Projected Gradient Method with Global Convergence for Spherically Constrained Nonlinear Eigenvalue Problems in Bose-Einstein Condensates

A spherically constrained nonlinear eigenvalue problem (NEPv) arising in Bose–Einstein condensates (BEC) is investigated. The Projected Gradient Method (PGM) is proposed and analyzed in detail. Rigorous theoretical analysis establishes its global convergence for both “easy” and “hard” cases, including Lipschitz continuity of the gradient, monotonic objective decrease, and convergence to optimality. Numerical experiments in 1D, 2D, and 3D BEC models demonstrate that PGM achieves competitive accuracy compared to other methods while offering significant advantages in computational efficiency and scalability, enabling large-scale simulations.