The local Borg–Marchenko uniqueness theorem of matrix‐valued Dirac‐type operators for coefficients locally smooth at the right endpoint

Abstract

AbstractWe provide an expression of the Weyl–Titchmarsh matrix associated with a self‐adjoint matrix‐valued Dirac‐type operator defined on [0, b) for . As a concrete application of it, we establish asymptotics of the difference of two Weyl–Titchmarsh matrices corresponding to two Dirac‐type operators , with a.e. on [0, a], in with , when the are sufficiently smooth in a right neighborhood of the point a and their right derivatives at a coincide up to a certain order. In addition to this, we also provide new proofs of the local Borg–Marchenko theorem and asymptotic high‐energy expansion of Weyl–Titchmarsh matrices associated with Dirac‐type operators.