Mathematical Aspects of Quantum Physics with NonSelfAdjoint Operators
Code  Completion  Credits  Range 

D01KTNO  ZK  2P 
 Garant předmětu:
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 Supervisor:
 Department of Mathematics
 Synopsis:

Motivated by the new concept in quantum mechanics where observables are represented by possibly nonselfadjoint operators, we intend to give an account of mathematical challenges arising in spectral theory of linear differential operators when the spectral theorem is not available.
 Requirements:
 Syllabus of lectures:

1.Motivations. QuasiHermitian and pseudoHermitian quantum mechanics. Open systems.2.Closed operators in Hilbert spaces. Point, continuous and residual spectra. Numerical range. Normal, symmetric and complex symmetric operators, physical symmetries.3.Definition of Schrödinger operators with complex potentials as closed operators in a Hilbert space. Sectorial operators and quadratic forms. Accretive operators and Kato's inequality. Beyond accretivity.4.Compactness and discrete spectra, operators with compact resolvent. Fredholm operators and various definitions of the essential spectrum. Stability of the essential spectrum.5.Spectral analysis. LiebThirringtype inequalities for Schrödinger operators with complex potentials. The method of multipliers.6.Similarity to normal and selfadjoint operators. Quasiselfadjoint operators. Basis properties of eigenfunctions.7.Pseudospectral analysis. Approximate eigenvalues and eigenfunctions. Microlocal techniques. WKB construction of pseudomodes.
 Syllabus of tutorials:
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Key references:
[1] D. Krejčiřík and P. Siegl: Elements of Spectral Theory without the Spectral Theorem, in „NonSelfadjoint Operators in Quantum Physics: Mathematical Aspects“, F. Bagarello, J.P. Gazeau, F. H. Szafraniec, and M. Znojil, Eds., WileyInterscience, 2015.
[2] T. Kato: Perturbation theory for linear operators, SpringerVerlag, Berlin, 1966.
 Note:
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 The course is a part of the following study plans: