Modified Ricatti approach to partially solvable quantum Hamiltonians: Finite Laurent-type potentials

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Abstract
Partial solubility in quantum mechanics is investigated by studying the logarithmic derivative of the wave function. By explicitly isolating the singularities of the logarithmic derivative, a modified Ricatti equation for the regular component is obtained. For the finite Laurent series potentials considered, we derive the constraints on the coupling constants to obtain closed-form solutions for a subset of eigenstates. With an appropriate change of variables the method is generalized in order to cope with more general potentials. In this way, new families of partially solvable potentials related to the already known exactly solvable ones are identified.