Schrödinger operators
Code  Completion  Credits  Range 

D01SO  ZK  2P 
 Garant předmětu:
 Lecturer:
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

This is an updated and extended version of the BCAM course given by the lecturer in Bilbao in 2010 (Schrödinger operators and their spectra). The goal of the course is to give an overview ofclassical as well as recent methods used in nonrelativistic quantum mechanics.
 Requirements:

Functional analysis welcome but not required.
 Syllabus of lectures:

1.Motivations. The crisis of classical physics and the rise of quantum mechanics. Mathematical formulation of quantum theory. The quantum stability of matter.2.Definition of Schrödinger operators as selfadjoint operators in a Hilbert space. Semibounded quadratic forms. Selfadjoint extensions of symmetric operators.3.Qualitative features of the spectrum. The discrete and essential spectra. The free Hamiltonian and dimensional aspects of the Euclidean space. The Hardy inequality and virtual bound states.4.Weakly coupled bound states. Analytic versus asymptotic perturbation theory. The BirmanSchwinger analysis.5.Strongly coupled bound states. The semiclassical limit.LiebThirring inequalities.6.The nature of the essential spectrum. The absolute and singular continuous spectra, embedded eigenvalues. The limiting absorption principle. Commutator methods and the Mourre theory. 7.Magnetic Schrödinger operators. The diamagnetic inequality. Magnetic Hardy inequalities. The largetime behaviour of the heat semigroup.
 Syllabus of tutorials:
 Study Objective:
 Study materials:

Key references:
[1] W. O. Amrein, A. Boutet de Monvel and V. Georgescu: C0groups, commutator methods and spectral theory of Nbody Hamiltonians, Progress in Math. Ser., vol. 135, Birkhäuser, 1996.
[2] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon: Schrödinger operators, with application to quantum mechanics and global geometry, SpringerVerlag, Berlin, 2008.
[3] N. Raymond: Bound states of the magnetic Schrödinger operator, EMS, 2017.
[4] M. Reed and B. Simon: Methods of modern mathematical physics, IIV, Academic Press, New York, 19721978.
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans: