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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.

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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.