CONTINUITY AND ANALYTICITY OF FAMILIES OF SELF-ADJOINT DIRAC OPERATORS ON A MANIFOLD WITH BOUNDARY

摘要

Given a continuous or analytic family D_t of self-adjoint elliptic operators on a manifold X, it is often useful to know whether the spectrum of D_t also varies continuously or analytically. If X is a closed manifold, the answer to this question is well known to be yes (assuming, in the analytic case, that the parameter space is an interval in the reals). The key point here is that because the domain of the operator is independent of the parameter t, one may apply standard theorems on deformations of self-adjoint operators to conclude that the spectrum varies in as nice a way as the operator. An excellent reference for these theorems is Kato's book, Perturbation Theory For Linear Operators [K].