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

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 both the spectrum and the approximate point spectrum equal the closure of the image of the open left half-plane under m. Points obtained from the open half-plane are genuine eigenvalues, while the image of the boundary line supplies approximate eigenvalues. The resolvent outside this closed image is constructed directly as a one-sided convolution operator. The auxiliary semigroup generated by the half-line operator has the sharp norm identity ∥T(τ)∥=exp((ReB+|C|)τ) and is exponentially stable exactly when ReB+|C|<0; this concerns an additional evolution parameter, not stability in the original pantograph variable. We transfer the spectral theorem back to Lp((0,1),dt/t): interior modes decay at t=0, while boundary approximate modes are Mellin waves concentrating at the singular endpoint. As a secondary representation result, logarithmic pullback of the Hermite basis gives a Mellin–Hermite basis on L2(R+,dt/t) with explicit Galerkin matrices for the Euler operator and dilations. The corresponding translation truncations converge strongly, and the shifted Gaussian admits an exact truncation-error formula.