Possible scenarios transgressing the nondegeneracy theorem

Abstract

In contrast to the nondegeneracy theorem, we present various scenarios in one-dimensional quantum mechanics that demonstrate how the Wronskian of two bound-state eigenfunctions with the same energy eigenvalue can be zero without implying that the eigenfunctions are linearly dependent. It is shown that the nondegeneracy theorem fails only when the potential makes different bound-state solutions corresponding to the same energy vanish at the singular point or region of singularity.

Revisiting the bound states of a confined delta potential

“…While twofold degeneracy is generally precluded in describing one-dimensional bound states, as ensured by the nondegeneracy theorem, it becomes a possibility for a strongly singular potentials (see, e.g. [16]). For a system confined by symmetric impenetrable walls around the origin…”