4,779 research outputs found
Semiclassical almost isometry
Let M be a complex projective manifold, and L an Hermitian ample line bundle
on it. A fundamental theorem of Gang Tian, reproved and strengthened by
Zelditch, implies that the Khaeler form of L can be recovered from the
asymptotics of the projective embeddings associated to large tensor powers of
L. More precisely, with the natural choice of metrics the projective embeddings
associated to the full linear series |kL| are asymptotically symplectic, in the
appropriate rescaled sense. In this article, we ask whether and how this result
extends to the semiclassical setting. Specifically, we relate the Weinstein
symplectic structure on a given isodrastic leaf of half-weighted
Bohr-Sommerfeld Lagrangian submanifolds of M to the asymptotics of the the
pull-back of the Fubini-Study form under the semiclassical projective maps
constructed by Borthwick, Paul and Uribe.Comment: exposition improve
Slow Diffeomorphisms of a Manifold with Two Dimensions Torus Action
The uniform norm of the differential of the n-th iteration of a
diffeomorphism is called the growth sequence of the diffeomorphism. In this
paper we show that there is no lower universal growth bound for volume
preserving diffeomorphisms on manifolds with an effective two dimensions torus
action by constructing a set of volume-preserving diffeomorphisms with
arbitrarily slow growth.Comment: 12 p
Lower order terms in Szego type limit theorems on Zoll manifolds
This is a detailed version of the paper math.FA/0212273. The main motivation
for this work was to find an explicit formula for a "Szego-regularized"
determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll
manifold. The idea of the Szego-regularization was suggested by V. Guillemin
and K. Okikiolu. They have computed the second term in a Szego type expansion
on a Zoll manifold of an arbitrary dimension. In the present work we compute
the third asymptotic term in any dimension. In the case of dimension 2, our
formula gives the above mentioned expression for the Szego-redularized
determinant of a zeroth order PsDO. The proof uses a new combinatorial
identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This
identity is related to the distribution of the maximum of a random walk with
i.i.d. steps on the real line. The proof of this combinatorial identity
together with historical remarks and a discussion of probabilistic and
algebraic connections has been published separately.Comment: 39 pages, full version, submitte
Legendrian Distributions with Applications to Poincar\'e Series
Let be a compact Kahler manifold and a quantizing holomorphic
Hermitian line bundle. To immersed Lagrangian submanifolds of
satisfying a Bohr-Sommerfeld condition we associate sequences , where is a
holomorphic section of . The terms in each sequence concentrate
on , and a sequence itself has a symbol which is a half-form,
, on . We prove estimates, as , of the norm
squares in terms of . More generally, we show that if and
are two Bohr-Sommerfeld Lagrangian submanifolds intersecting
cleanly, the inner products have an
asymptotic expansion as , the leading coefficient being an integral
over the intersection . Our construction is a
quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of . We prove
that the Poincar\'e series on hyperbolic surfaces are a particular case, and
therefore obtain estimates of their norms and inner products.Comment: 41 pages, LaTe
Topological constraints on spiral wave dynamics in spherical geometries with inhomogeneous excitability
We analyze the way topological constraints and inhomogeneity in the
excitability influence the dynamics of spiral waves on spheres and punctured
spheres of excitable media. We generalize the definition of an index such that
it characterizes not only each spiral but also each hole in punctured,
oriented, compact, two-dimensional differentiable manifolds and show that the
sum of the indices is conserved and zero. We also show that heterogeneity and
geometry are responsible for the formation of various spiral wave attractors,
in particular, pairs of spirals in which one spiral acts as a source and a
second as a sink -- the latter similar to an antispiral. The results provide a
basis for the analysis of the propagation of waves in heterogeneous excitable
media in physical and biological systems.Comment: 5 pages, 6 Figures, major revisions, accepted for publication in
Phys. Rev.
