I need to understand self-adjoint extensions in quantum mechanics to solve some problems of scattering and bound states in Aharonov-Bohm potentials. There are some referencies that present the math concepts in a more friendly way for someone with a physics background as myself? And I would like to see more applications on the resolutions of problems for better understanding. Any suggestions?
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$\begingroup$ The book "Mathematical Methods in Quantum Mechanics, With Applications to Schrödinger Operators" by Gerald Teschl is probably a good choice. $\endgroup$– Abdelmalek AbdesselamCommented Aug 20, 2020 at 21:10
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$\begingroup$ Also the book "Sturm-Liouville Theory" by Anton Zettl would have a wealth of material on self-adjoint extensions and their relation to the choice of boundary conditions, on more manageable 1d or ODE-like problems. $\endgroup$– Abdelmalek AbdesselamCommented Aug 20, 2020 at 21:17
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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.