Justification in assuming a perturbative expansion Almost every resource regarding perturbation theory initially introduces the concept by assuming that, given a perturbation of the form $\lambda H^1$ with $\lambda$ small,  the energy eigenvalues and eigenkets of the perturbed states can be written as a power series of the unperturbed values. That is, using Shankar's notation (which somewhat annoyingly avoids introducing $\lambda$ altogether):
$$|n \rangle = |n^0 \rangle + |n^1 \rangle + |n^2 \rangle ... \\
   E_n = E^0_n + E^1_n + E^2_n ...$$
I have a couple of questions that have been holding me back:


*

*Most resources will write the above expansions as $|n \rangle = |n^0 \rangle + \lambda|n^1 \rangle + \lambda^2|n^2 \rangle ...$. That is, as a power series of lambda centered at 0. My main confusion is concerning what exactly $\lambda$ is - my current understanding is that $\lambda$ is just a small parameter introduced to indicate that the perturbation is small ($0 < \lambda < 1$). If that is the case, what explicitly is the function $|n \rangle (\lambda)$ that we are expanding? The only thing I can think of is that if $H(\lambda) = H^0 + \lambda H^1 $, then $|n \rangle(\lambda)$ is an eigentket of the same. Then we expand $|n \rangle (\lambda)$ as a power series and say $|n^k \rangle = \frac{1}{k!}\frac{d^k |n\rangle}{d\lambda^n}$ to recover the original power series. Is this correct or am I missing the big picture?

*If my assumption above is correct. Since we do not know $|n \rangle$ a prior (and the point of this whole thing is to find it), we then determine $|n^k \rangle$ using the methods of perturbation theory in order to fill out the above expansion. This gives us a reasonable expression for $|n \rangle$.

*Why is it okay to assume that such an expansion in $\lambda$ converges? Many texts will also choose $\lambda = 1$ at the end of all of this in order to eliminate it. Why is this valid? I though $\lambda$ determined the size of the perturbation.
I think I am just bogged down in details here and it's preventing me from understanding what is going on with these perturbations. If anyone could help me clear some of this confusion up and to explain what's going on here, I would greatly appreciate it. In general, I understand how to find each $|n^k \rangle$ as Shankar does an alright job of explaining it, I think I am just confused as to what exactly it signifies. 
 A: For 1. and 2., your ideas are correct. The $\lambda$ should correspond to a small parameter in the full Hamiltonian so its eigenstates formally depend on $\lambda$ and we can Taylor expand them as functions of $\lambda$. That the $\lambda$ is a "small perturbation" means that in the limit $\lambda\to 0$ one recovers the unperturbed system.
Generically, perturbative expansions do not necessarily converge, but are so-called asymptotic series which may or may not converge. For instance, the QFT perturbation series around vanishing coupling is generally believed to be non-convergent because otherwise it would also converge for some negative value of the coupling, which is physically non-sensical.
Additionally, there are effects that are invisible to perturbation theory - those that are described by functions whose Taylor series vanishes although the functions are smooth and non-zero. In the end, the justification for any given perturbative expansion is mainly that it matches experiment well enough to not bother with the trouble of the often significantly more difficult or even intractable non-perturbative computations.
A: I used to have the same confusion. I think it in two ways. 


*

*lambda is used as an infinitesimal number to denote different levels of Taylor expansion, therefore we reach the final perturbation expansion formula for energy and wavefunction to different orders. 
In practice, the infinitesimal variable is included in the Hamilitonian, like a weak magnetic field and a weak electric field. The E or B is put into the perturbation formula while we take lambda as 1.

*In theory of lattice dynamics, and density functional perturbation theory, lambda is not set to 1, but indeed used as an infinitesimal number in the Tayler expansion. 
For example, in the displacement of atoms with lattice vibration, displacement is set as a very small perturbation value lambda and the second-order derivative term as you wrote in (1) is used to achieve an equation of motion for displacement lambda. The solution of this equation is the phonon perturbation frequency.  
