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It is shown that the numerical range of a linear operator on a Hilbert space is a (complete) (1+√2)-spectral set. The proof relies, among other things, on the behavior of the Cauchy transform of the conjugates of holomorphic functions. [ABSTRACT FROM AUTHOR]

We approach the analytical resolution of Richardson's equations from which the exact energy of N Cooper pairs can be obtained, through an expansion in the dimensionless parameter associated to sample volume. We first derive the explicit solutions of these Richardson's equations for the lowest N's, up to fourth order in this parameter. From them, we deduce a compact expression of the N-pair energy, that we recover using a more general procedure. Our results support the recently found fact that the expression of the N-pair energy obtained at the lowest order stays valid up to the dense limit, the next order term being here shown to be underextensive due to a set of non-trivial cancellations. [ABSTRACT FROM AUTHOR]