A Unified Theory of Nuclear Reactions, II

Feshbach, Herman

doi:10.1006/aphy.2000.6016

Cited by 48 publications

(53 citation statements)

References 20 publications

(29 reference statements)

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“…We conclude that with probability 1 an element F in the ensemble will satisfy (11), consequently (13) and (14) will hold. However, the centering constants (12) can not be left out when the ξ k have the distribution defined by (16).…”

Section: An Examplementioning

confidence: 80%

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Fermi’s Golden Rule and Exponential Decay as a RG Fixed Point

Langmann

Lindblad

2009

J Stat Phys

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We discuss the decay of unstable states into a quasicontinuum using models of the effective Hamiltonian type. The goal is to show that exponential decay and the golden rule are exact in a suitable scaling limit, and that there is an associated renormalization group (RG) with these properties as a fixed point. The method is inspired by a limit theorem for infinitely divisible distributions in probability theory, where there is a RG with a Cauchy distribution, i.e. a Lorentz line shape, as a fixed point. Our method of solving for the spectrum is well known; it does not involve a perturbation expansion in the interaction, and needs no assumption of a weak interaction. We use random matrices for the interaction, and show that the ensemble fluctuations vanish in the scaling limit. Thus the limit is the same for every model in the ensemble with probability one. IntroductionThe standard textbook derivation of Fermi's golden rule starts from a perturbation expansion of the unitary evolution, keeping the lowest nontrivial order, and then sums over a dense set of final states, to get a decay or reaction rate for an unstable stateHere v represents the (average) transition matrix element and ρ is the density of final states. (We will use the convention = 1 throughout.) The formula is essentially contained in Dirac [8], see also [23]. If the rate Γ is a constant there results an exponential decay of the occupation number p(t) = exp(−Γt)p(0). The corresponding quantum amplitudes are the Fourier transforms of a Lorentz line shape function, see equation (5).It has been known for a long time that there are models where the golden rule and the exponential decay can be obtained without a perturbation expansion. It is our goal to show that such results hold for a large ensemble of models, and are exact in a suitable limit which leaves Γ invariant. This paper treats only the mathematical aspects of the *

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Section: An Examplementioning

confidence: 80%

“…We note that the formulas derived in this section are known, the main ideas going back at least to Feshbach [12,13,18]. The mathematical background can be traced from Remark 2.1 of [16].…”

Section: Matrix Models and Resolventsmentioning

confidence: 89%

Fermi’s Golden Rule and Exponential Decay as a RG Fixed Point

Langmann

Lindblad

2009

J Stat Phys

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“…b) When explicit reference to some particles is suppressed, and is replaced in terms of expressions involving integrals over Green functions between the remaining particles [10], then the resulting nonlocal potential becomes nonsemi-separable. An example given below refers to the interaction of two nucleons (protons or neutrons), mediated by the exchange of mesons, whose coordinates are suppressed.…”

Section: Discussionmentioning

confidence: 99%

Nyström-Clenshaw-Curtis quadrature for integral equations with discontinuous kernels

2002

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Abstract. A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis quadrature for Fredholm integral equations of the second kindwhose kernel k(t, s) is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when k(t, s) is infinitely differentiable away from the diagonal t = s. Relation to the singular value decomposition is indicated. Application to integro-differential Schrödinger equations with nonlocal potentials is given.

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“…The time scales of these two different types of resonance states are well separated from one another: up to 6 10 neutron resonances are overlapped by one single-particle resonance. The Feshbach unified theory of nuclear reactions gives a good description of this situation [8,9].…”

Section: Introductionmentioning

confidence: 99%

Dynamical Phase Transitions in Quantum Systems

Rotter¹

2010

JMP

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Many years ago Bohr characterized the fundamental differences between the two extreme cases of quantum mechanical many-body problems known at that time: between the compound states in nuclei at extremely high level density and the shell-model states in atoms at low level density. It is shown in the present paper that the compound nucleus states at high level density are the result of a dynamical phase transition due to which they have lost any spectroscopic relation to the individual states of the nucleus. The last ones are shell-model states which are of the same type as the shell-model states in atoms. Mathematically, dynamical phase transitions are caused by singular (exceptional) points at which the trajectories of the eigenvalues of the non-Hermitian Hamilton operator cross. In the neighborhood of these singular points, the phases of the eigenfunctions are not rigid. It is possible therefore that some eigenfunctions of the system align to the scattering wavefunctions of the environment by decoupling (trapping) the remaining ones from the environment. In the Schrödinger equation, nonlinear terms appear in the neighborhood of the singular points.

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A Unified Theory of Nuclear Reactions, II

Cited by 48 publications

References 20 publications

Fermi’s Golden Rule and Exponential Decay as a RG Fixed Point

Fermi’s Golden Rule and Exponential Decay as a RG Fixed Point

Nyström-Clenshaw-Curtis quadrature for integral equations with discontinuous kernels

Dynamical Phase Transitions in Quantum Systems

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