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Aharonov and Bohm versus Welsh eigenvalues.

P Exner1,2, S Kondej3

  • 11Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University in Prague, Břehová 7, 11519 Prague, Czechia.

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Investigating Schrödinger operators with singular interactions on concentric circles reveals a critical flux value. Below this threshold, discrete spectrums accumulate; above it, they become finite or empty.

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Area of Science:

  • Mathematical Physics
  • Quantum Mechanics

Background:

  • Schrödinger operators model quantum systems.
  • Singular interactions and Aharonov-Bohm flux introduce complex behaviors.

Purpose of the Study:

  • Analyze the spectral properties of a 2D Schrödinger operator.
  • Investigate the impact of Aharonov-Bohm flux on singular interactions.

Main Methods:

  • Considered a 2D Schrödinger operator with type singular interaction.
  • Introduced an Aharonov-Bohm flux to the system.
  • Analyzed the behavior of the discrete spectrum below the essential spectrum.

Main Results:

  • Identified a critical flux value when .
  • Observed accumulation points in the discrete spectrum for .
  • Found a finite or empty discrete spectrum for and small .

Conclusions:

  • The Aharonov-Bohm flux significantly alters spectral properties.
  • A critical flux exists, distinguishing between accumulation and finiteness of eigenvalues.
  • The system exhibits rich spectral behavior dependent on flux strength and interaction parameters.