Concentration of the eigenfunctions of Schrödinger operators

We consider a Schrödinger operator $ A = - \frac{d^2}{dx^2} + Q(x)$, where $Q(x) \in C^2(\mathbb{R})$ is a nonnegative, even, convex, slowly changing potential, and consider the limit of distributions of its eigenfunctions. Let $A \psi_k = \lambda_k \psi_k$, $\Vert \psi_k \Vert = 1$, $k \in \mathbb{N}$ be a complete system of eigenfunctions, and let the turning points $x_k > 0$ be defined by $Q(x_k) = \lambda_k$. Assume that $Q(x)$ satisfies
$$\lim_{x \to \infty} \frac{Q(tx)}{Q(x)} = t^{\beta}, \quad \beta \geq 2.$$
Rescale measures, or their densities, on $\mathbb{R}$ by
$$\varphi_k(x) = x_k \psi_k^2(x_k x).$$

The behavior of measures $\psi_k(x)^2\, dx$ determines the asymptotics of the norms of spectral 1D-projections of non-self-adjoint perturbations of $A$.
For any $f$ in the Schwartz space on $\mathbb{R}$,
$$ \lim_{k \to \infty} \int_{-\infty}^{\infty}f(x) \varphi_k(x) \, dx = c(\beta) \int_{-1}^1 f(x) \frac{ dx}{(1 - |x|^{\beta})^{1/2}} $$
where $c(\beta) = \frac{\Gamma( \frac{1}{2} + \frac{1}{\beta})}{2 \pi^{1/2} \Gamma(1 + \frac{1}{\beta})}$.
Such statements, in the context of the theory of orthogonal polynomials, are well known (Rakhmanov, Mhaskar–Saff, Lubinsky). In the algebraic case, i.e., when $Q(x)$ is a polynomial potential, the limit distributions were given by A. Eremenko, A. Gabrielov, and B. Shapiro.
The talks is based on our joint work with Petr Siegl (Queen's University Belfast, UK) and Joseph Viola (University of Nantes, France) [1], [2].