Mathematical properties of gap-functions in strongly coupled superconductors |
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Authors: | R Englman M Weger |
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Institution: | (1) Soreq Nuclear Research Center, 70600 Yavne, Israel;(2) Racah Institute for Physics, The Hebrew University, Jerusalem, Israel |
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Abstract: | We consider the Dyson equation associated with the BCS superconducting state from a mathematical point of view. The Dyson equation gives rise to a modified gap equation that is similar to the BCS gap equation, but with a different kernel. We first show that for strong coupling (such that the McMillan parameter | | 1) both the real and imaginary parts of the solution (E) of the modified gap equation alternate in sign as function of the excitation energyE, the periods
being 4 0 for positive and 4 0/3 for negative . ( 0 is the frequency of an Einstein spectrum of phonons). A closed, algebraic approximation to (E) is 2| | 0logcotan( E/
)]. Finally, the poles of the kernel of the integral equation are located in the complex-E plane. For the new-type, oscillatory solution of the modified gap equation the analogue of the causal (zero-temperature) Green's function is shown to have different analytic properties from those of the smooth Eliashberg solution of BCS theory. |
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