Change of representation and the rigged Hilbert space formalism in quantum mechanics

Abstract

Generalized eigenvectors are key tools in the theory of rigged Hilbert spaces. Let H be a Hilbert space and let Φ be a dense subspace of H. Let A be a densely defined linear operator in H such that Φ ⊂ DA and AΦ ⊂ Φ. The generalized eigenvectors of A are the eigenvectors of the algebraic dual of A |Φ. In the case where Φ is endowed with a topology τ finer than the norm topology inherited from H, generalized eigenvectors that are τ-continuous may be of great interest. We discuss conditions which ensure the existence of representations associated to generalized eigenvectors of A. As an application, we review and refine Böhm's study of the algebra of the quantum harmonic oscillator.