Is there a book on the purely mathematical version of perturbation theory?

Question

Is there a book on the purely mathematical version of perturbation theory, or all current references just in relation to applied fields like statistics and quantum mechanics? I remember first coming across this theory in my Quantum Mechanics class, and have been trying to late textbooks, video lectures on youtube, and articles detailing the pure mathematics of Perturbation Theory. However, all that came up were applied mathematical examples like solving the Schrodinger equation by making corrections to the bit we know in order to get as close as possible to finding the total Hamiltonian of the system.

Answer 1

Having encountered perturbation theory for the first time in my physics lectures as well (and wondering what the mathematical underpinnings would be), I understand where you're coming from. Fortunately, there are quite a lot of books that treat perturbation theory from a mathematical viewpoint. I would take a look at the resources mentioned in this answer and/or this answer, which I'll quote here for clarity:

I would recommend

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer, 2006, ISBN 978-3-540-60934-6;

starting with chapter 9, several methods of multiple scale analysis are treated, with extensive use of examples. For another slightly more general reference, see

C. Kuehn, Multiple Time Scale Dynamics, Springer, 2015, ISBN 978-3-319-12316-5.

and

A very good, albeit somewhat mathematical, source for these techniques is

C. Kuehn, Multiple Time Scale Dynamics, Springer, 2015.

For a more gentle introduction, you could look at

M.H. Holmes, Introduction to Perturbation Methods, Springer, 1995,

or the golden oldie

J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathematics, Springer, 1985 (2nd ed).