Morse potential, symmetric Morse potential and bracketed bound-state energies

For the needs of non-perturbative quantum theory, an upgraded concept of solvability is proposed. In a broader methodical context, the innovation involves Schrödinger equations which are piecewise analytic and piecewise solvable in terms of special (in our illustrative example, Whittaker) functions. In a practical implementation of our symbolic-manipulation-based approach, we work with a non-analyticity in the origin. A persuasive advantage is then found in the both-sidedness of our iterative localization of the energies.