Eulerian Pantograph Equations, Half-Line Spectra, and Hermite Representations

<jats:p>We study an Eulerian pantograph equation with proportional delay by means of a logarithmic change of variables, which transforms the problem into a constant-shift equation on the half-line. The main object is the maximal left-transport realization on Lp(0,∞), with derivative domain W1,p(0,∞) and no boundary condition at the endpoint. For the characteristic function m(μ)=vμ+B+Cexp(Lμ), we prove that the spectrum is the closure of the image of the open left half-plane under m: the open image gives the point spectrum, the boundary image gives the approximate spectrum, and the closure gives the full spectrum. The associated semigroup has a sharp norm identity with exponent ReB+|C|, yielding the exact stability threshold ReB+|C|&amp;lt;0. We compare this half-line result with the full-line Fourier multiplier realization and with the Mellin-root description of pantograph modes. Hermite functions are used only as a computational representation tool for histories and translations; they do not replace the half-line spectral theorem. The same logarithmic translation structure also suggests a natural connection with Fourier Neural Operator parameterizations of finite-history input-output maps, although the spectral results proved here are independent of any neural approximation scheme.</jats:p>