Let $L = -\Delta + u$ operate on $C^\infty_c(\mathbb{R})\subset L^2$ and define its domain by taking the selfadjoint extension of $C^\infty_c(\mathbb{R})$, for $u:\mathbb{R}\to\mathbb{R}$ a quasi periodic function.
For $H=L$ with boundary condition $$ dom(H) = \{ g\in L^2(\mathbb{R}| g,g' \in AC_{loc}(\mathbb{R}), Lg\in L^2(\mathbb{R})\}. $$ Assume that $spec(H) = \bigcup_{j=0}^{n-1} [E_{2j},E_{2j+1}] \cup [E_{2n},\infty)$ with $E_{2i+1} < E_{2i+2}$
Is there a way to show that in the vicinity of $u$ $E_i$ can be extended continuously and even differentiated?