The course will start on February 26, 2025, with the following time-table:
Wednesday 2.00 p.m. - 4.00 p.m. in aula Conversi
Thursday 8.00 a.m. - 10.00 a.m. in aula Conversi
Hamiltonian systems: basic elements on Hamilton's equations and Hamilton-Jacobi equation.
Continuous systems, Euler equation, Navier-Stokes equation, examples and applications.
Quantum Mechanics: mathematical structure, introduction to scattering theory.
Hamilton's equations, variational principle, constant of motion, Poisson brackets,
canonical transformations, generating functions, Hamilton-Jacobi equation. ([T])
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.
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.
Variational principle for Euler Equations ([MP], page 5); Examples: Couette flow, gravity waves [N].
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.
Stability of stationary solutions. Liapunov function; Arnold stability theorem. [MP] chapter 3 (pages 102-110).
Bounded and unbounded operators, adjoint operator, symmetric operator, self-adjoint operator, self-adjointness criterion.
Definition of resolvent and spectrum, the spectrum of a self-adjoint operator is real.
Isometric and unitary operator.
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)
Definition of the free Hamiltonian.
Resolvent of the free Hamiltonian, unitary group generated by the free Hamiltonian and its asymptotic behavior for t large. ([T1], chapter 6)
Kato-Rellich theorem, application to the hydrogen atom. ([T1], sections 4.5 and 9.1)
Point interaction: heuristic formulation of Bethe-Peierls, rigorous definition, self-adjointness, resolvent, point spectrum. ([T3])
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 eigenfunctions and eigenfunction expansion theorem (w.p.).
Representation of wave operators and asymptotic completeness.
Representation of the scattering operator, scattering amplitude and scattering cross section.
Connection with stationary scattering theory.
Perturbative solution of the Lippmann-Schwinger equation and Born approximation. ([T2])
[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 (6/4/2025)
[T3] Teta, A., Introduction to point interaction (17/3/2025)
[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
Further (optional) reading: Bhaduri et al., An elementary exposition of the Efimov effect
The exam consists of an interview on the most relevant topics presented in the course.
For further information on the course see: https://corsidilaurea.uniroma1.it/it/corso/2023/32384/programmazione