Cantor spectrum for the quasi-periodic Schrödinger equation
The paper is devoted to study the nature of the spectrum of the (Bohr) almost periodic Schrödinger operator and its discrete analogue. The following result is established: There is a residual subset \(F\subset {\mathbb{R}}^ k\) such that, if \(\gamma =(\gamma_ 1,\gamma_ 2,...,\gamma_ k)\in F\), then the following statement holds. There is a residual subset \(V=V(\gamma)\subset C^{\delta}(T^ k)\) such that, if \(Q\in V\) and \(\psi =(\psi_ 1,\psi_ 2,...,\psi_ k)\in T^ k\), then the operator \(H=H_{\psi}=-(d^ 2/dt^ 2)+Q(\psi +\gamma t)\) has Cantor spectrum. \([0\leq \delta <1\) a real number and \(C^{\delta}(T^ k)\) the corresponding Hölder space of real-valued functions on the torus \(T^ k={\mathbb{R}}^ k/{\mathbb{Z}}^ k\), so that \(0\leq \psi_ i\leq 1\) (1\(\leq i\leq k)\). Cantor spectrum for each \(\psi \in T^ k\) means the spectrum of H regarded as a selfadjoint operator on \(L^ 2({\mathbb{R}})\) is a closed nowhere dense subset of the real line.] Moser's technique is heavily used in the paper. Further the author conjectures that the statement is true for every rationally independent rotation vector \(\gamma \in {\mathbb{R}}^ k\).
scientific article; zbMATH DE number 4214455
scientific article