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Internal Lifschitz singularities for one dimensional Schrödinger operators. (English) Zbl 0783.60062

Summary: The integrated density of states of the periodic plus random one- dimensional Schrödinger operator \(H_ \omega=- \Delta+V_{\text{per}}+\sum_ iq_ i(\omega)f(\circ-i)\); \(f\geq 0\), \(q_ i(\omega)\geq 0\), has Lifschitz singularities at the edges of the gaps in \(Sp(H_ \omega)\). We use Dirichlet-Neumann bracketing based on a specifically one-dimensional construction of bracketing operators without eigenvalues in a given gap of the periodic ones.

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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