Operators with singular continuous spectrum. I: General operators. (English) Zbl 0851.47003

It is shown that certain generic sets of selfadjoint operators have purely singular continuous spectrum. The main abstract result is the spectacular “Wonderland Theorem”: Let \(X\) be a complete metric space of selfadjoint operators in which convergence in the metric topology implies strong resolvent convergence (such an \(X\) is said to be regular), and suppose that the sets \(\{A\mid A\) has a purely absolutely continuous spectrum} and \(\{A\mid A\) has purely point spectrum} are dense in \(X\). Then \(\{A\mid A\) has only singular continuous spectrum} is “Baire typical” in the sense that it is a dense \(G_\delta\) set.
A number of remarkable results follow. For instance, for a Baire typical set of \(V\) in \(C_\infty (\mathbb{R}^\gamma)\), the set of continuous functions in \(\mathbb{R}^\gamma\) which vanish at \(\infty\) in the uniform norm, the Schrödinger operator \(-\Delta +V\) has purely singular continuous spectrum on \((0, \infty)\). Consequences for random Schrödinger operators, Jacobi matrices and a celebrated theorem of Weyl and von Neumann (that given a selfadjoint operator \(A\) and \(\varepsilon >0\), there exists a Hilbert-Schmidt operator \(B\) with \(|B|_2< \varepsilon\) such that \(A+B\) has only point spectrum) are also given.


47A10 Spectrum, resolvent
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