In this presentation, we talk about the eigenfunction behaviors of nonlinear eigenvalue problems in quantum physics. We first prove that the eigenfunction cannot be a polynomial in any open set, which is a refinement of the standard unique continuation property. Then we apply non-polynomial behaviors of eigenfunctions to prove that the adaptive finite element approximation is convergent even if the initial mesh is not fine enough. We finally remark that the adaptive finite element method has linear convergence rate and optimal complexity.