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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)

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 +iε… (More)

- R. de la Madrid
- 2001

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 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… (More)

We study the selfadjoint time operator recently constructed by one of the authors. We will show that this time operator must be interpreted as a “selfadjoint variant” of the time operator.

- Rafael de la Madrid, Gastón Garćıa-Calderón, Juan Gonzalo Muga
- 2005

The Gamow (or resonance) states are the wave functions of resonances. These states are eigensolutions of the Schrödinger equation subject to a “purely outgoing boundary condition.” The Gamow states were introduced by Gamow [1] in 1928 to describe α decay (see also [2]). Some years later, in 1939, Siegert made use of the Gamow states to obtain a resonance… (More)

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)

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 localization 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. PACS numbers: 03.65.-w

We first present two possible analytic continuations of the LippmannSchwinger eigenfunctions to the second sheet of the Riemann surface, and then we compare the different Gamow vectors that are obtained through each analytic continuation.

We explicitly construct the Rigged Hilbert Space (RHS) of the free Hamiltonian H0. The construction of the RHS of H0 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. PACS numbers: 03.65.-w, 02.30.Hq