While reading the article Finite element approximations for Schrödinger equations with applications to electronic structure computations, I don't understand part of the proof of regularity. For the Schrodinger eigenvalue problem,

\begin{cases} -\Delta u+Vu=\lambda u, \ \text{in}\ \Omega \\ \Vert u\Vert_{0,\Omega}=1 \end{cases} where $\Omega$ is a bounded domain in $\mathbb{R}^3$, $V =\frac{1}{|x|}$.

Define $$a(u,v)=\int_\Omega \nabla u\nabla v+Vuv,\ u,v\in H_0^1(\Omega)$$ A number $\lambda$ is called an eigenvalue of the form $a(\cdot, \cdot)$ relative to the form $(\cdot, \cdot)$ if there is a nonzero vector u ∈ H1 0 (Ω), called an associated eigenfunction, satisfying Gong, Xingao; Shen, Lihua; Zhang, Dier; Zhou, Aihui, Finite element approximations for Schrödinger equations with applications to electronic structure computations, J. Comput. Math. 26, No. 3, 310-323 (2008). ZBL1174.65047.