Question on perturbation theory

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Hey everyone,

I'm studying quantum mechanics from Griffiths (Introduction to Quantum Mechanics, 2nd edition), and I'm puzzling over his derivation of the nth order corrections to the energies and corresponding eigenstates for a perturbed Hamiltonian. The steps that are outlined in Griffiths are all very clear, but the derivation seems to dodge some of the deeper mathematical ideas, which could really give more insight into the physics of perturbation theory.

Firstly, if I am interpreting the text correctly, Griffiths starts off the non-degenerate case by asserting that for each n, there exists fixed ψnj and Enj 0≤j≤∞, such that for any λ in [0,1], (H°+λH')ψn,λ = En,λ, where
ψn,λ = Ʃ λjψnj, and
En,λ = Ʃ λjEnj

Although I can intuitively see how this could be true, I am not sure how to make the justification precise.
Also, I noticed that the technique for moving eigenstates from the non-perturbed Hamiltonian to the full Hamiltonian looks a lot like the continuity argument when proving Rouche's theorem (when you "perturb" a holomorphic function f by some holomorphic function g [with norm(g) less than norm(f) on a circle whose interior is contained in the open set on which f is holomorphic], the number of zeros of f inside the circle is equal to the number of zeros of f+g inside the same circle). Is the justification for perturbation theory related to this?

Thanks!
 

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