Internal Lifshits tails for random perturbations of periodic Schrödinger operators. (English) Zbl 1060.82509

Duke Math. J. 98, No. 2, 335-396 (1999); correction ibid. 109, No. 2, 411-412 (2001).
The author studies the Lifshits tails for random perturbations of periodic Schrödinger operators. He proves that, at the edge of a gap, a true Lifshits tail for the random operator occurs if and only if the integrated density of states of the background periodic operator has a nondegenerate behavior. More precisely, he first considers periodic operators on \(L^2(\mathbb{R}^d)\) of the form \(H=-\Delta + W\), where \(W\) is a bounded, \(\mathbb{Z}^d\)-periodic, real-valued potential on \(\mathbb{R}^d\), and he assumes that the spectrum of \(H\) has a gap below energy zero. Then he considers random operators of the form \(H_{\omega}=H+V_{\omega}\), where \(V_{\omega}= \sum_{\gamma \in \mathbb{Z}^d}\omega_{\gamma}V(\cdot-{\gamma})\), where \(V\) is a nonnegative, bounded, compactly supported real-valued potential, and \(\omega=(\omega_{\gamma})_{\gamma \in \mathbb{Z}^d}\) is a collection of i.i.d. random variables on \([0,1]\), and he assumes that the \(\omega\)-almost sure spectrum of \(H_{\omega}\) has a gap below energy zero. Under these assumptions, he proves that \(H_{\omega}\) exhibits a Lifshits tail, that is, that the difference of integrated densities of states \(N(E)-N(0)\) of \(H_{\omega}\) has the same exponential decay of the form \(e^{-E^{-d/2}}\), when \(E\) tends to zero from above, that is, inside the spectrum, as the difference of integrated densities of states \(n(E)-n(0)\) of \(H\).


82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47N55 Applications of operator theory in statistical physics (MSC2000)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47B80 Random linear operators
Full Text: DOI


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