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Analytic Eigenbranches in the Semi-classical Limit.

Stefan Haller1

  • 1Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Complex Analysis and Operator Theory
|July 11, 2020
PubMed
Summary
This summary is machine-generated.

This study analyzes the semi-classical behavior of eigenvalues for Laplacians on manifolds. It extends existing theorems to vector-valued settings, including geometric operators like Witten's Laplacian.

Keywords:
Analytic eigenbranchesSemi-classical limitWitten Laplacian

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Area of Science:

  • Spectral theory
  • Differential geometry
  • Mathematical physics

Background:

  • Laplacians on manifolds are fundamental operators in geometry and analysis.
  • Understanding the semi-classical limit of eigenvalues provides insights into operator behavior.

Purpose of the Study:

  • To investigate the semi-classical limit of analytically parametrized eigenvalues for a one-parameter family of Laplacians on a closed manifold.
  • To establish a vector-valued analogue of Luc Hillairet's theorem for scalar Schrödinger operators.

Main Methods:

  • Analysis of a one-parameter family of Laplacians.
  • Study of the semi-classical limit of eigenvalues.
  • Extension of scalar operator theorems to vector-valued settings.

Main Results:

  • A vector-valued analogue of a key theorem for scalar Schrödinger operators is established.
  • The results are applicable to geometric operators, specifically mentioning Witten's Laplacian on differential forms.

Conclusions:

  • The study successfully extends spectral theory results to a more general vector-valued context.
  • The findings have implications for understanding geometric operators in semi-classical analysis.