Herglotz property

Gqo( ) is analytic and ImC7oo( ) < 0. We also see that any approximant of (5.3), where a are real and are positive, satisfy the Herglotz property since it represents a half linear chain of variables of energies a and nearest-neighbor interaction b . 

This integral, to be taken in the Stieltjes-Lebesgue sense, is known as the Stieltjes transform of the spectral density of states n( ). In general the Stieltjes transform defines two analytic fimctions, one in the upper complex plane and the other in the lower complex plane, and we refer to the former consistently with the Herglotz properties of Gqq(E). 

An important property which is preserved in the relaxation case is the Herglotz property of the Green s function Goo( ). It can be expanded in the form... 

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