From the physical point of view, an undergraduate course in Quantum Mechanics may be helpful to figure out the general framework and the physical meaning of the course, although it is not strictly required, since such points are also mentioned along the way. Mathematically, the course requires only very basic notions and tools of functional analysis (e.g., Hilbert spaces; linear operators; Fourier transform), which may also depend on the audience, since most of the needed mathematics is discussed in detail.
Introduction. Basics of Quantum Mechanics; operator theory: self-adjointness, ground and excited states; many-body systems: non-interacting bosons and fermions; anyons.
Stability of the first kind. Boundedness from below of the enegy; existence and uniqueness of the ground state; Schroedinger equation; excited states.
Stability of the second kind. Lieb-Thierring inequalities; electrostatics: potential theory and Baxter inequality; Thomas-Fermi theory; proof of stability of the second kind; bosonic instability; Bose-Einstein condensation.
The main goal of the course is to provide a simple and impressive application of modern mathematical physics methods to relevant questions in the physics of realistic quantum systems. More precisely, the aim is to give a complete and self-contained proof of the stability of matter, i.e., the boundedness from below of the energy of atoms (stability of the first kind) and its extensive behavior (stability of the second kind). Along the way, several interesting mathematical tools are discussed and developed in order to tackle the main questions above.
The Stability of Matter in Quantum Mechanics, E.H. Lieb, R. Seiringer, Cambridge University Press, 2010.
Analysis, E.H. Lieb, M. Loss, Second Edition, AMS, 2001.