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Groups, continuous eigenvalues

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... [Pg.99]

In addition to the continuous spectrum there is an infinite number of discrete eigenvalues which axe essentially lined up along a parallel to the imaginary axis— this is another illustration of the fact that the underlying semi-group is not analytic. [Pg.222]

With certain additional restrictions T t) is completely continuous for t 3t, T as described in the first paragraph of this section. This is true, for example, when F is a three-dimensional set, K is given by (3.2) and h is integrable over D x V x V and bounded. It is also true for ri-velocity group theory, n 1. When T t) is completely continuous it has a discrete spectrum, plus possibly the point zero, where the Ay are the eigenvalues of A, The spectrum may be empty. [Pg.149]

The fields are constructed from the solution of Eq. (30-32b) for hy and Eq. (30-33). This leads to the components in Table 12-10, where U, V and W are now defined in terms of n, nl, and The differences with the TM mode fields in Table 12-1 arise solely due to the changes in refractive-index values. This is evident from comparing Eqs. (30-32b) and (30-33) with Eq. (12-20) and (12-23), respectively. The eigenvalue equations in Table 12-11 follow by demanding continuity of and hy at the interfaces, and the remaining expressions follow from Table 11-1, with the exception of the group velocity, which is calculated from the modified form of Eq. (31-31). [Pg.273]


See other pages where Groups, continuous eigenvalues is mentioned: [Pg.66]    [Pg.110]    [Pg.53]    [Pg.208]    [Pg.131]    [Pg.301]   
See also in sourсe #XX -- [ Pg.61 , Pg.62 ]




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Eigenvalue

Group (continued

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