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arXiv:math-ph/9910009

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

Toshiki Shimbori, Tsunehiro Kobayashi

Published Oct 4, 1999• 7 citations
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Abstract

The 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.

DOI
10.48550/arxiv.math-ph/9910009

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The paper deals with the one-dimensional parabolic potential barrier V(x)=V0−mγ2x2/2V(x)={V_0-mγ^2 x^2/2}V(x)=V0​−mγ2x2/2, 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 V0∓i\Gammavn{V_0\mp i\Gammav_n}V0​∓i\Gammavn​ whose imaginary parts are quantized as \Gammavn=(n+1/2)ℏγ{\Gammav_n=(n+1/2)\hslashγ}\Gammavn​=(n+1/2)ℏγ. 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 V0−i\Gammavn{V_0-i\Gammav_n}V0​−i\Gammavn​ exist on the future part and all those belonging to the complex energy eigenvalues V0+i\Gammavn{V_0+i\Gammav_n}V0​+i\Gammavn​ 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.
THE dissertation deals with the two-dimensional isotropic parabolic potential barrier as a solvable model of a two-dimensional unstable system in non-relativistic quantum mechanics. The time-independent Schrodinger equation for this model is set up as the eigenvalue problem in Gel'fand triplet and its exact solutions are expressed by generalized eigen-functions belonging to complex energy eigenvalues. There are stationary states with a real energy eigenvalues. There are statinary states with a real energy eigenvalue involved in those solutions, and they are infinitely degenerate. A physical picture of the generalized eigenstates of the unstable system is obtained from the hydrodynamical point of view. For the first few stationary states of the two-dimensional parabolic potential barrier, the complex velocity potentials in the hydrodynamical formulation of quantum mechanics express the two-dimensional irrotational flows round a right angle. We find that this result can be deduced from the general connection between complex velocity potentials and singular potentials in two-dimensional quantum systems.
Statistical mechanics for states with complex eigenvalues, which are described by Gel'fand triplet and represent unstable states like resonances, are discussed on the basis of principle of equal apriori{\it a priori}apriori probability. A new entropy corresponding to the freedom for the imaginary eigenvalues appears in the theory. In equilibriums it induces a new physical observable which can be identified as a common time scale. It is remarkable that in spaces with more than 2 dimensions we find out existence of stable and quasi-stable systems, even though all constituents are unstable. In such systems all constituents are connected by stationary flows which are generally observable and then we can say that they are semiclassical systems. Examples for such semiclassical systems are constructed in parabolic potential barriers. The flexible structure of the systems is also pointed out.
Time-dependent processes accompanied by vortex creations and annihilations are investigated in terms of the eigenstates in conjugate spaces of Gel'fand triplets in 2-dimensions. Creations and annihilations of vortices are described by the insertions of unstable eigenstates with complex-energy eigenvalues into stable states written by the superposition of eigenstates with zero-energy eigenvalues. Some concrete examples are presented in terms of the eigenfunctions of the 2-dimensional parabolic potential barrier, i.e., −mγ2(x2+y2)/2-m γ^2 (x^2+y^2)/2−mγ2(x2+y2)/2. We show that the processes accompanied by vortex creations and annihilations can be analyzed in terms of the eigenfunctions in the conjugate spaces of Gel'fand triplets. Throughout these examinations we point out three interesting properties of the zero-energy flows. (i) Mechanisms using the zero-energy flows are absolutely economical from the viewpoint of energy consumption. (ii) An enormous amount of informations can be discriminated in terms of the infinite variety of the zero-energy flows. (iii) The zero-energy flow patterns are absolutely stable in any disturbance by inserting arbitrary decaying flows with complex-energy eigenvalues.
In the two-dimensional isotropic parabolic potential barrier V ( x , y ) = V 0 - m γ 2 ( x 2 + y 2 )/2, though it is a model of an unstable system in quantum mechanics, we can obtain the stationary states corresponding to the real energy eigenvalue V 0 . Further, they are infinitely degenerate. For the first few eigenstates, we will find the stationary flows round a right angle that are expressed by the complex velocity potentials W = ±γ z 2 /2. From the hydrodynamical point of view vortex structures for the general solutions are also studied.
We show that zero-energy flows appear in many particle systems as well as in single particle cases in 2-dimensions. Vortex patterns constructed from the zero-energy flows can be investigated in terms of the eigenstates in conjugate spaces of Gel'fand triplets. Stable patterns are written by the superposition of zero-energy eigenstates. On the other hand vortex creations and annihilations are described by the insertions of unstable eigenstates with complex-energy eigenvalues into the stable patterns. Some concrete examples are presented in the 2-dimensional parabolic potential barrier case. We point out three interesting properties of the zero-energy flows (i) the absolute economy as for the energy consumption, (ii) the infinite variety of the vortex patterns, and (iii) the perfect stability of the vortex patterns.
We show that zero-energy flows appear in many particle systems as same as in single particle cases in 2-dimensions. Vortex patterns constructed from the zeroenergy flows can be investigated in terms of the eigenstates in conjugate spaces of Gel’fand triplets. Stable patterns are written by the superposition of zero-energy eigenstates. On the other hand vortex creations and annihilations are described by the insertions of unstable eigenstates with complex-energy eigenvalues into the stable patterns. Some concrete examples are presented in the 2-dimensional parabolic potential barrier case. We point out three interesting properties of the zero-energy flows; (i) the absolute economy as for the energy consumption, (ii) the infinite variety of the vortex patterns, and (iii) the absolute stability of the vortex patterns Keywords: Zero-energy solutions, vortex creations and annihilations, quantum mechanics, Gel’fand triplets,