Renormalization theory has sidestepped these divergence difficulties, which may be indicative of the failure of the perturbation expansion.

Unphysical numbers are a sign of a divergent perturbation expansion.

Momentum-space RG is usually performed on a perturbation expansion.

The Hubbard model is in general intractable under usual perturbation expansion techniques.

The natural orbitals of the perturbation expansion are a useful diagnostic for detecting intruder state effects.

Borel summation finds application in perturbation expansions in quantum field theory.

Unlike the usual perturbation expansion, it converges in some non-trivial regions, in particular when the interaction is small.

These correct perturbed eigenstates in D are now the basis for the perturbation expansion:

Krylov-Bogoliubov averaging can be used to approximate oscillatory problems when a classical perturbation expansion fails.

However, perturbation expansions are generally not reliable for questions of long-term existence and stability, in the case of nonlinear equations.