Continuous spectrum, of eigenvalues

Equation (3.73) may also be obtained from (3.42). The contribution of Xi to the ratio of determinants in (3.70) is equal to. Apart from the bound states, there is a continuous spectrum of eigenvalues e whose contribution to (3.70) may be shown [Vainshtein et al. 1982] to equal Finally (3.70) gives... 

We notice that it is the analytic continuation which has the effect of breaking the time-reversal symmetry. If we contented ourselves with the continuous spectrum of eigenvalues with Re = 0, we would obtain the unitary group of time evolution valid for positive and negative times. The unitary spectral decomposition is as valid as the spectral decompositions of the forward or... 

An important example of a maximal set of commuting observables with a continuous spectrum of eigenvalues is provided by the operators 4r representing the position coordinates of a set of particles. The state function in terms of the coordinate representation is given by... 

The net effect of these restrictions is that in some cases, E may have any value then we speak of a continuous spectrum of eigenvalues of E. In other cases, E may be restricted to certain particular values then we have a discrete spectrum of eigenvalues. In these latter cases we say that E is quantized. Ordinarily, for each boundary condition we impose, we introduce a quantization of some observable. 

In the case of a continuous spectrum of eigenvalues, in other words when the eigenvalues k ) are defined by some continuous variable the relaxation spectrum H(r) can rewritten as (see [12] and also Sect. 7.5.1) ... 

For a continuous spectrum of eigenkets with non-degenerate eigenvalues, it is more convenient to write the eigenvalue equation (3.45) in the form... 

In general, relation (21) is false one only has exp(tpoint spectrum (constituted of eigenvalues), and for the residual spectrum, but it is false for the continuous spectrum there exist abstract operators A with empty spectrum, but such that the continuous spectrum of S t) is the circle of radius exp(Trt) thus the solutions of equation (20) grow exponentially in time. 

It should be noted that the spectrum of eigenvalues of this problem is continuous, 0 < P < oo. 

Note that the spectrum of eigenvalues of the problem (the integration variable a in (6.10.14)) is continuous. 

The discreteness in this new theory refers to the discrete nature of all measurements. Each measurement fixes a particle s position Xn and time Both and tn are allowed to take on any value in the spectrum of continuous eigenvalues of the operators (xn)op and (tn)op. Notice also that in this discrete theory there is no Hamiltonian and no Lagrangian only Action. 

The spectrum of the operator q consists of the points in euclidean three space. The eigenfunctions x > are not normalizable in the usual way as they correspond to eigenvalues in the continuous spectrum, but are normalized to a 8-function... 

In (10-124) and (10-125), n and m refer to the eigenvalues of a complete set of commuting observables so Snm stands for a delta function m those observables in the set that have a continuous spectrum, and a Kronecker 8 in those that have a discrete spectrum. 

For our purpose, however, it is necessary to have equality sign in (7.13), For otherwise the multiplicity of XQ would abruptly increase or X0 would diffuse into continuous spectrum when the perturbation kII0) is switched on, or roughly speaking, there would appear too many eigenvalues, and p.m. would be invalidated. Thus we are obliged to introduce the following further assumption ... 

It should be remarked that Ex. 3 is analogous to the case of the Hamiltonian of the Stark effect. In these cases the diffusion of the eigenvalue into continuous spectrum takes place, and p.m. is not valid in the sense hitherto considered. Nevertheless p.m. is applied to the Stark effect with successful results. In order to justify the application of p.m. to such problems, more profound study is necessary than that given here. We shall discuss the subject in the next chapter. 

Whether or not such a tc exists depends on the nature of the bath. It is certainly necessary that the bath has a continuous, or at least very dense, spectrum with eigenvalues that contribute equally to the interaction in order to have a smooth strength function g(k). Also this function must be virtually constant over the range over which the interaction is felt, i.e., the line width, as seen in 2. It would be desirable to have more concrete criteria, but it is hard to formulate them. 

For the case of a point spectrum, oPr the spectral function,p, is a constant function of E, with discrete steps dco,t at each point eigenvalue Ek. For the continuous spectrum, aAC, one obtains... 

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