Principle: The ground state eigenfunction of a Schrodinger operator is localized near the minima of the potential.
I learned this principle in basic quantum mechanics courses (when solving the Schrödinger equation for various potentials, e.g. double well, periodic box, etc.). My question concerns how to quantify this statement.
Let $I=[-1,1]$ and $V:I\to \mathbb{R}$ be a smooth periodic potential and let $$ L=-\frac{d^2}{dx^2}+V(x) $$ be the corresponding Schrödinger operator on $I$ with periodic boundary conditions. Let $\lambda_1$ be the lowest eigenvalue of the eigenvalue problem $$ Lu=\lambda u $$ on $I$ with periodic boundary conditions and let $u_1$ be the corresponding eigenfunction (the ground state eigenfunction). By the nodal domain theorem, $u_1$ has no zeros, hence we may assume $u_1>0$. Suppose also that $\lambda_1\leq 0$, which implies that $V_{\min}\leq 0$. Let $x_0\in I$ be the point at which the potential is maximized $$ V(x_0)=V_{\max}. $$
Question: Is it true that $x_0$ minimizes $u$, i.e. that $$ u(x_0)=u_{\min}? $$
Numerical solutions I generated using Mathematica for various potentials indicate that the answer to the question is affirmative. How can one prove or disprove this. If the answer is negative, how can the "principle" written above be quantified? Any references would be appreciated. Thank you.