The Scaling Component of an Adelic Pseudodifferential Operator

We construct and analyse the symmetric scaling component D_scaling of a global adelic pseudodifferential operator acting on the adele-class space A_Q / Q^×. A complete functional–analytic framework is developed: we prove density of a natural domain, establish symmetry and essential self-adjointness, and compute the full adelic symbol by combining archimedean Mellin analysis with p-adic Fourier methods. The operator lies in the class Psi^{1+epsilon} for every epsilon > 0; a unitary Mellin transform yields an explicit spectral resolution, and a truncated diagonalisation confirms the predicted linear eigenvalue growth. By isolating the core scaling dynamics we obtain a rigorous, self-contained platform for future perturbative inclusions of representation-theoretic terms.