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The spectrum of a quantum system has in general bound, scattering and resonant parts. The Hilbert space includes only the bound and scattering spectra, and discards the resonances. One must therefore enlarge the Hilbert space to a rigged Hilbert space, within which the physical bound, scattering and resonance spectra are included on the same footing. In… (More)
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… (More)
The goal of this contribution is to discuss various resonance expansions that have been proposed in the literature.
This paper is a contribution to the problem of particle localization in non-relativistic Quantum Mechanics. Our main results will be (1) to formulate the problem of lo-calization in terms of invariant subspaces of the Hilbert space, and (2) to show that the rigged Hilbert space incorporates particle localization in a natural manner.
The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann-Schwinger equation are functionals over spaces of Hardy functions. The preparation-registration arrow of time provides the physical justification for the Hardy… (More)
We discuss the role of boundary conditions in determining the physical content of the solutions of the Schrödinger equation. We study the standing-wave, the " in, " the " out, " and the purely outgoing boundary conditions. As well, we rephrase Feynman's +iε prescription as a time-asymmetric, causal boundary condition, and discuss the connection of Feynman's… (More)
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 non-sensical 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… (More)
In this work we present an application developed in Derive 6 to compose counterpoints for a given melody (" cantus firmus "). The result is non deterministic, so different counterpoints can be generated for a fixed melody, all of them obeying classical rules of counterpoint. It is also possible that the counterpoint could not be generated, in this case,… (More)
We explicitly construct the Rigged Hilbert Space (RHS) of the free Hamiltonian H 0. The construction of the RHS of H 0 provides yet another opportunity to see that when continuous spectrum is present, the solutions of the Schrödinger equation lie in a RHS rather than just in a Hilbert space.
There is compelling evidence that, when continuous spectrum is present, the natural mathematical setting for Quantum Mechanics is the rigged Hilbert space rather than just the Hilbert space. In particular, Dirac's bra-ket formalism is fully implemented by the rigged Hilbert space rather than just by the Hilbert space. In this paper, we provide a pedestrian… (More)