Spectral asymptotics of the Dirac operator in a thin shell

TL;DR

This paper analyzes how the eigenvalues of the Dirac operator behave asymptotically in a thin shell around a smooth hypersurface, revealing their connection to a Schrödinger operator with geometric potentials.

Contribution

It establishes the spectral asymptotics of the Dirac operator in a shrinking tubular neighborhood, linking it to a Schrödinger operator with geometric electric and Yang-Mills potentials.

Findings

Eigenvalues asymptotically driven by a Schrödinger operator

As the shell shrinks, spectral behavior is governed by geometric potentials

Results connect Dirac spectrum to geometric analysis in thin domains

Abstract

We investigate the spectrum of the Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a smooth compact hypersurface in $\mathbb{R}^n$ without boundary. We prove that when the tubular neighborhood shrinks to the hypersurface, the asymptotic behavior of the eigenvalues is driven by a Schr\"odinger operator involving electric and Yang-Mills potentials, both of geometric nature.