Abstract
New classes of exact solutions of the Schrödinger equation with simple explicit analytic expressions for potential fields, energy levels, and wave functions of stationary states are considered. The solutions are discovered with the help of new original methods elaborated in the quantum theory of spin systems. The corresponding effective potentials are compared to similar models of soliton origin. The main attention is paid to peculiar phenomena such as quasi-exact solvability, potentials with multiple and flexible profiles, fourth-order extrema, finite-band spectra and structural transformations in energy bands, and the spin–soliton analogy.
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