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... 

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. 

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. 

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). 

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