Eigenvalue boundary problems for the Dirac operator
Let \(M\) be a compact Riemannian spin manifold with non-empty boundary. The authors study the spectrum of the Dirac operator under four different boundary conditions: (1) the global Atiyah-Patodi-Singer condition associated with the spectral resolution of the intrinsic boundary Dirac operator, (2) the local condition associated with a chirality operator, (3) MIT bag boundary conditions, and (4) a new global boundary condition that is a modification of the APS condition. The first 3 boundary conditions give rise to self-adjoint realization of the Dirac operator and spectra tending to \(\pm\infty\); the final boundary condition gives rise to a spectra of complex numbers with positive imaginary part. The authors also analyse the classical Friedrich lower bound for the spectrum in terms of scalar curvature provided that the mean curvature of the boundary hypersurface is non-negative.
scientific article; zbMATH DE number 1889175