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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2024/2025

Mathematical Aspects of Quantum Physics with Non-Self-Adjoint Operators

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Code Completion Credits Range
D01KTNO ZK 2P
Garant předmětu:
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Tutor:
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Department of Mathematics
Synopsis:

Motivated by the new concept in quantum mechanics where observables are represented by possibly non-self-adjoint 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. Quasi-Hermitian and pseudo-Hermitian 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. Lieb-Thirring-type inequalities for Schrödinger operators with complex potentials. The method of multipliers.6.Similarity to normal and self-adjoint operators. Quasi-self-adjoint 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 „Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects“, F. Bagarello, J.-P. Gazeau, F. H. Szafraniec, and M. Znojil, Eds., Wiley-Interscience, 2015.

[2] T. Kato: Perturbation theory for linear operators, Springer-Verlag, Berlin, 1966.

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Data valid to 2024-05-27
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