The Schrödinger–Newton Ground State: A Complete Asymptotic Theory

We analyse the unique positive, spherically symmetric ground state of the stationary Schrödinger–Newton system. Using the regularity and symmetry of solutions, we construct the full even-power Taylor expansions of the wave function and the Newtonian potential at the origin, and we express all coefficients through explicit recurrence relations depending polynomially on the initial data (a₀,b₀)=(y(0),V(0)) with a₀>0 and b₀>1. In the far field we derive the complete Coulomb-corrected asymptotic expansion of the ground state, y(r)=Ce^−rr−^1/2(1+O(r⁻¹) and V(r)=1/r+O(e^−2r), and we show that the Newtonian potential admits no algebraic corrections beyond the leading Coulomb term. The mass constraint and the virial identity provide global relations that uniquely determine the admissible pair (a₀,b₀) and fix the asymptotic amplitude C, thereby linking the origin expansion to the far-field behaviour. These results complement the classical analysis of Moroz–Penrose–Tod and the asymptotic theory of Moroz–Van Schaftingen for the Choquard equation. In the Schrödinger–Newton setting, the Coulomb tail and the mass-preserving scaling interact in a subtle way, and the present work clarifies this interaction by giving explicit expansions, precise recurrence relations, and a unified analytic description of the ground state across all spatial scales.