A Correction Note on Sobolev Propagation and Polynomial Truncation for a Dirichlet-to-Neumann Approximation in Thin Dielectric Multilayers

Abstract

This correction note discusses two technical points in the derivation of the Dirichlet-to-Neumann operator in B. Alliti, T. Laadj and K. M'hamed-Messaoud, Mathematical Methods in the Applied Sciences 46 (2023), 6843–6856. First, the proof of the exact transfer formula is based on successive abstract Cauchy evolutions and operator exponentials. In the transverse-electric specialization considered by the authors, the tangential Fourier symbol possesses eigenvalues of the form ±√(ξ²−κ). High tangential frequencies therefore contain an exponentially growing branch, showing that unrestricted normal Cauchy propagation cannot define a C₀-semigroup of bounded linear operators on the natural finite-order Sobolev trace spaces. The obstruction is demonstrated explicitly by computing the matrix exponential and constructing normalized high-frequency wave packets. The note also shows that this obstruction does not, by itself, rule out the existence of the corresponding Dirichlet-to-Neumann map: in a one-layer model, imposition of the terminal boundary condition cancels the exponentially growing/decaying components and yields an ordinary first-order Dirichlet-to-Neumann symbol. Second, an exact polynomial identity used in passing from a product of Taylor polynomials to an order-n transfer matrix omits terms of degrees n+1 through pn. The corrected noncommutative product identity is provided, together with an explanation of why the coefficients through order n remain unchanged when explicit truncation is performed. The note therefore identifies a functional-analytic gap in the abstract-Cauchy justification and an algebraic truncation error, while not claiming that the resulting low-order impedance approximations are necessarily invalid.