Spectral Transition for Dirac Operators with Electrostatic -Shell Potentials Supported on the Straight Line.

Area of Science:

• Quantum mechanics
• Mathematical physics
• Relativistic quantum mechanics

Background:

• The Dirac operator is fundamental in relativistic quantum mechanics.
• Investigating spectral properties of operators with interactions is crucial.
• Shell interactions introduce unique mathematical challenges.

Purpose of the Study:

• To analyze the spectral properties of the 2D Dirac operator with a specific electrostatic shell interaction.
• To identify and characterize spectral transitions under varying interaction strengths.
• To understand the behavior of the continuous spectrum near critical interaction strengths.

Main Methods:

• The study employs mathematical analysis of the 2D Dirac operator.
• Focus is on the spectral properties, particularly the continuous spectrum.
• The behavior is examined at critical strengths of the electrostatic shell interaction.

Main Results:

• A spectral transition is observed in the Dirac operator.
• For critical interaction strengths, the continuous spectrum collapses.
• This collapse occurs to a single point within the spectral gap of the free Dirac operator.

Conclusions:

• The electrostatic shell interaction significantly alters the Dirac operator's spectrum.
• Critical interaction strengths lead to a dramatic spectral collapse.
• This finding provides insight into the behavior of relativistic quantum systems with localized interactions.