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Let $\lambda \in \mathbb{R}$ and consider some function $q$ on the interval $[0,1].$

What is known to be the least restrictive condition on $q$ such that there are two linearly independent $H^2$ solutions (standard Sobolev space) for the following equation on $[0,1]$ $-y''(x)+q(x)y(x)= \lambda y(x)$ spanning the space of all solutions? Thoughts: Something like $q$ Lipschitz for example would be sufficient, but I have a paper where it is implicitly claimed that $q \in L^2$ suffices, but no argument is given why this could be true. I assume that this is wrong, but maybe somebody here knows more about this.

Thus, I would like to know: what is known to be a weak condition on $q$ so that this still holds?

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    $\begingroup$ A standard assumption to obtain solutions with two locally integrable derivatives is $q\in L^1_{loc}$ (see any advanced ODE text, for example Coddington-Levinson). So if you make the stronger assumption that $q\in L^2$, then it's obvious from the equation that $y''\in L^2$, as claimed. $\endgroup$
    – Christian Remling
    Commented Aug 29, 2016 at 21:50

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