1,143 research outputs found
The Analytic Continuation of the Lippmann-Schwinger Eigenfunctions, and Antiunitary Symmetries
We review the way to analytically continue the Lippmann-Schwinger bras and
kets into the complex plane. We will see that a naive analytic continuation
leads to nonsensical results in resonance theory, and we will explain how the
non-obvious but correct analytical continuation is done. We will see that the
physical basis for the non-obvious but correct analytic continuation lies in
the invariance of the Hamiltonian under anti-unitary symmetries such as time
reversal or PT
Reply to ``Comment on `On the inconsistency of the Bohm-Gadella theory with quantum mechanics'''
In this reply, we show that when we apply standard distribution theory to the
Lippmann-Schwinger equation, the resulting spaces of test functions would
comply with the Hardy axiom only if classic results of Paley and Wiener, of
Gelfand and Shilov, and of the theory of ultradistributions were wrong. As
well, we point out several differences between the ``standard method'' of
constructing rigged Hilbert spaces in quantum mechanics and the method used in
Time Asymmetric Quantum Theory.Comment: 13 page
The rigged Hilbert space approach to the Lippmann-Schwinger equation. Part I
We exemplify the way the rigged Hilbert space deals with the
Lippmann-Schwinger equation by way of the spherical shell potential. We
explicitly construct the Lippmann-Schwinger bras and kets along with their
energy representation, their time evolution and the rigged Hilbert spaces to
which they belong. It will be concluded that the natural setting for the
solutions of the Lippmann-Schwinger equation--and therefore for scattering
theory--is the rigged Hilbert space rather than just the Hilbert space.Comment: 34 pages, 1 figur
The resonance amplitude associated with the Gamow states
The Gamow states describe the quasinormal modes of quantum systems. It is
shown that the resonance amplitude associated with the Gamow states is given by
the complex delta function. It is also shown that under the near-resonance
approximation of neglecting the lower bound of the energy, such resonance
amplitude becomes the Breit-Wigner amplitude. This result establishes the
precise connection between the Gamow states, Nakanishi's complex delta function
and the Breit-Wigner amplitude. In addition, this result provides another
theoretical basis for the phenomenological fact that the almost-Lorentzian
peaks in cross sections are produced by intermediate, unstable particles
The Gamow-state description of the decay energy spectrum of neutron-unbound 25O
We show the feasibility of calculating the decay energy spectrum of neutron emitting nuclei within the Gamow-state description of resonances by obtaining the decay energy spectrum of 25O. We model this nucleus as a valence neutron interacting with an 24O inert core, and we obtain the resulting resonant energies, widths and decay energy spectra for the ground and first excited states. We also discuss the similarities and differences between the decay energy spectrum of a Gamow state and the Breit–Wigner distribution with energy-dependent width.Fil: Id Betan, Rodolfo Mohamed. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Instituto de Física de Rosario. Universidad Nacional de Rosario. Instituto de Física de Rosario; ArgentinaFil: de la Madrid, Rafael. Lamar University; Estados Unido
The rigged Hilbert space approach to the Lippmann-Schwinger equation. Part II: The analytic continuation of the Lippmann-Schwinger bras and kets
The analytic continuation of the Lippmann-Schwinger bras and kets is obtained
and characterized. It is shown that the natural mathematical setting for the
analytic continuation of the solutions of the Lippmann-Schwinger equation is
the rigged Hilbert space rather than just the Hilbert space. It is also argued
that this analytic continuation entails the imposition of a time asymmetric
boundary condition upon the group time evolution, resulting into a semigroup
time evolution. Physically, the semigroup time evolution is simply a (retarded
or advanced) propagator.Comment: 32 pages, 3 figure
- …