94 research outputs found
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 pairs via parameters coupled through
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 -pair
energy depends on as 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 be an open convex domain of the complex plane. We study
constants K such that is K-spectral or complete K-spectral for each
continuous linear Hilbert space operator with numerical range included in
. 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 . 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 -radius is discussed for
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 closed disks , of the Riemann sphere
are spectral sets for a bounded linear operator on a Hilbert space, then
their intersection is a complete -spectral set for
, with . When and the intersection 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 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
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