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Nonrelativistic Limit of Generalized MIT Bag Models and Spectral Inequalities
Jussi Behrndt1, Dale Frymark1, Markus Holzmann1
1Institut für Angewandte Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria.
Mathematical Physics, Analysis, and Geometry
|July 25, 2024
Summary
This study demonstrates that the nonrelativistic limit of Dirac operators with generalized MIT bag boundary conditions is the Dirichlet Laplacian. This finding facilitates the transfer of spectral geometry results to Dirac operators for high speeds (large c).
Area of Science:
- Mathematical Physics
- Quantum Mechanics
- Spectral Theory
Background:
- Dirac operators are fundamental in relativistic quantum mechanics.
- MIT bag boundary conditions are used to model particle confinement.
- Understanding the nonrelativistic limit is crucial for connecting relativistic and nonrelativistic quantum theories.
Purpose of the Study:
- To analyze the nonrelativistic limit of self-adjoint Dirac operators.
- To investigate the behavior of these operators under generalized MIT bag boundary conditions.
- To establish a connection between Dirac operators and the Dirichlet Laplacian in a specific limit.
Main Methods:
- Analysis of self-adjoint Dirac operators on specified domains.
- Application of generalized MIT bag boundary conditions.
- Utilizing the norm resolvent sense for convergence analysis.
- Mathematical derivation of the nonrelativistic limit.
Main Results:
- The nonrelativistic limit of the studied Dirac operators is rigorously shown to be the Dirichlet Laplacian.
- This convergence is established in the norm resolvent sense.
- The results are valid for Dirac operators on various domains in Euclidean space.
Conclusions:
- The study provides a rigorous mathematical link between relativistic (Dirac) and nonrelativistic (Laplacian) quantum mechanical operators.
- This connection enables the transfer of established spectral geometry results from Dirichlet Laplacians to Dirac operators.
- The findings are particularly relevant for understanding Dirac operators in the regime of large speeds (large c).
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