Università Roma Tre

1) The crisis of Classical Physics and the postulates of Quantum Mechanics.
2) Elements of the theory of distributions: Fourier transform; L^p spaces; Sobolev spaces.
3) Elements of operator theory: bounded and unbounded operators; adjoint, symmetric, self-adjoint and unitary operators; orthogonal projectors; self-adjointness criteria and the Kato-Rellich theorem; resolvent and spectrum of an operator; the spectral theorem for self-adjoint operators; point spectrum, continuous spectrum, absolutely continuous and singular continuous spectrum; discrete and essential spectrum; proper and generalized eigenfunctions; variational characterization of the spectrum; compact, trace class and Hilbert-Schmidt operators; the Weyl theorem on the stability of the essential spectrum; the Stone theorem.
4) Mathematical formulation of Quantum Mechanics: fundamental axioms; elementary observables; compatibility criterion and the Heisenberg uncertainty principle; time evolution; constants of motion; bound states and scattering states; density matrix and mixed states.
5) One particle models with exact solution: free particle, harmonic oscillator; hydrogen atom; point interaction.
6) Advanced topics (depending on the preferences of the students): scattering theory; stabilty of matter; classical limit.

[1] A. Teta, A Mathematical Primer on Quantum Mechanics, Springer (2018).
[2] M. Correggi, Aspetti Matematici della Meccanica Quantistica, lecture notes available at [https://sites.google.com/view/michele-correggi/teaching].
[3] Lecture notes.