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

TL;DR

This paper analyzes the complex eigenvalues of a quantum parabolic potential barrier using Gel'fand triplet formalism, revealing quantized imaginary parts and a probabilistic interpretation of unstable system dynamics.

Contribution

It provides exact solutions for complex energy eigenvalues in a quantum unstable system within the Gel'fand triplet framework, including a probabilistic interpretation and quantization of resonance widths.

Findings

Complex eigenvalues are expressed with quantized imaginary parts.

Physical energy distributions follow Breit-Wigner resonance formulas.

Imaginary parts of eigenvalues determine quantized resonance widths.

Abstract

The paper deals with the one-dimensional parabolic potential barrier $V(x)={V_0-m\gamma^2 x^2/2}$, as a model of an unstable system in quantum mechanics. The time-independent Schr\"{o}dinger 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 ${\Gammav_n=(n+1/2)\hslash\gamma}$. 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.