Alberto Maspero

Introduction to pseudodifferential operators and applications in dynamical systems

Aim of the course is to introduce the basic tools of pseudodifferential calculus, and apply them to analyze the long time dynamics of linear, time dependent Schr\"odinger equations. A particular emphasis will be given to the problem of growth of Sobolev norms.

Timetable

Program

Textbooks

1. L. Hörmander, The Analysis of Linear Partial Differential Operators I-III (Springer) [H]
2. M. Shubin, Pseudodifferential Operators and Spectral Theory (Springer) [S]
3. S. Alinhac and P. Gérard, Pseudo-differential Operators and the Nash-Moser Theorem, (AMS, Graduate Studies in Mathematics, vol. 82, 2007). [AG]
4. X. Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential Operators} (Studies in Advanced Mathematics, CRC Press, Boca Raton, 1991.) [SR]
5. M. M. Wong, An Introduction to Pseudo-differential Operators (World Scientific, Singapore, 2nd ed., 1999.)
6. D. Robert, Autour de l'Approximation Semi-Classique (Boston etc., Birkh\"auser 1987).
7. M. Taylor, Pseudo Differential Operators (Princeton Univ. Press, Princeton, N.J., 1981)
8. 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.
9. D. Bambusi, B. Grebert, A. Maspero, D. Robert: Growth of Sobolev norms for abstract linear Schrödinger Equations. J. Eur. Math. Soc. (JEMS), in press.
10. A. Maspero: Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations. Math. Res. Lett, in press 2018.
11. A. Weinstein: Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44 (1977), no. 4, 883--892.

Additional material

1. Calderon-Vaillancourt theorem in $S^0_{0,0}$ link
2. Hörmander construction of almost analytic functions link