Analytic expression of the DOS for a new model of 1d-potential and its random perturbation

Abstract

In this article we present comparisons between the spectrum of a one-dimensional Schrödinger operator for a particular periodic potential and for its restriction to a finite number of sites. We deduce from this finite, but large, number of sites, the Integrated Density of States (IDS) associated to the Hamiltonian operator whose derivate is the DOS. The exact formula for the IDS is given and the expression of the DOS is analytical. All our calculations are done on the particular periodic Airy-potential, which is a new case for which one has an analytical expression of the DOS. It is a continuous, periodic potential, piecewise affine. As a periodic operator, the spectrum is a band spectrum.

Figures (5)

• Figure 1: The considered periodic potential $\mathbf{V}$
• Figure 2: The DOS in the Hydrogen case ($\mathbf{V_0}=13,6$eV, $L_0=2\mathring{A}$ and $\kappa=1.526$)
• Figure 3: The DOS in the Carbon case ($\mathbf{V_0}=489.99$eV, $L_0=3.08\mathring{A}$ and $\kappa=10.682$)
• Figure 4: Integrated density of states for $\kappa=2.8$
• Figure 5: The DOS associated to the potential V for $\kappa=2.8$