Complex Eigenvalues of the Parabolic Potential Barrier and Gel'fand Triplet

Simple item page
dc.creatorShimbori, Toshiki
dc.creatorKobayashi, Tsunehiro
dc.date1999-10-04
dc.date.accessioned2026-07-07T04:32:59Z
dc.date.available2026-07-07T04:32:59Z
dc.descriptionThe paper deals with the one-dimensional parabolic potential barrier $V(x)={V_0-mγ^2 x^2/2}$, as a model of an unstable system in quantum mechanics. The time-independent Schrödinger equation for this model is set up as the eigenvalue problem in Gel'fand triplet and its exact solutions are expressed by generalized eigenfunctions belonging to complex energy eigenvalues ${V_0\mp i\Gammav_n}$ whose imaginary parts are quantized as ${\Gammav_n=(n+1/2)\hslashγ}$. Under the assumption that time factors of an unstable system are square integrable, we provide a probabilistic interpretation of them. This assumption leads to the separation of the domain of the time evolution, namely all the time factors belonging to the complex energy eigenvalues ${V_0-i\Gammav_n}$ exist on the future part and all those belonging to the complex energy eigenvalues ${V_0+i\Gammav_n}$ exist on the past part. In this model the physical energy distributions worked out from these time factors are found to be the Breit-Wigner resonance formulas. The half-widths of these physical energy distributions are determined by the imaginary parts of complex energy eigenvalues, and hence they are also quantized.
dc.description18 pages
dc.identifierhttps://arxiv.org/abs/math-ph/9910009
dc.identifierhttp://arxiv.org/abs/math-ph/9910009
dc.identifierNuovo Cim. B115 (2000) 325-342
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/58407
dc.subjectMathematical Physics
dc.titleComplex Eigenvalues of the Parabolic Potential Barrier and Gel'fand Triplet
dc.typetext

Files

Collections

arXiv