The 2nd order perturbation of the ground state energy is of the following expression
$$ E^{(2)} = \sum_{n\neq 0 } \frac{|\langle n | H_1 | 0\rangle |^2}{E_0-E_n} . $$
Can this series diverge in some problem? I mean a non-relatitivstic problem. In quantum field theory, such divergence seems quite common.
When treating perturbatively the electron-electron electrostatic repulsion on a free electron gas, the first-order correction to the energy is finite but the second-order diverges. To treat the problem, one needs to resum contributions of all orders in the Random Phase Approximation.
Note that $E_n-E_0$ is strictly positive and increasing, so the convergence follows from the convergence of the sums of the numerators.
When there are degenerate energy states, i.e. distinct states of the quantum system that have the same energy. When your Hamiltonian has degenerate states, degenerate state perturbation theory is utilized to derive the first order and second order, etc... equations for the perturbation Hamilonian.
There are several classic examples where this is utilized. For example, shifting and splitting of spectral lines of atoms and molecules due to the presence of external magnetic (Zeeman effect) or electric (Stark effect) field.
If you're interested in reading a textbook about the basics of this, the intro book by David MacIntyre has an exquisitely written approach to learning about perturbation theory in (non-relativistic) quantum mechanics.
