Let $V:\mathbb{R}\rightarrow \mathbb{R}^{+ \*}$ a real positive function such that $\displaystyle \lim\_{ x \to \pm\infty} V(x)= +\infty $. 


Then the Schrödinger operator $H=-\frac{d^2}{dx^2}+V(x)$ has compact resolvant, in particular it has a pure discrete spectrum 
$(\lambda\_i)\_{i\geq 0}$ such that $\displaystyle \lim\_{i\to +\infty} \lambda\_i = + \infty$. Associated to each eigenvalue, there is an eigenfunction $\phi\_i\in L^2(\mathbb{R})$, ie
 $$-\phi\_i''(x)+V(x)\phi\_i(x)=\lambda\_i \phi\_i(x), \quad \forall x\in\mathbb{R} $$


satisfying $||\phi\_i||\_{L^2(\mathbb{R})}=1$, 


**My question is:** Are eigenfunctions uniformly bounded? i.e. Does exist $M>0$ such that for all $n\geq 0$, 


$$|| \phi\_i ||\_\infty <M $$



Schrödinger eigenfunctions are bounded
In general, there won't be a uniform bound on all eigenfunctions simultaneously. If $[a,b]$ is a short interval with Dirichlet boundary conditions $y(a)=y(b)=0$ and constant potential $V=c$, then the ground state (normalized) eigenfunction
$$
\phi(x)=\sqrt{\frac{2}{b-a}}\, \sin\pi\frac{x-a}{b-a}
$$
is quite large pointwise.


A very high steep potential wall has approximately the same effect as a Dirichlet boundary condition, so we can easily cook up a $V$ where we are close to this situation infinitely many times, with arbitrarily small interval lengths.


