I need to study the eigenvalues and the eigenfunctions of a perturbated operator knowing the eigenvalues and the eigenfunctions of the initial operator, using the theory of perturbation and writing formulas as asymptotic formulas.
The initial operator is self-adjoint, has compact resolvent and admits simple eigenvalue $\lambda_1\leq \lambda_2\leq \ldots ...$ where $\operatorname{lim}_{n\to+\infty} \lambda_n =+\infty.$
I'm totally new to the theory of perturbation. Could you please give me a reference that explain how should I proceed or eventually if someone could summarize what we do in such case.