Asymptotic Recovery in Fourier Spectral Methods for the Schrödinger Equation with Point Singularities

This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials V in H^s, where s > max\{d/2-2,-1\} and d denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argument to derive sharp convergence orders for FSM, yielding order 2s+2 for eigenvalues and order s+1 for eigenfunctions in the H¹ norm. More importantly, the H¹ error with respect to the projected eigenfunction converges with a higher order s+1+b, where b=min\{s+2-d/2-varepsilon, s+1, 2\}>0 for arbitrarily small varepsilon>0, revealing a super-convergence phenomenon. Second, in the presence of potentials with isolated point singularities, we develop an asymptotic-recovery (AR) technique to post-process the FSM solutions. The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders 2s+2+2b for eigenvalues and s+1+b for eigenfunctions in the H¹ norm, while the AR post-processing requires only a computational cost that is linear in the number of FSM degrees of freedom. The analysis introduces a rigorous definition of point singularities and develops a foundational framework for their study. It further establishes an asymptotic expansion of eigenfunctions consisting of a regular component in H^s+4 together with d+1 asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.