lectures: Thu at 14:00 - 16:00 (Oct 13 - Dec 15, Jan 12 - Jan 26)
seminar: tba

 
The aim of this course is to analyse the spectrum (mainly the bottom) of the Schrödinger operator with magnetic fields. The main focus will be on semi-classical analysis, but emphasis will also be given on the analysis of superconductivity of the phenomenon of nucleation at the boundary. This will enable us to discover how abstract spectral analysis can be used in connection with asymptotic methods (mainly semi-classical methods).

Basic spectral theory; On the Schrödinger Operators with magnetic fields: -selfadjointness, -rough spectral estimates, -diamagnetism, -magnetic bottles; Models with constant magnetic fields in dimension 2 or 3; Harmonic approximation; Decay estimates for eigenstates; On some questions coming from superconductivity; Sharp asymptotics for the ground state energy of the Neumann problem; About the splitting between the two lowest eigenvalues. For the bibliography, students can look at the webpage (http://www.math.u-psud.fr/~helffer) for various manuscripts in spectral theory. They can also look at one of the two books devoted to semi-classical analysis:

• M.Dimassi and J. Sjöstrand. /Spectral Asymptotics in the semi-classical limit/. London Mathematical Society. Lecture Note Series 268. Cambridge University Press (1999).
• B. Helffer. /Semi-classical analysis for the Schrödinger operator and / applications. Lecture Notes in Mathematics 1336. Springer Verlag 1988.