New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces
A generalized version of Bertrand's theorem on spherically symmetric curved
spaces is presented. This result is based on the classification of
(3+1)-dimensional (Lorentzian) Bertrand spacetimes, that gives rise to two
families of Hamiltonian systems defined on certain 3-dimensional (Riemannian)
spaces. These two systems are shown to be either the Kepler or the oscillator
potentials on the corresponding Bertrand spaces, and both of them are maximally
superintegrable. Afterwards, the relationship between such Bertrand
Hamiltonians and position-dependent mass systems is explicitly established.
These results are illustrated through the example of a superintegrable
(nonlinear) oscillator on a Bertrand-Darboux space, whose quantization and
physical features are also briefly addressed.Comment: 13 pages; based in the contribution to the 28th International
Colloquium on Group Theoretical Methods in Physics, Northumbria University
(U.K.), 26-30th July 201
Three natural mechanical systems on Stiefel varieties
We consider integrable generalizations of the spherical pendulum system to
the Stiefel variety for a certain metric. For the case
of V(n,2) an alternative integrable model of the pendulum is presented.
We also describe a system on the Stiefel variety with a four-degree
potential. The latter has invariant relations on which provide the
complete integrability of the flow reduced on the oriented Grassmannian variety
.Comment: 14 page
Complex zeros of real ergodic eigenfunctions
We determine the limit distribution (as ) of complex
zeros for holomorphic continuations \phi_{\lambda}^{\C} to Grauert tubes of
real eigenfunctions of the Laplacian on a real analytic compact Riemannian
manifold with ergodic geodesic flow. If is an
ergodic sequence of eigenfunctions, we prove the weak limit formula
\frac{1}{\lambda_j} [Z_{\phi_{j_k}^{\C}}] \to \frac{i}{\pi} \bar{\partial}
{\partial} |\xi|_g, where [Z_{\phi_{j_k}^{\C}}] is the current of
integration over the complex zeros and where is with respect
to the adapted complex structure of Lempert-Sz\"oke and Guillemin-Stenzel.Comment: Added some examples and references. Also added a new Corollary, and
corrected some typo
On Bohr-Sommerfeld bases
This paper combines algebraic and Lagrangian geometry to construct a special
basis in every space of conformal blocks, the Bohr-Sommerfeld (BS) basis. We
use the method of [D. Borthwick, T. Paul and A. Uribe, Legendrian distributions
with applications to the non-vanishing of Poincar\'e series of large weight,
Invent. math, 122 (1995), 359-402, preprint hep-th/9406036], whereby every
vector of a BS basis is defined by some half-weighted Legendrian distribution
coming from a Bohr-Sommerfeld fibre of a real polarization of the underlying
symplectic manifold. The advantage of BS bases (compared to bases of theta
functions in [A. Tyurin, Quantization and ``theta functions'', Jussieu preprint
216 (Apr 1999), e-print math.AG/9904046, 32pp.]) is that we can use information
from the skillful analysis of the asymptotics of quantum states. This gives
that Bohr-Sommerfeld bases are unitary quasi-classically. Thus we can apply
these bases to compare the Hitchin connection with the KZ connection defined by
the monodromy of the Knizhnik-Zamolodchikov equation in combinatorial theory
(see, for example, [T. Kohno, Topological invariants for 3-manifolds using
representations of mapping class group I, Topology 31 (1992), 203-230; II,
Contemp. math 175} (1994), 193-217]).Comment: 43 pages, uses: latex2e with amsmath,amsfonts,theore
Small oscillations and the Heisenberg Lie algebra
The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems
for quadratic Hamiltonians of on coadjoint orbits of the
Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie
algebra that admits an ad-invariant metric. Its quadratic induces
the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that
one on . This system is a Lax pair equation whose solution can
be computed with help of the Adjoint representation. For a certain class of
functions, the Poisson commutativity on the coadjoint orbits in
is related to the commutativity of a family of derivations of the
2n+1-dimensional Heisenberg Lie algebra . Therefore the complete
integrability is related to the existence of an n-dimensional abelian
subalgebra of certain derivations in . For instance, the motion
of n-uncoupled harmonic oscillators near an equilibrium position can be
described with this setting.Comment: 17 pages, it contains a theory about small oscillations in terms of
the AKS schem
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