21,829 research outputs found
A compactness lemma and its application to the existence of minimizers for the liquid drop model
The ancient Gamow liquid drop model of nuclear energies has had a renewed
life as an interesting problem in the calculus of variations: Find a set
with given volume A that minimizes the sum of its
surface area and its Coulomb self energy. A ball minimizes the former and
maximizes the latter, but the conjecture is that a ball is always a minimizer
-- when there is a minimizer. Even the existence of minimizers for this
interesting geometric problem has not been shown in general. We prove the
existence of the absolute minimizer (over all ) of the energy divided by
(the binding energy per particle). A second result of our work is a general
method for showing the existence of optimal sets in geometric minimization
problems, which we call the `method of the missing mass'. A third point is the
extension of the pulling back compactness lemma from to .Comment: 16 page
Sharp constants in several inequalities on the Heisenberg group
We derive the sharp constants for the inequalities on the Heisenberg group
H^n whose analogues on Euclidean space R^n are the well known
Hardy-Littlewood-Sobolev inequalities. Only one special case had been known
previously, due to Jerison-Lee more than twenty years ago. From these
inequalities we obtain the sharp constants for their duals, which are the
Sobolev inequalities for the Laplacian and conformally invariant fractional
Laplacians. By considering limiting cases of these inequalities sharp constants
for the analogues of the Onofri and log-Sobolev inequalities on H^n are
obtained. The methodology is completely different from that used to obtain the
R^n inequalities and can be (and has been) used to give a new, rearrangement
free, proof of the HLS inequalities.Comment: 30 pages; addition of Corollary 2.3 and some minor changes; to appear
in Annals of Mathematic
Norms of quantum Gaussian multi-mode channels
We compute the norm of a general Gaussian
gauge-covariant multi-mode channel for any , where is a Schatten space. As a consequence, we verify the Gaussian optimizer
conjecture and the multiplicativity conjecture in these cases.Comment: 9 pages; minor changes; to appear in J. Math. Phy
Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality
We give a new proof of certain cases of the sharp HLS inequality. Instead of
symmetric decreasing rearrangement it uses the reflection positivity of
inversions in spheres. In doing this we extend a characterization of the
minimizing functions due to Li and Zhu.Comment: 15 pages; references added and minor change
Maximizers for the Stein-Tomas inequality
We give a necessary and sufficient condition for the precompactness of all
optimizing sequences for the Stein-Tomas inequality. In particular, if a
well-known conjecture about the optimal constant in the Strichartz inequality
is true, we obtain the existence of an optimizer in the Stein-Tomas inequality.
Our result is valid in any dimension.Comment: 37 page
Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators
We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schrödinger-like operators remain true, with possibly different constants, when the critical Hardy-weight C │x│^(-2) is subtracted from the Laplace operator. We do so by first establishing a Sobolev inequality for such operators. Similar results are true for fractional powers of the Laplacian and the Hardy-weight and, in particular, for relativistic Schrödinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend, for the first time, the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge Zɑ = 2/π, for ɑ less than some critical value
On an Extension Problem for Density Matrices
We investigate the problem of the existence of a density matrix rho on the
product of three Hilbert spaces with given marginals on the pair (1,2) and the
pair (2,3). While we do not solve this problem completely we offer partial
results in the form of some necessary and some sufficient conditions on the two
marginals. The quantum case differs markedly from the classical (commutative)
case, where the obvious necessary compatibility condition suffices, namely,
trace_1 (rho_{12}) = \trace_3 (rho_{23}).Comment: 12 pages late
Ground state energy of large polaron systems
The last unsolved problem about the many-polaron system, in the
Pekar-Tomasevich approximation, is the case of bosons with the
electron-electron Coulomb repulsion of strength exactly 1 (the 'neutral case').
We prove that the ground state energy, for large , goes exactly as
, and we give upper and lower bounds on the asymptotic coefficient
that agree to within a factor of .Comment: 16 page
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