Taylor Recurrences and Coulomb-Corrected Asymptotics for the Schrödinger–Newton Ground State

We study the positive, radial ground-state profile of the stationary Schrödinger–Newton system in the fixed-energy normalization μ=1 with V(∞)=0. Using regularity and radial symmetry of solutions, we derive convergent even Taylor expansions for the wave function and Newtonian potential near the origin, with explicit recurrence relations expressing all the coefficients in terms of the initial data (a0,b0)=(y(0),V(0)) with a0>0 and b0>1. Global existence and uniqueness of the positive radial ground state are taken from the known Schrödinger–Newton/Choquard theory, while the present work focuses on the local coefficient structure and the far-field expansion. In the far field, the Poisson equation yields the Coulomb tail V(r)=M^/r+O(e−2rrM^−2) with no algebraic corrections at any order, where M^=∫0∞r2y2dr is the determined reduced mass. The decaying wave profile admits the Coulomb-corrected asymptotic expansion y(r)=Ce−rrM^/2−1∑m≥0cmr−m, obtained by reducing the radial equation to a Whittaker equation with an exponentially small perturbation controlled by asymptotic integration. The inverse-power series is divergent and interpreted in the Poincaré sense. The mass, energy and virial identities serve as compatibility conditions for the globally selected profile.