S-matrix eigenchannels and eigenphases

A simple parameterization of the S matrix in the form (33) or (34) might appear impractical when many channels, and hence many fitting parameters T, and Er, are involved. However, the diagonalization of the S matrix greatly simplifies such parametrization and also provides a clear physical idea. 

The background S matrix (35) is also diagonalized by an orthogonal matrix Ob  

The background eigenphase sum is defined as 8b(E) = X V The diago-nalization (37) may be achieved by choosing Ob to be 

we note that the determinants of matrices have a property that det(AB) = det(BA) = (detA)(detB), and hence, the determinant of a matrix remains intact by a unitary transformation of this matrix. Using this fact, one may show that the eigenphase sum and the background eigenphase sum may be related in a simple way to the determinants of the S and Sb matrices as 

To study the properties of the eigenphase sum in resonance scattering, we further note that det(f + cP) = 1 + c, where c is any constant. This 

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