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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2024/2025

Schrödinger operators

The course is not on the list Without time-table
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 non-relativistic 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 self-adjoint operators in a Hilbert space. Semibounded quadratic forms. Self-adjoint 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 Birman-Schwinger analysis.5.Strongly coupled bound states. The semiclassical limit.Lieb-Thirring 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 large-time behaviour of the heat semigroup.

Syllabus of tutorials:
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Study materials:

Key references:

[1] W. O. Amrein, A. Boutet de Monvel and V. Georgescu: C0-groups, commutator methods and spectral theory of N-body 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, Springer-Verlag, 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, I-IV, Academic Press, New York, 1972-1978.

Note:
Further information:
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2024-05-28
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