2001/10/29 by Rafael de la Madrid, R. de la Madrid · 36 citations
Mathematics · Physics and Astronomy · #Absolute continuity #Computer science #Continuous spectrum #Discrete spectrum #Eigenvalues and eigenvectors #Energy spectrum #Hamiltonian (control theory) #Hilbert space #Laser-Matter Interactions and Applications #Mathematical analysis #Mathematics #Physics #Projective Hilbert space #Quantum chaos and dynamical systems #Quantum mechanics #Quantum optics and atomic interactions #Reproducing kernel Hilbert space #Rigged Hilbert space #Space (punctuation) #Spectrum (functional analysis) #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1088/0305-4470/35/2/311
published in Journal of Physics A Mathematical and General 35(2), 319-342 (Institute of Physics) · 27 RevTex pages
arxiv created 2001/10/29 · openalex publication_date 2002/01/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
It is shown that the natural framework for the solutions of any Schrödinger equation whose spectrum has a continuous part is the rigged Hilbert space (RHS) rather than just the Hilbert space. The difficulties of using only the Hilbert space to handle unbounded Schrödinger Hamiltonians whose spectrum has a continuous part are disclosed. Those difficulties are overcome by using an appropriate RHS. The RHS is able to associate an eigenket with each energy in the spectrum of the Hamiltonian, regardless of whether the energy belongs to the discrete or to the continuous part of the spectrum. The collection of eigenkets corresponding to both discrete and continuous spectra forms a basis system that can be used to expand any physical wavefunction. Thus the RHS treats discrete energies (discrete spectrum) and scattering energies (continuous spectrum) on the same footing.