-based
practical implementation of the method a serendipitous advantage of the
construction of bound states is found in the both-sided nature of the numerical
localization of their energies.
