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 |λ|≫1) both the real and imaginary parts of the solution Δ(E) of the modified gap equation alternate in sign as function of the excitation energy E, the periods {Mathematical expression} being 4ω0 for positive λ and 4ω0/3 for negative λ. (ω0 is the frequency of an Einstein spectrum of phonons). A closed, algebraic approximation to Δ(E) is 2|λ|ω0log[cotan(πE/ {Mathematical expression})]. 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.
Original language | English |
---|---|
Pages (from-to) | 417-424 |
Number of pages | 8 |
Journal | European Physical Journal B |
Volume | 81 |
Issue number | 3 |
DOIs | |
State | Published - Oct 1990 |