Effective disappearance of degenerating energy levels

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Abstract
Inverse spectral theory provides a theorem on the shifting of a single level, leaving all others unchanged, in a quantum-mechanical system over a finite range in one dimension. We show that the mechanism can be used to remove a pair of states from the spectrum by shifting the energy of one state onto that of a neighboring state. We extend the procedure to potentials defined on the whole axis possessing a purely discrete or a partly discrete and partly continuous spectrum.