Absolute continuity of eigenvectors of time-varying operators

Abstract

If K ( t ) K(t) is a compact, selfadjoint operator function of a real variable t with distinct eigenvalues at each t, we show that the eigenvalues and eigenvectors are absolutely continuous and that { K ( t ) } \{ K(t)\} is a commuting set provided that K ( t ) K(t) commutes with its time derivative K ′ ( t ) K’(t) at each t. The distinct eigenvalue condition is shown to be necessary.