955 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
(Convex) level sets integration
The paper addresses the problem of recovering a pseudoconvex function from the normal cones to its level sets that we call the convex level sets integration problem. An important application is the revealed preference problem. Our main result can be described as integrating a maximally cyclically pseudoconvex multivalued map that sends vectors or âbundlesâ of a Euclidean space to convex sets in that space. That is, we are seeking a pseudoconvex (real) function such that the normal cone at each boundary point of each of its lower level sets contains the set value of the multivalued map at the same point. This raises the question of uniqueness of that function up to rescaling. Even after normalizing the function long an orienting direction, we give a counterexample to its uniqueness. We are, however, able to show uniqueness under a condition motivated by the classical theory of ordinary differential equations
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
A-stable Runge-Kutta methods for semilinear evolution equations
We consider semilinear evolution equations for which the linear part
generates a strongly continuous semigroup and the nonlinear part is
sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the
existence of solutions which are temporally smooth in the norm of the lowest
rung of the scale for an open set of initial data on the highest rung of the
scale. Under the same assumptions, we prove that a class of implicit,
-stable Runge--Kutta semidiscretizations in time of such equations are
smooth as maps from open subsets of the highest rung into the lowest rung of
the scale. Under the additional assumption that the linear part of the
evolution equation is normal or sectorial, we prove full order convergence of
the semidiscretization in time for initial data on open sets. Our results
apply, in particular, to the semilinear wave equation and to the nonlinear
Schr\"odinger equation
- âŠ