2010/02/16 by J. Julve, F. J. de Urries, F. J. de Urríes · 10 citations
Mathematics · Physics and Astronomy · #Bound state #Cold Atom Physics and Bose-Einstein Condensates #Dirac (video compression format) #Dirac delta function #Gaussian #Inner product space #Kronecker delta #Mathematical analysis #Mathematical physics #Mathematics #Norm (philosophy) #Orthogonality #Physics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Resonance (particle physics) #Scattering #math-ph #math.MP
paper · pdf · doi:10.1088/1751-8113/43/17/175301
published in Journal of Physics A Mathematical and Theoretical 43(17), 175301 (Institute of Physics) · 13 pages, 2 figures
arxiv created 2010/02/16 · openalex publication_date 2010/04/13 · arxiv updated 2015/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
The properties of a prescription for the inner products of resonance (Gamow states), scattering (Dirac kets) and bound states for one-dimensional quantum barriers are worked out. The divergent asypmtotic behaviour of the Gamow states is regularized using a Gaussian convergence factor first introduced by Zel'dovich. With this prescription, most of these states (with discrete complex energies) are found to be orthogonal to each other and to the Dirac kets, except when they are neighbours, in which case the inner product is divergent. Therefore, as it happens for the continuum scattering states, the norm of the resonant ones remains non-calculable. Thus, they exhibit properties halfway between the (continuum real) Dirac-δ orthogonality and the (discrete real) Kronecker-δ orthogonality of the bound states.