Eckart singularity

It should be noted that the A2 operators of (160) and (167) display Eckart singularities [58], but for the results reported in Sec. V these singularities have not produced special difficulties. At higher energies and/or for other systems difficulties may arise and will have to be dealt with accordingly. 

It can be proved that the decomposition is always possible and that the solution is unique (except for the algebraic signs of the columns of U and V) [3]. Singular value decomposition of a rectangular table is an extension of the classical work of Eckart and Young [4] on the decomposition of matrices. The decomposition of X into U, V and A is illustrated below using a 4x3 data table which has been adapted from V an Borm [5]. (A similar example has been used for the introduction to PC A in Chapter 17.)... 

The other is of the singularities of the complex classical trajectories [19], which is peculiar to time-continuous systems and plays an important role to understand tunnehng phenomena of barrier potentials [25]. First, we briefly explain the role of singularities of classical trajectory by taking the static Eckart barrier as a simple example. 

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