Aim of the course is to introduce the basic tools of pseudodifferential calculus, and to show applications of such techniques in the analysis of dispersive PDEs, spectral theory, or other areas. A particular emphasis will be given to the problem of growth of Sobolev norms.

**Course Contents:**

Part 1: Review of Fourier calculus

- Fourier transform on Schwartz space and distributions
- Fourier multipliers
- Littlewood-Paley decomposition
- Dynamics of linear dispersive constant coefficients PDEs

Part 2: Pseudodifferential operators

- Symbolic calculus: composition, adjoints and quantizations
- Continuity in $L^2$ and the Calderon-Vaillancourt theorem
- Garding inequality
- Flow generation and Egorov theorem
- Functional calculus
- Pseudodifferential operators on a manifold and global quantization

Part 3: Applications (a selection from these topics according to the remaining time and the interest of the audience)

- Upper and lower bounds on the growth of Sobolev norms in linear, time depedent Schrödinger equations
- Asymptotics of eigenvalues
- Introduction to quantum chaos
- Burq-Gerard-Tzvetkov result on well-posedness of Schrödinger equations on compact manifold
- Introduction to paradifferential calculus

**References**

[1] S. Alinhac and P. Gérard, *Pseudo-differential Operators and the Nash-Moser Theorem* (AMS, Graduate Studies in Mathematics, vol. 82, 2007).

[2] H. Kumano-go, Pseudodifferential operators (MIT Press, Cambridge, Mass.-London, 1981)

[3] X. Saint Raymond, *Elementary Introduction to the Theory of Pseudodifferential Operators* (Studies in Advanced Mathematics, CRC Press, Boca Raton, 1991.)

[4] D. Robert,* Autour de l’Approximation Semi-Classique* (Boston etc., Birkhäuser 1987).

[5] M. Taylor, *Pseudo Differential Operators* (Princeton Univ. Press, Princeton, N.J., 1981)

[6] A. Maspero, D. Robert: *On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms*. J. Funct. Anal., 273(2):721–781, 2017.

[7] D. Bambusi, B. Grebert, A. Maspero, D. Robert: *Growth of Sobolev norms for abstract linear Schrödinger Equations*. J. Eur. Math. Soc. (JEMS), in press.

[8] A. Maspero: *Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations*. Math. Res. Lett, in press 2018.

[9] A. Weinstein:* Asymptotics of eigenvalue clusters for the Laplacian plus a potential*. Duke Math. J. 44 (1977), no. 4, 883–892

**Other link: **https://people.sissa.it/~amaspero/teaching/pseudodiff/pseudo20.html