Rigged Hilbert space approach to the Schrödinger equation

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.