This question is about my lets say lack of understanding. I can not make the connection between what i study and what i saw in lectures.
For example in our QM lectures we saw perturbation theory like this $$ H_0 |n_0\rangle=E_0^{(n)}|n_0\rangle $$ where $E_0^{(n)}$ are eigenvalues and $|n_0\rangle$ are eigenvectors of operator $H_0$.(Of course there are some other conditions like non-degeneracy etc but it doesn't matter for my question.) then we perturb it like this $$ (H_0+ \lambda V) |n\rangle=E_n|n\rangle$$ This is just standart perturbation theory in any physics book or (first) quantum mechanics lecture.
But i study quantum mechanics and related mathematical topics in more rigorous manner as much as i can. Actually from these two books "Teschl G. - Mathematical methods in quantum mechanics" and "Konrad Schmüdgen (auth.) - Unbounded Self-adjoint Operators on Hilbert Space".
In these books they talk about Kato-Rellich theorem, Weyl theorem, KLMN theorem, Aronszajn–Donoghue Theory of Rank One Perturbations etc. but i cannot connect my knowledge from these "perturbation theory" to other "perturbation theory". I just cannot connect them i tried to look at some other books (Of course i just skim over them) but cannot find the connection. If someone is able to share or point me to this wonderful knowledge i would be very grateful.
Thanks in advance for your answers