# Is the variational method valid when discrete spectrum eigenfunctions are in finite number?


I encountered, however, in some exercises some potentials that do not even have an infinite number of eigenstates, let alone a complete set, e.g. finite rectangular wells in one dimension, or the potential $$V(x)=-\frac{\hbar^2}{ma^2\cosh^2(x/a)}$$ which has just one bound state.

In these cases and similar ones the assumptions for the variational method don't hold, so is that theorem always valid (but the proof is different) or do I have to check that indeed a complete set of eigenfunctions exists associated to the discrete spectrum?

• The precise range of applicability is given by the min-max theorem. Essentially, it holds for eigenvalues of self-adjoint operators below the essential spectrum, or to bound the infimum of the essential spectrum. – yuggib Jul 2 '16 at 12:00
