We will follow these lecture notes. Note that they are currently being written and will be updated continuously during the course.
For a recent and solid textbook on the topic (though lacking some of the techniques introduced in the course which are even more recent), consult:
Elliot H. Lieb and Robert Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge University Press, 2010. [errata etc]
Other useful books:
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Volumes I - IV, 1972-2010 ca
Gerald Teschl, Mathematical Methods in Quantum Mechanics - With Applications to Schrödinger Operators, Second Edition, AMS, 2014
Walter Thirring, Classical Mathematical Physics, Third Edition, Springer, 2003
Walter Thirring, Quantum Mathematical Physics, Second Edition, Springer, 2003