An approximation theorem of Runge type for kernels of certain non-elliptic partial differential operators

Abstract

For a constant coefficient partial differential operator P(D) with a single characteristic direction, such as the time-dependent free Schrödinger operator as well as non-degenerate parabolic differential operators like the heat operator, we characterize when open subsets X1⊆X2 of Rd form a P-Runge pair. The presented condition does not require any kind of regularity of the boundaries of X1 nor X2. As part of our result we prove that for a large class of non-elliptic operators P(D) there are smooth solutions u of the equation P(D)u=0 on Rd with support contained in an arbitarily narrow slab bounded by two parallel characteristic hyperplanes for P(D).