# Uniform boundedness Schrödinger operator eigenfunctions with Dirichlet conditions

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

$$\begin{equation*} \sup_n \lvert\lvert \phi_n \rvert\rvert_\infty\leq M \end{equation*}$$ ?

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 $$\|\phi_n\|_{\infty}$$.)
• The book by Poschel and Trubowitz discusses asymptotics in some detail (though with an $L^2$ assumption on $V$, which isn't necessary; you can adapt the proofs if required) and it's what I usually quote when I use these things, but there are many other references, so keep searching if you don't like this (Marchenko's book is another option). May 15, 2022 at 20:25
• @Christian Remling Christian, do you know examples where $\sup_n \|\phi_n\|_\infty$ really depends on $\|V\|_1$ ($V \geq 0$)? May 16, 2022 at 17:10
• @GiorgioMetafune: I think one way of doing this is to take $V=0$ on $(0,a)$ and $V=N\to\infty$ on $(a,L)$. In the limit $N\to\infty$, this will simulate a Dirichlet boundary condition at $x=a$, and if $a$ is small, then $\|\phi_n\|_{\infty}/\|\phi_n\|_2$ will become large. May 16, 2022 at 17:22