Spectral asymptotics of the Dirac operator in a thin shell

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 Schrödinger operator involving electric and Yang-Mills potentials, both of geometric nature.