I learned this subject by reading "Principles of Advanced Mathematical Physics, Vol. 1" by Robert D. Richtmyer. This is a great book for filling in the gaps in math knowlege left by the traditional undergraduate math-for-physics course. Richtmeyer was a down-to-earth computational physicist and there is nothing in this book that is there just for mathematics sake. Everything has applications to real-world physics settings, but ones where more sophistication (eg self-adjoint extensions) is needed.

For a less sophisticated and simpler account of why extensions are sometimes required, and how you figure out what they are, you can try chapter 4 in "Mathematics for Physics: A guided tour for graduate students" by myself and Paul Goldbart (our book is publsihed by CUP, but a freely-available draft version is available here.) but I still recommend Richtmeyer.