Spectral flow of exterior Landau-Robin hamiltonians

TL;DR

This paper investigates how the spectrum of exterior Landau-Robin Hamiltonians varies with Robin boundary conditions, providing insights into spectral flow and asymptotic behavior related to Dirichlet problems.

Contribution

It introduces a method to analyze spectral flow dependence on Robin data and establishes continuous dependence of Hamiltonians on Robin parameters in the gap topology.

Findings

Spectral flow can be localized to the boundary for detailed analysis.

The spectrum asymptotically approaches that of the Dirichlet problem.

Continuous dependence on Robin data is proven in the gap topology.

Abstract

We study the spectral flow of Landau-Robin hamiltonians in the exterior of a compact domain with smooth boundary. This provides a method to study the spectrum of the exterior Landau-Robin hamiltonian's dependence on the choice of Robin data, even explaining the heuristics of how the spectrum of the Robin problem asymptotically tends to the spectrum of the Dirichlet problem. The main technical result concerns the continuous dependence of Landau-Robin hamiltonians on the Robin data in the gap topology. The problem can be localized to the compact boundary where the asymptotic behavior of the spectral flow in some special cases can be described.