  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 low-lying 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-B\"{a}cklund transform.
