A spectral alternative for continuous families of self-adjoint operators

Abstract

We consider a continuous family of self-adjoint operators $A_s$ in a separable Hilbert space, the parameter $s$ being a point of a complete metric space $S$. It is well known that isolated simple eigenvalues (assuming that the operators are bounded and the mapping $s\mapsto A_s$ is continuous in the norm sense) behave “well”: under small changes of the parameter they do not disappear and change continuously. Unlike this, the eigenvalues embedded in the essential spectrum can display a very “bad” behavior. It turns out, nevertheless, that if the set of eigenvalues is non-empty for a topologically rich (e.g., open) set of values of the parameter, then the (multi-valued) eigenvalue function has continuous branches. One application is as follows. Suppose a one-dimensional quasi-periodic Schrödinger operator has Cantor spectrum; then a Baire generic operator in its hull does not have eigenvalues.