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Self-adjoint operators in Mathematical Physics

Course Type: 
PhD Course
Academic Year: 
2014-2015
Period: 
February-April
Duration: 
40 h
Description: 

venue and schedule: room A-136, Monday 11-13, Tuesday 11-13
start: Monday 9 February 2015
end: Tuesday 14 April 2015

Synopsis:

Self-adjointness of (in general) unbounded operators on a Hilbert space is an ubiquitous tool in the mathematical methods for quantum mechanics and in the theory of linear and non-linear PDEs of mathematical physics. It determines the interpretation of Hamiltonians (of atoms, molecules, many-body systems) as quantum observables and the existence and uniqueness of the corresponding dynamics.

This course will present the main features of the general theory of self-adjoint operators (unboundedness, spectrum, spectral measures and spectral integrals, the spectral theorem, perturbation theory, quadratic forms, self-adjoint extensions) and will provide a survey of the most significant applications. This includes "classical" results on the self-adjointness of physically relevant Hamiltonians, as well as topics that are currently at the centre of an intense research activity, such as

  1. rigorous models for cold-atom systems with zero-range interactions ("unitary gases", in the condensed matter physics jargon)
  2. rigorous models for particles constrained on surfaces, on metric graphs, on hedgehog manifolds, etc.
  3. energy methods for the well-posedness of non-relativistic and semi-relativistic non-linear Schrödinger equations with strong and singular external electromagnetic fields, for which standard methods based on Strichartz estimates are not applicable.


Pre-requisites: Only a basic knowledge of Hilbert spaces and bounded operators on a Hilbert space is expected. A number of relevant notions taught in last fall's graduate course "Topics in the Mathematics of Quantum Mechanics" (by G. Dell'Antonio) will be revisited. The course is also designed to intersect with a few seminar talks on the subject, scheduled within the Analysis, Math-Phys, and Quantum series.

Literature:

Amrein, "Hilbert Space Methods in Quantum Mechanics", EPFL Press (2009)
Dell'Antonio, "Lectures on the Mathematics of Quantum Mechanics I", Springer (2015)
De Oliveira, "Intermediate Spectral Theory and Quantum Dynamics", Birkhäuser (2009)
Gitman, Tyutin, Voronov, "Self-adjoint Extensions in Quantum Mechanics", Birkhäuser (2010)
Schmüdgen, "Unbounded Self-adjoint Operators on Hilbert Space", Springer (2012)
Teschl, "Mathematical Methods in Quantum Mechanics", AMS (2009)

Links:

Research Group: 
Geometry and Mathematical Physics
Location: 
A-136
Next Lectures: 

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