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regularity of solutions of PDEs.

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A question about the regularity of the Schrödinger equation

While reading the article [1], I noticed I don't understand part of the proof of regularity. … associated eigenfunction, satisfying $$ a(u,v) = \lambda(u,v)\quad\forall v \in H_0^1(\Omega).$$ Now, if $(\lambda, u) \in \mathbb{R}\times H_0^1(\Omega)$ and $Vu \in L^2(\Omega)$, can we deduce from the regularity