The confluent second-order supersymmetric quantum mechanics, with factorization energies ǫ 1 , ǫ 2 tending to a single ǫ-value, is studied. We show that the Wronskian formula remains valid if generalized eigenfunctions are taken as seed solutions. The confluent algorithm is used to generate SUSY partners of the Coulomb potential.

The hyperconfluent third-order supersymmetric quantum mechanics, in which all the factorization energies tend to a common value, is analyzed. It will be shown that the final potential as well can be achieved by applying consecutively a confluent second-order and a first-order SUSY transformations, both with the same factorization energy. The technique will be applied to the free particle and the Coulomb potential.

Shallow one-dimensional double well potentials appear in atomic and molecular physics and other fields. Unlike the "deep" wells of macroscopic quantum coherent systems, shallow double wells need not present low-lying two-level systems. We argue that this feature, the absence of a lowlying two-level system in certain shallow double wells, may allow the finding of new test grounds for quantum mechanics in mesoscopic systems. We illustrate the above ideas with a family of shallow double wells obtained from Bargman potentials through the Darboux-Bäcklund transform.

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