Expansion of Continuous Spectrum Operators in Terms of Eigenprojections

Abstract

Publisher Summary This chapter describes the expansion of continuous spectrum operators in terms of Eigenprojections. Eigenfunction expansions are at the heart of the picture of quantum mechanics that was developed by Dirac. The idea is to expand states which change with time as they evolve under the Schrodinger equation in terms of those which do not, in the sense that they give the same expectation values for all observables. However, in quantum mechanics, observables in the physical sense correspond to operators in a Hilbert space. The chapter concentrates on the approximation of spectral projections by finitely many Eigenprojections, because once this is done the spectral theorem can be used to do the rest. The approach is self-contained, and involves developing the theory of continuous spectrum Eigenfunctions afresh and paying very careful attention to convergence; in fact, new results on convergence are contained in the chapter. The chapter gives new convergence results, which hold even in situations where no reasonable a priori estimates on the domain of the self-adjoint operator are available; one such situation would be the Laplace-Beltrami operator on a semi-Riemannian manifold.