In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\varepsilon \rightarrow 0$ and the convergence of the corresponding eigenvalues. This holds if $u_\varepsilon \rightarrow u$ with $u_\varepsilon$ smooth and $u\in L^2$ and where $L_\varepsilon, L$ are definded by $$ L_\varepsilon=-\frac{d^2}{dx^2}+u_\varepsilon'\quad\text{ and }\quad L=-\frac{d^2}{dx^2}+q $$ for $q=u'$ in the sense of distributions. The one I am referring to precisely is Theorem 3, page 7 in the paper.
I was wondering if this also provides me with the convergence of the corresponding eigenfunctions in some usable sense like $L^2$ or pointwise. The last line of the proof seems to suggest as much, since $f$ seems to be independent of $\varepsilon$.
Also they seem to use the terms norm resolvent convergence and uniform resolvent convergence interchangeably. Does anyone know if these terms are used as synonyms?
Also, I am a physics bachelor, sadly, so don't go to hard on me, thanks!
