A spectral alternative for continuous families of self-adjoint operators
While eigenvalues embedded in the essential spectrum of a continuous family of self-adjoint operators in a separable Hilbert space can display very bad behaviour, it is proved in this paper that, if the set of eigenvalues is nonempty for a topologiaclly rich set of values of the parameter, then the multi-valued eigenvalue function has continuous branches. The following is proved as an application. Suppose that a one-dimensional quasi-periodic Schrödinger operator has Cantor spectrum; then a Baire generic operator in its hull does not have eigenvalues.
scientific article; zbMATH DE number 6160609
scientific article