Integrated density of states: From the finite range to the periodic Airy–Schrödinger operator

Abstract

We compute, in the semiclassical regime, an explicit formula for the integrated density of states of the periodic Airy–Schrödinger operator on the real line. The potential of this Schrödinger operator is periodic, continuous, and piecewise linear. For this purpose, we study precisely the spectrum of the Schrödinger operator whose potential is the restriction of the periodic Airy–Schrödinger potential to a finite number of periods. We prove that all the eigenvalues of the operator corresponding to the restricted potential are in the spectral bands of the periodic Airy–Schrödinger operator and none of them are in their spectral gaps. In the semiclassical regime, we count the number of these eigenvalues in each of the spectral bands. Note that in our results, we have explicit constants that characterize the semiclassical regime.