Generic sets in spaces of measures and generic singular continuous spectrum for Delone Hamiltonians

2004/10/07 by Daniel Lenz, Lenz, Daniel, Peter Stollmann +1
Computer Science · Mathematics · Physics and Astronomy · #35J10 #81Q10 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.FA #math.MP #msc:35J10 #msc:81Q10

paper · pdf · doi:10.48550/arxiv.math-ph/0410021

arxiv created 2004/10/07 · openalex publication_date 2004/10/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that geometric disorder leads to purely singular continuous spectrum generically. The main input is a result of Simon known as the ``Wonderland theorem''. Here, we provide an alternative approach and actually a slight strengthening by showing that various sets of measures defined by regularity properties are generic in the set of all measures on a locally compact metric space. As a byproduct we obtain that a generic measure on euclidean space is singular continuous.

Citations

• Delone dynamical systems and associated random operators

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