Some constants related to numerical ranges

In an attempt to progress towards proving the conjecture the numerical range W (A) is a 2--spectral set for the matrix A, we propose a study of various constants. We review some partial results, many problems are still open. We describe our corresponding numerical tests

Energy of N Cooper pair by analytically solving Richardson-Gaudin equations

This Letter provides the solution to a yet unsolved basic problem of Solid State Physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 year old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these $N$ pairs via $N$ parameters coupled through $N$ non-linear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the $N$-pair energy depends on $N$ as $N(N-1)$ only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only

K-spectral sets and intersections of disks of the Riemann sphere

We prove that if two closed disks X_1 and X_2 of the Riemann sphere are spectral sets for a bounded linear operator A on a Hilbert space, then the intersection X_1\cap X_2 is a complete (2+2/\sqrt{3})-spectral set for A. When the intersection of X_1 and X_2 is an annulus, this result gives a positive answer to a question of A.L. Shields (1974).Comment: 10 page

Convex domains and K-spectral sets

Let $\Omega$ be an open convex domain of the complex plane. We study constants K such that $\Omega$ is K-spectral or complete K-spectral for each continuous linear Hilbert space operator with numerical range included in $\Omega$. Several approaches are discussed.Comment: the introduction was changed and some remarks have been added. 26 pages ; to appear in Math.

A lenticular version of a von Neumann inequality

International audienceWe generalize to lens-shaped domains the classical von Neumann inequality for the disk

Numerical radius and distance from unitary operators

Denote by w(A) the numerical radius of a bounded linear operator A acting on Hilbert space. Suppose that A is invertible and that the numerical radius of A and of its inverse are no greater than 1+e for some non-negative e. It is shown that the distance of A from unitary operators is less or equal than a constant times $e^{1/4}$. This generalizes a result due to J.G. Stampfli, which is obtained for e = 0. An example is given showing that the exponent 1/4 is optimal. The more general case of the operator $\rho$-radius is discussed for $\rho$ between 1 and 2.Comment: Final version : new title and several other change

Intersections of several disks of the Riemann sphere as K-spectral sets

We prove that if $n$ closed disks $D_1, D_2, ..., D_n$, of the Riemann sphere are spectral sets for a bounded linear operator $A$ on a Hilbert space, then their intersection $D_1\cap D_2...\cap D_n$ is a complete $K$-spectral set for $A$, with $K\leq n+n(n-1)/\sqrt3$. When $n=2$ and the intersection $X_1\cap X_2$ is an annulus, this result gives a positive answer to a question of A.L. Shields (1974).Comment: 4 figures, a remark suggested by Vern Paulsen was adde

Two-Cooper-pair problem and the Pauli exclusion principle

While the one-Cooper pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for N=2 already. The two-Cooper pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to $N$ pairs, but no precise description of this continuous change has been given so far. Using Richardson procedure, we here show that Pauli blocking increases the free part of the two-pair ground state energy, but decreases the binding part when compared to two isolated pairs - the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease of the pair binding while the gap increases strongly indicates that, at odd with common belief, the average pair binding energy cannot be of the order of the gap.Comment: 9 pages, no figures, final versio