Individual energy level distributions for one-dimensional diagonal and off-diagonal disorder

Abstract

We study the distribution of the $n$-th energy level for two different one-dimensional random potentials. This distribution is shown to be related to the distribution of the distance between two consecutive nodes of the wave function. We first consider the case of a white noise potential and study the distributions of energy level both in the positive and the negative part of the spectrum. It is demonstrated that, in the limit of a large system ($L\to \infty$), the distribution of the $n$-th energy level is given by a scaling law which is shown to be related to the extreme value statistics of a set of independent variables. In the second part we consider the case of a supersymmetric random Hamiltonian (potential $V(x)=\varphi (x{)}^2+{\varphi }^{\prime }(x)$). We study first the case of $\varphi (x)$ being a white noise with zero mean. It is in particular shown that the ground state energy, which behaves on average like $\exp -L^{1/3}$ in agreement with previous work, is not a self averaging quantity in the limit $L\to \infty$ as is seen in the case of diagonal disorder. Then we consider the case when $\varphi (x)$ has a non zero mean value.

LaTeX, 33 pages, 9 figures

References in corpus (4)

• Single parameter scaling in one-dimensional localization revisited
• Density of states in coupled chains with off-diagonal disorder
• Time Delay Correlations and Resonances in 1D Disordered Systems
• Imaginary Potential as a Counter of Delay Time for Wave Reflection from a 1D Random Potential

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