GROUND STATE IN THE ENERGY SUPER-CRITICAL GROSS-PITAEVSKII EQUATION WITH A HARMONIC POTENTIAL

The paper rigorously proves the large‑b asymptotics of λ(b) near λ∞ for d ≥ 5 under explicit non‑degeneracy assumptions, using Emden–Fowler reduction, three solution families, a weighted Wronskian identity, and an implicit‑function/matching scheme. The candidate solution develops the same core structure and conclusions (including the oscillatory law for 5 ≤ d ≤ 12 and the monotone law for d ≥ 13), invoking the same non‑degeneracy and producing an equivalent solvability condition. Minor technical details (e.g., the paper’s use of intermediate‑time matching and small‑parameter a) are not fully spelled out by the model, but the arguments align in substance.