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arXiv:quant-ph/9705012

Resonances from Higher Order S-Matrix Poles with Exponential Decay

C. Püntmann, P. Patuleanu

Published May 8, 1997• 0 citations
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Abstract

In analogy to Gamow vectors describing resonance states from first order S-matrix poles, one can define Gamow vectors from higher order poles of the S-matrix. With these vectors we are going to discuss a density operator that describes exponentially decaying resonances from higher order poles.
DOI
10.48550/arxiv.quant-ph/9705012

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In the first part, Gamow vectors from S-matrix poles of higher multiplicity are derived in analogy to the Gamow vectors describing resonances from first-order poles. With these vectors a density operator is constructed that describes resonances associated with higher-order poles that obey an exponential decay law. It turns out that this operator formed by higher-order Gamow vectors has a unique structure. In the second part, the Gamow vectors are used in order to describe decaying and resonance states at a potential barrier of finite width. The exact numerical solutions of the S-matrix poles are presented and the out-going boundary condition for the Gamow vector description and consequences from that are discussed.
Resonances and decaying states have been associated with first-order poles of the S-matrix and represented by exponentially decaying Gamow kets using the Rigged Hilbert Space formulation of quantum mechanics. In analogy, higher-order S- matrix poles, represented by a mixture of Gamow-Jordan vectors, are associated with higher-order decaying states. The exact Golden Rule for decaying states associated to a higher-order pole of the S-matrix is calculated. These higher-order decaying states operators also obey the exponential decay law, but they have a more complicated linewidth than the Breit-Wigner energy distribution. This linewidth depends on the interaction V. To test whether these higher-order decaying state operators represent possible physical states, we discuss their linewidth for various assumptions about the matrix elements of V.
After a summary of the Rigged Hilbert space formulation of quantum mechanics and a brief statement of its advantages over von Neumann’s formulation, a mathematically correct definition of Gamow’s exponentially decaying vectors as generalized energy eigenvectors is suggested. It is shown that exponentially decaying vectors are obtained from the S-matrix poles in the lower half of the second sheet and exponentially growing vectors from the S-matrix poles in the upper half of the second sheet. Decaying ’’state’’ vectors are defined as functionals over half of the space of physical states and growing ’’state’’ vectors are defined as functionals over the other half. On functionals over these subspaces, the dynamical group of time development splits into two semigroups, one for t ≳ 0 and the other for t < 0. The generalized basis system connected with the spectrum of the Hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated.
A state vector description for relativistic resonances is derived from the first order pole of the jjj-th partial SSS-matrix at the invariant square mass value $\sm_R=(m-iΓ/2)^2$ in the second sheet of the Riemann energy surface. To associate a ket, called Gamow vector, to the pole, we use the generalized eigenvectors of the four-velocity operators in place of the customary momentum eigenkets of Wigner, and we replace the conventional Hilbert space assumptions for the in- and out-scattering states with the new hypothesis that in- and out-states are described by two different Hardy spaces with complementary analyticity properties. The Gamow vectors have the following properties: - They are simultaneous generalized eigenvectors of the four velocity operators with real eigenvalues and of the self-adjoint invariant mass operator M=(PμPμ)1/2M=(P_μP^μ)^{1/2}M=(Pμ​Pμ)1/2 with complex eigenvalue $\sqrt{\sm_R}$. - They have a Breit-Wigner distribution in the invariant square mass variable $\sm$ and lead to an exactly exponential law for the decay rates and probabilities.
In this paper we construct Gamow vectors for resonances given by poles of the analytic continuation of the S matrix of any finite order. We study their modes of decay (or growth). We obtain Jordan block structures for the extended Hamiltonians and evolution operators on the subspaces spanned by these Gamow vectors. We perform this study within the context of the rigged Hilbert space extension of quantum theory. We construct an explicit Friedrichs model with a double pole resonance to illustrate the general formulation.