Fluid Mechanics: continuous systems, Euler equation, Navier-Stokes equation, examples and applications.
Quantum Mechanics: mathematical structure, introduction to scattering theory.
Fluid Mechanics
i) Euler’s Equations. Conservation of mass; Balance of Momentum; Conservation of energy; Transport Theorem; Incompressible flows; Isentropic fluids; Bernoulli’s Theorem; Example: Euler flow in a channel; Rotation and Vorticity; Kelvin’s Circulation Theorem; Evolution equation for the vorticity; Stream lines and Stream function; Example: Stream function with radial symmetry. [CM] chapter 1.
ii) Navier-Stokes Equations. Stress tensor; Cauchy’s Theorem; Navier-Stokes Equations in dimensionless variables; Reynolds number; Helmholtz–Hodge Decomposition Theorem; Stokes Equations; Dissipation of kinetic energy; Example: flow in a channel (Poiseuille flow). [CM] chapter 1.
iii) Variational principle for Euler Equations ([MP], page 5); Examples: Couette flow, gravity waves [N].
iv) Potential Flow. Complex velocity and complex potential; Blausius Theorem; Kutta–Joukowski Theorem; Examples, flow around a disk, potential vortex; D’Alembert’s Paradox in Three Dimensions (see also [MP] page 52); Point vortex Hamiltonian system (see also [MP] chapter 4). [CM] chapter 2.
v) Stability of stationary solutions. Lyapunov function; Arnold stability theorem. [MP] chapter 3 (pages 102-110).
Quantum Mechanics
i) Linear operators in Hilbert spaces
Definition of bounded and unbounded operators, adjoint operator, symmetric operator, self-adjoint operator.
Self-adjointness criterion. Examples: position operator Q, momentum operator P, free Hamiltonian H_0.
Definition of small perturbation in the sense of operators, Kato-Rellich theorem, application to the hydrogen atom.
Definition of resolvent and spectrum, the spectrum of a self-adjoint operator is real (w.p.). Examples: spectrum of Q, P, H_0.
Definition of isometric and unitary operator, characterization of an isometric operator (prop. 4.16). Examples.
Definition of spectral family and spectral measure, spectral theorem (w.p.), functional calculus (w.p.), strongly continuous unitary groups.
Definition of H_{p}, H_{ac}, H_{sing} and \sigma_{p}, \sigma_{ac}, \sigma_{sing}.
([T1], chapter 4 and section 9.1)
ii) Formulation of QM
Formulation of the rules, comments
([T1], sect. 5.1, 5.2)
iii) Free Hamiltonian
Self-adjointness, spectrum, resolvent in d=3.
Unitary group, asymptotic behaviour for t large.
([T1] sect. 6.1, 6.2 and exercise)
iv) Hamiltonian with a point interaction
Notion of point interaction, definition of the Hamiltonian.
Proof of self-adjointness, resolvent, spectrum.
([T3])
v) Scattering theory
Formulation of a scattering problem, definition and main properties of wave operators, definition of asymptotic completeness, definition and main properties of the scattering operator, scattering into cones theorem.
Existence of wave operators.
Generalized eigenfunction and eigenfunction expansion theorem (w.p.).
Representation of wave operators and asymptotic completeness.
Representation of the scattering operators, scattering amplitude and scattering cross section.
Connection with stationary scattering theory.
Perturbative solution of the Lippmann-Schwinger equation and Born approximation.
([T2])
Further optional reading
Hamiltonian with N point interaction, self-adjointness, characterization of the eigenvalues.
The case N=2 with local or with non local regularizing boundary conditions.
Bound state problem for \alpha_1=\alpha_2=\alpha.
For \alpha=0, study of the negative eigenvalue as function of the distance between the two centers.
Generalities on the Efimov effect.
Efimov effect in a three-particle system composed of two (non-interacting) bosons of mass M plus a different particle of mass m, considering zero-range interactions and Born-Oppenheimer approximation.
([T3], [BFT]).
Introduction to the classical limit ([T1] Appendix A)
[CM] A. Chorin, J.E. Marsden: A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, 2000.
[MP] C. Marchioro, M. Pulvirenti: Mathematical Theory of Incompressible Nonviscous Fluids, Springer-Verlag, 1994.
[N] notes available on elearning
[T] Teta, A., Notes on Hamiltonian Mechanics (16/4/2024)
[E] R. Esposito, Appunti dalle lezioni di Meccanica Razionale, ed. Aracne
[T1] Teta, A., A Mathematical Primer on Quantum Mechanics, Springer 2018
[T2] Teta, A., Notes on Scattering Theory (2/6/2026)
[T3] Teta, A., Introduction to Point Interactions (5/5/2026)
[BFT] Basti G., Ferretti D., Teta A., A zero-range model for the Efimov effect in the Born-Oppenheimer approximation, https://arxiv.org/abs/2601.20762
The exam consists of an interview on the most relevant topics presented in the course.