On the structure of the energy spectrum for the one-dimensional Schr\"odinger operator with random potential

L. N. Grenkova

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (1982), pp. 6-10

Résumé

We study the distance between high energetic levels (eigenvalues) of the one-dimensional Schrödinger operator $H(\omega)=-\frac{d^2}{dt^2}+q(t,\omega)$, $t\in R_+^1$, $\omega\in\Omega$, where $q(t,\omega)$ is a stationary random process with certain conditions on smoothness and the rate of the decreasing of correlations. We obtain an asymptotic decomposition for the spectral split $\Delta_k=\sqrt{E_{k+1}}-\sqrt{E_k}$ when $k\to\infty$ where $E_k$, $E_{k+1}$ are neighbouring energetic levels. We find the appearence of repulsion in the case of high energetic levels. We find the appearence of repulsion in the case of high energetic levels.