Gamow states as continuous linear functionals over analytical test functions. (English) Zbl 0892.46039
The space of analytical test functions \(\zeta \), rapidly decreasing on the real axis (i.e., Schwartz test functions of the type \(S\) on the real axis), is used to construct a rigged Hilbert space (RHS) or a Gelfand triplet (\(\zeta,\pi,\zeta'\)). Gamow resonant states (GS) is defined in RHS starting from Dirac’s formula. Then Dirac’s formulation of quantum mechanics in RHS is shown, the structure of GS is given explicitly, and the norm of GS in RHS is calculated. The contribution of GS to \(P(E)\), the probability distribution of a system at energy \(E\), is obtained and the relation with the Breit-Wigner weighted energy distribution is studied. As well, some examples of GS as analytical functionals are given.
Reviewer: D.N.Zarnadze (Tbilisi)
MSC:
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
| 47A70 | (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces |
| 81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |
| 46F15 | Hyperfunctions, analytic functionals |
| 46N50 | Applications of functional analysis in quantum physics |
Keywords:
space of analytical test functions; Schwartz test functions; rigged Hilbert space; Gelfand triplet; Gamow resonant states; Dirac’s formula; probability distribution of a system; Breit-Wigner weighted energy distribution; analytical functionalsReferences:
| [1] | DOI: 10.1007/BF01339451 · doi:10.1007/BF01339451 |
| [2] | DOI: 10.1007/BF01339451 · doi:10.1007/BF01339451 |
| [3] | DOI: 10.1063/1.1703665 · Zbl 0090.19303 · doi:10.1063/1.1703665 |
| [4] | DOI: 10.1016/0375-9474(71)90095-9 · doi:10.1016/0375-9474(71)90095-9 |
| [5] | DOI: 10.1016/0375-9474(82)90519-X · doi:10.1016/0375-9474(82)90519-X |
| [6] | DOI: 10.1016/0375-9474(82)90519-X · doi:10.1016/0375-9474(82)90519-X |
| [7] | DOI: 10.1016/0370-2693(91)90692-J · doi:10.1016/0370-2693(91)90692-J |
| [8] | DOI: 10.1016/0370-2693(91)90692-J · doi:10.1016/0370-2693(91)90692-J |
| [9] | DOI: 10.1016/0370-2693(91)90692-J · doi:10.1016/0370-2693(91)90692-J |
| [10] | DOI: 10.1063/1.524655 · Zbl 0462.47011 · doi:10.1063/1.524655 |
| [11] | DOI: 10.1063/1.524871 · doi:10.1063/1.524871 |
| [12] | DOI: 10.1063/1.524553 · doi:10.1063/1.524553 |
| [13] | DOI: 10.1063/1.525883 · Zbl 0526.47007 · doi:10.1063/1.525883 |
| [14] | DOI: 10.1063/1.525966 · Zbl 0527.35065 · doi:10.1063/1.525966 |
| [15] | DOI: 10.1063/1.526468 · doi:10.1063/1.526468 |
| [16] | DOI: 10.1119/1.15797 · doi:10.1119/1.15797 |
| [17] | Zel’dovich Ya. B., JETP 12 pp 542– (1961) |
| [18] | DOI: 10.1007/BF01350287 · Zbl 0195.41302 · doi:10.1007/BF01350287 |
| [19] | DOI: 10.2748/tmj/1178244354 · Zbl 0103.09201 · doi:10.2748/tmj/1178244354 |
| [20] | DOI: 10.3792/pja/1195518691 · Zbl 0348.46029 · doi:10.3792/pja/1195518691 |
| [21] | DOI: 10.3792/pja/1195518621 · Zbl 0348.46030 · doi:10.3792/pja/1195518621 |
| [22] | DOI: 10.1063/1.530862 · Zbl 0841.46054 · doi:10.1063/1.530862 |
| [23] | DOI: 10.1007/BF00672852 · Zbl 0790.53055 · doi:10.1007/BF00672852 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
