Comment on ‘On the inconsistency of the Bohm–Gadella theory with quantum mechanics’

Abstract

Rigged Hilbert spaces of Hardy functions lead to a consistent theory of resonance scattering and decay. Contrary to the claims of a recent article [8], the theory holds for a wide range of potentials and rigorously describes the asymmetric time evolution of resonances.

Reply to ‘Comment on ‘On the inconsistency of the Bohm–Gadella theory with quantum mechanics’’

“…With some variations, this is the "standard method" followed by [7][8][9][10][11][12][13][14][15] to introduce spaces of test functions in quantum mechanics. Thus, contrary to what the authors of [1] assert, the method followed by the present author runs (somewhat) parallel to [8], not to TAQT. In order to use (2.13) to construct the rigged Hilbert spaces for the analytically continued Lippmann-Schwinger eigenfunctions and for the Gamow states, we need to obtain the growth of χ ± (r; z), χ 0 (r; z) and u(r; z n ).…”

“…The authors of [1] argue that one cannot draw any conclusion on the limit |z| → ∞ from estimates such as (2.19) or (2.20), and therefore they conclude that nothing prevents ϕ + (z) from tending to zero and therefore from being Hardy functions. Their conclusion is not true, because their argument does not take the nature of (2.19) and (2.20) into account.…”

“…The authors of [1] claim that the present author has inadvertently constructed an example of TAQT. That such is not the case can be seen not only from the differences between the "standard method" and the method used in TAQT to introduce rigged Hilbert spaces, but also from the outcomes.…”

“…The authors of [1] inadvertently acknowledge that there is no example of TAQT when they say that they still need "to identify the form and properties" of the functions of (3.3), see the last paragraph in section 2 of [1]. By saying so, they are acknowledging that they don't know whether the standard Gamow states defined in the position representation are well defined as functionals acting on Φ BG± .…”

“…In this reply, we will show that the arguments of [1] are missing essential aspects of [2], and that therefore the conclusions of [2] still stand.…”