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Journal of Physics A Mathematical and General

On the one-point Friedrichs model

G Duerinckx

Published Feb 1, 1984• 11 citations
Publisher

Abstract

The author investigates both analytically and numerically the one-point Friedrichs model characterised by a lambda V interaction with a non-square integrable coupling function nu ( omega )=1. If the continuous spectrum of the Hamiltonian is bounded from below, a form factor is needed to avoid some infinities and the solution of the spontaneous emission problem then depends on an unphysical cut-off parameter alpha . This solution is denoted by p(t; alpha ) and its alpha -dependence may be removed by taking the appropriate limit. Indeed, the author finds that: Lim alpha to infinity p(t; alpha )=1. If, on the contrary, the continuous spectrum is assumed to be infinite, then no form factor is needed and p(t)=exp(-2 pi lambda 2 t).

DOI
10.1088/0305-4470/17/2/024

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The author investigates both analytically and numerically the one-point Friedrichs model characterised by a lambda V interaction with a non-square integrable coupling function nu ( omega )=1. If the continuous spectrum of the Hamiltonian is bounded from below, a form factor is needed to avoid some infinities and the solution of the spontaneous emission problem then depends on an unphysical cut-off parameter alpha . This solution is denoted by p(t; alpha ) and its alpha -dependence may be removed by taking the appropriate limit. Indeed, the author finds that: Lim alpha to infinity p(t; alpha )=1. If, on the contrary, the continuous spectrum is assumed to be infinite, then no form factor is needed and p(t)=exp(-2 pi lambda 2 t).
The Friedrichs model consists in an interaction between the bound state of a quantum system (say, an atom) and an external field, interaction which produces a resonance. This interaction intertwines the bound state with the eigenvectors of the continuous spectrum of the field Hamiltonian and depends on a coupling parameter. The resonance appears as a pole of the analytic continuation of the reduced resolvent corresponding to the interacting Hamiltonian. This model admits generalizations like, for instance, adding bound states and/or degeneracy to the field Hamiltonian. We examine what happens as the imaginary component of the eigenvalue associated with a resonant energy approaches zero.
The effect of quantum fluctuations on the Peierls instability is investigated for two models, the Su-Schrieffer-Heeger model and the molecular model introduced by Holstein. The author finds for both models that when the ionic mass M not=0 the beta function has the form beta ( lambda R )=-(1-(1/N)f( lambda R / alpha R )) lambda R 2 / pi with alpha R being an invariant quantity alpha R =M R lambda R exp(-2 pi / lambda R ). This allows the study of the crossover behaviour M to 0 to M to infinity . In the limit M to 0 he obtains exactly the Gross-Neveau model (M=O) which has a Peierls gap for spin 1 / 2 (N=2). In the limit M to infinity the mean field Peierls gap is found. The molecular model with M=0 is different from the Gross-Neveau model (M=0). It is similar to the non-linear sigma model with the beta -function: beta ( lambda R )=-(1-2/N) lambda R 2 / pi . In this case there is no Peierls gap for spin 1 / 2 (N=2) or for spinless (N=1) electrons. The analysis is performed in a 1/N expansion. The renormalisation group method is used to compute the beta -function with an anisotropic parameter alpha R which controls the crossover behaviour. The renormalisation functions are obtained by a dimensional regularisation method in 2- in dimensions where simple poles and in (1/ in ) are cancelled.
We study the spectral properties of a spin-boson Hamiltonian that depends on two continuous parameters 0 < or = Lambda < infinity (interaction strength) and 0 < or = alpha < or = pi/2 (integrability switch). In the classical limit, this system has two distinct integrable regimes, alpha=0 and alpha=pi/2 . For each integrable regime we can express the quantum Hamiltonian as a function of two action operators. Their eigenvalues (multiples of variant Planck's over 2pi ) are the natural quantum numbers for the complete level spectrum. This functional dependence cannot be extended into the nonintegrable regime (0<alpha<pi/2) . Here level crossings are prohibited and the level spectrum is naturally described by a single (energy sorting) quantum number. In consequence, the tracking of individual eigenstates along closed paths through both regimes leads to conflicting assignments of quantum numbers. This effect is a useful and reliable indicator of quantum chaos-a diagnostic tool that is independent of any level-statistical analysis.
The authors have studied the behaviour of two- and three-dimensional self avoiding walks confined to a wedge of wedge angle alpha . Series have been obtained and analysed for the (angular dependent) critical exponents characterising various edge susceptibilities. In terms of a general scaling form for the edge free energy, f e approximately t (d-2) nu psi e (ht -y0 nu , h 1 t -y1 nu , h 2 t -y2 nu ), they find for the two-dimensional case the following scaling indices: y0=91/48, y1=3/8, y2( alpha )=-5 pi /8 alpha . They argue that these results are exact, from which follow all exponents for the bulk, surface and edge problem. In three dimensions the authors obtain y0 approximately=2.488, y1=0.65+or-0.02, y2( alpha )= alpha +b pi / alpha where a=0.51+or-0.04, b=-0.847+or-0.017, which, for y2, is precisely of the functional form given by mean-field theory, y2( alpha )= 1 / 2 - pi /2. They argue that a= 1 / 2 for all three-dimensional O(N) models. This simple angular dependence of y2 is different from that suggested by Cardy's one-loop epsilon -expansion, (1983). For the square lattice, they have also studied the case in which the wedge is rotated through an angle of pi /4, and find that the various exponents are unchanged. For the three-dimensional Ising model in a wedge, analogy with the SAW results, plus mean-field results in conjunction with RG and series work yield y0 approximately=2.485, y1=0.71+or-0.02 and y2=a+b pi / alpha with a= 1 / 2 and b=-0.79+or-0.02.
The compatibility of irreversibility, as expressed by the second law of thermodynamics, with ordinary quantum mechanics requires an extension of the latter. Such an extension of a reversible formalism is made possible by the presence of self-energy processes (already found in quantum electrodynamics), which allows for the widening of the dynamics through the introduction of specific additional variables leading to the recognition of such processes. A projection technique modeled on that introduced long ago by the Brussels group defines a subdynamics that is shown to contain the starting formulation in its entirety. (It corresponds to a restricted class of initial conditions, namely the reversible ones.) A new kinetic description is obtained in which, e.g., collision processes for privileged components arise from the original interaction. This method is applied successively to various models, and we focus in this paper on the possible presence of temporal behaviour excluded in ordinary quantum mechanics. Among the differences between this framework and that of ordinary quantum mechanics, the possibility appears for the purely exponential decay of a particle, while ordinary quantum mechanics requires long time tails associated with the finite lower bound in the spectrum of the Hamiltonian.