Asymptotic series in field theory and quantum mechanics It is well known that perturbation theory in quantum field theory leads to a series that is (at best) asymptotic. Dyson's famous argument for quantum electrodynamics is a good justification for this.
What about in non relativistic quantum mechanics - are there examples where (time independent, say) perturbation theory leads to a convergent series / a divergent series / an asymptotic series?
As an example, say one were to study the anharmonic oscillator, with some perturbation $\mu x^{4}$. Perturbation theory for the ground state energy would lead to a power series in $\mu$. I don't think Dyson applies here as the sign of $\mu$, for sufficiently small $\mu$ should not lead to such dramatic change in the physics as in negating the electric charge. Are there examples were perturbative series such as this diverge, converge, are asymptotic?
Thanks for any input. 
 A: For certain systems, there indeed are convergent perturbation series. In the case of the anharmonic oscillator described by a Hamiltonian,
$$H = \frac12 p^2 + \frac12 m^2x^2 + \frac14 gx^4$$
one may construct a convergent series which is convergent for any $g>0$ and arbitrary harmonic term, valid in both the weak and strong coupling limits.
Similarly, using a procedure involving splitting the Hamiltonian in a particular way and applying perturbation theory, a system of coupled harmonic oscillators admitted a convergent series.
Furthermore, there exists a generalisation of a convergent perturbation series for a $q$-deformed anharmonic oscillator, which is to say a system based on a $q$-deformed Heisenberg algebra, with a Hamiltonian based on operators with modified commutation relations.

In a more general setting, it has been shown that for a class of hyperbolic differential equations, there exists convergent perturbation series for particular families under certain conditions. (As an aside this is used to show the Einstein field equations in a particular scheme yield a divergent perturbation series as opposed to asymptotic.)
A: You should take a look at Francisco Fernandez "Introduction to Perturbation Theory in Quantum Mechanics", in chapter 6 he notes that almost all perturbation series are divergent, including fan-favorites Stark and Zeeman effect, so that having a convergent series is an exception. This has nothing to do with the infinite degrees of freedom of QFT, is just a fact of QM. 
As an addendum, let me just note that although almost all perturbation series are indeed divergent the problem only arises (or arised) in QFT. That's because in finite degrees of freedom QM we can define rigorously the hamiltonian, for example the Zeeman effect one, we just can't compute exactly the eigenvalues.
In the early days of QFT people only knew how to define free fields, there was no satisfactory definition of a interacting QFT. For some time there was some hope of defining QFT to BE the results of perturbation series. That's why Dyson's argument was significant, it shows that one cannot define interacting QFT via perturbation expansion.
(The last couple of paragraphs we're my extrapolation of why you we're curious about convergence in QM, and so I thought appropriate to include. I'll be happy to delete them if people find it not pertaining to the questions as such)
