## Mathematics for Quantum Mechanics

### Prof. Gianfausto Dell'Antonio

**Notice:**

- here below you can find a tentative table of contents
- the effective contents will depend on the background of the students
- part B will depend on the interest of the students
- notes will be given
- standard reference: Reed Simon II and IV
- schedule: January-February and May-June, two 2h-lectures per week

**PART A**

- Some elements of phenomenology
- De Broglie - Scrödinger's differential (wave) approach. Scrödinger equation.
- Heisenberg-Born-Jordan's algebraic approach. Matrix algebra.
- States, observables, automorphisms, derivations. Dirac-von Neumann's formulations.
- Elements of C*-algebras and semigroups theory. Conditional expectations. Positivity preserving semigroups.
- Weyl's algebra. Bargman's and Fock's representations. Harmonic oscillator. Coherent states.
- Comparison with Hamiltonian Mechanics. Semiclassical limit (o.d.e. viewpoint).
- Free Scrödinger equation as partial differential propagation equation. Dispersive and smoothing properties.
- Scrödinger equation with bounded potentials. Existence and uniqueness of solutions.
- Elements of operator theory on a Hilbert space: selfadjoint operators, spectral theorem, Stone's theorem, unitary groups, role of the resolvent.
- Quadratic forms and selfadjoint extensions.
- Elements of Scrödinger operators theory. Functional spaces. Sobolev spaces. Rollnik class potentials.
- Perturbation of operators and of quadratic forms. Kato's theorems.
- Examples of perturbations. Hydrogen atom. Anharmonic oscillator.
- Resolvents and Weyl's theorem.
- Periodic potentials. Bloch-Floquet theory. Elements of theory of crystals.
- Wigners's functions. Pseudodifferential operators. Semiclassical limit.
- Contraction semigroups. Feymann-Kac's formula.
- Asymptotic behaviour in time of solutions. Elements of quantum
scattering.

**PART B**

- Extension of the formalism to Quantum Statistical Mechanics and Quantum Field Theory. C*-algebras and von Neumann algebras. GNS construction.
- Conditions for the existence of a unitary evolution. KMS condition.
- Friedrichs extension. Modular operators.