The inverse approach to Dirac-type systems based on the A-function concept

Abstract

The principal objective in this paper is a new inverse approach to general Dirac-type systems of the formy′(x,z)=i(zJ+JV(x))y(x,z)(x≥0), where y=(y1,…,ym)⊤ and (for m1,m2∈N)J=[Im10m1×m20m2×m1−Im2],V=[0m1vv⁎0m2],m1+m2=:m, for v∈[C1([0,∞))]m1×m2, modeled after B. Simon's 1999 inverse approach to half-line Schrödinger operators. In particular, we derive the A-equation associated to this Dirac-type system in the (z-independent) form∂∂ℓA(x,ℓ)=∂∂xA(x,ℓ)+∫0xA(x−t,ℓ)A(0,ℓ)⁎A(t,ℓ)dt(x≥0,ℓ≥0). Given the fundamental positivity condition ST>0 in (1.14) (cf. (1.13) for details), we prove that this integro-differential equation for A(⋅,⋅) is uniquely solvable for initial conditionsA(⋅,0)=A(⋅)∈[C1([0,∞))]m2×m1, and the corresponding potential coefficient v∈[C1([0,∞))]m1×m2 can be recovered from A(⋅,⋅) viav(ℓ)=−iA(0,ℓ)⁎(ℓ≥0).