I would like to ask a question with possibly a reference. If we have a Schrödinger operator $-\Delta+V$ on an interval $[0,L]$ with $V$ continous and Dirichlet conditions, can we state that the eigenfunctions of such operator are uniformly bounded, i.e. there exists $M>0$ such that the eigenfunctions $\{\phi\_n\}\_n$ satisfy


\begin{equation\*} \sup\_n \lvert\lvert \phi\_n \rvert\rvert\_\infty\leq M \end{equation\*}
?



Uniform boundedness Schrödinger operator eigenfunctions with Dirichlet conditions
Yes, this follows because asymptotically, as $|z|\to\infty$, the solutions of $-y''+Vy=zy$ look like those of the free equation $V\equiv 0$, and the eigenfunctions of $-y''=zy$, $\phi\_n=(2/(L\pi ) )^{1/2}\sin n\pi x/L$, are uniformly bounded.


In fact, they are uniformly bounded not just in $n$, but also in the potential $V$ as long as we impose a uniform bound on $\|V\|\_1$.


(I'm assuming here that you normalize your eigenfunctions as usual, $\|\phi\_n\|\_2=1$, and then you are asking about $\|\phi\_n\|\_{\infty}$.)


