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Propagation for Schrödinger Operators with Potentials Singular Along a Hypersurface.
Jeffrey Galkowski1, Jared Wunsch2
1Department of Mathematics, University College London, London, UK.
Archive for Rational Mechanics and Analysis
|July 10, 2024
Summary
This study examines defect measures for Schrödinger operators on Riemannian manifolds. The standard propagation theorem holds even with potentials having singularities along a hypersurface, provided certain continuity conditions are met.
Area of Science:
- Mathematical physics
- Differential geometry
- Harmonic analysis
Background:
- Schrödinger operators are fundamental in quantum mechanics.
- Riemannian manifolds provide a framework for studying geometry and analysis.
- Defect measures and propagation theorems are crucial for understanding wave phenomena.
Purpose of the Study:
- To investigate the propagation of defect measures for Schrödinger operators on Riemannian manifolds.
- To analyze the impact of potentials with conormal singularities on propagation theorems.
- To extend the applicability of standard propagation theorems under specific regularity conditions.
Main Methods:
- Analysis of Schrödinger operators on Riemannian manifolds.
- Study of defect measures and their propagation properties.
- Application of microlocal analysis techniques.
- Investigation of potentials with conormal singularities.
Main Results:
- The standard propagation theorem is shown to hold for bicharacteristics traveling transversally to a hypersurface when the potential is absolutely continuous.
- Propagation continues to hold even for bicharacteristics tangential to the hypersurface to first order, given an absolutely continuous first derivative of the potential.
Conclusions:
- The regularity of the potential plays a critical role in the propagation of defect measures.
- The results extend the validity of standard propagation theorems to cases with potentials exhibiting specific types of singularities.
- This work contributes to a deeper understanding of wave propagation in the presence of singularities on curved spaces.
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