Associated matrix construction for a third-order expression with two independent distribution coefficients

Construct a (3×3) associated matrix function F(x; τ1, τ0) with locally integrable entries for the third-order formal differential expression y''' + (τ1 y)' + τ1 y' + τ0 y on (0,T), where τ1 ∈ W^{-1}_{3,loc}(0,T) and τ0 ∈ W^{-2}_{3,loc}(0,T), such that the quasi-derivatives defined via F satisfy y^{[3]} = y''' + (τ1 y)' + τ1 y' + τ0 y for all test functions and thereby yield a well-defined first-order system equivalent to the original differential equation.

Background

The paper investigates third-order differential operators with distribution coefficients by introducing an associated matrix F(x) that enables reduction of the differential equation to a first-order system with integrable coefficients. This framework is used to paper inverse spectral problems and prove uniqueness theorems for recovering the distribution coefficient σ from spectral data.

While reconstruction formulas are known for a related third-order expression with coefficients in L2 and W2^{-1}, extending the associated matrix construction to the more singular setting with τ1 ∈ W^{-1}_{3,loc} and τ0 ∈ W^{-2}_{3,loc} remains unresolved. This gap prevents applying the same spectral mapping techniques to broader classes of distributional coefficients and motivates the stated open problem.