1,171 research outputs found
Note on the paper of Fu and Wong on strictly pseudoconvex domains with K\"ahler--Einstein Bergman metrics
It is shown that the Ramadanov conjecture implies the Cheng conjecture. In
particular it follows that the Cheng conjecture holds in dimension two
Local trace formulae and scaling asymptotics in Toeplitz quantization
A trace formula for Toeplitz operators was proved by Boutet de Monvel and
Guillemin in the setting of general Toeplitz structures. Here we give a local
version of this result for a class of Toeplitz operators related to continuous
groups of symmetries on quantizable compact symplectic manifolds. The local
trace formula involves certain scaling asymptotics along the clean fixed locus
of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics
of the equivariant components of the Szeg\"o kernel along the diagonal
Scaling asymptotics for quantized Hamiltonian flows
In recent years, the near diagonal asymptotics of the equivariant components
of the Szeg\"{o} kernel of a positive line bundle on a compact symplectic
manifold have been studied extensively by many authors. As a natural
generalization of this theme, here we consider the local scaling asymptotics of
the Toeplitz quantization of a Hamiltonian symplectomorphism, and specifically
how they concentrate on the graph of the underlying classical map
Energy nonequipartition in a sheared granular mixture
The kinetic granular temperatures of a binary granular mixture in simple
shear flow are determined from the Boltzmann kinetic theory by using a Sonine
polynomial expansion. The results show that the temperature ratio is clearly
different from unity (as may be expected since the system is out of
equilibrium) and strongly depends on the restitution coefficients as well as on
the parameters of the mixture. The approximate analytical calculations are
compared with those obtained from Monte Carlo simulations of the Boltzmann
equation showing an excellent agreement over the range of parameters
investigated. Finally, the influence of the temperature differences on the
rheological properties is also discussed.Comment: 3 figure
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
The Generalized Dirichlet to Neumann map for the KdV equation on the half-line
For the two versions of the KdV equation on the positive half-line an
initial-boundary value problem is well posed if one prescribes an initial
condition plus either one boundary condition if and have the
same sign (KdVI) or two boundary conditions if and have
opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map
for the above problems means characterizing the unknown boundary values in
terms of the given initial and boundary conditions. For example, if
and are given for the KdVI
and KdVII equations, respectively, then one must construct the unknown boundary
values and , respectively. We
show that this can be achieved without solving for by analysing a
certain ``global relation'' which couples the given initial and boundary
conditions with the unknown boundary values, as well as with the function
, where satisifies the -part of the associated
Lax pair evaluated at . Indeed, by employing a Gelfand--Levitan--Marchenko
triangular representation for , the global relation can be solved
\emph{explicitly} for the unknown boundary values in terms of the given initial
and boundary conditions and the function . This yields the unknown
boundary values in terms of a nonlinear Volterra integral equation.Comment: 21 pages, 3 figure
The Unified Method: II NLS on the Half-Line with -Periodic Boundary Conditions
Boundary value problems for integrable nonlinear evolution PDEs formulated on
the half-line can be analyzed by the unified method introduced by one of the
authors and used extensively in the literature. The implementation of this
general method to this particular class of problems yields the solution in
terms of the unique solution of a matrix Riemann-Hilbert problem formulated in
the complex -plane (the Fourier plane), which has a jump matrix with
explicit -dependence involving four scalar functions of , called
spectral functions. Two of these functions depend on the initial data, whereas
the other two depend on all boundary values. The most difficult step of the new
method is the characterization of the latter two spectral functions in terms of
the given initial and boundary data, i.e. the elimination of the unknown
boundary values. For certain boundary conditions, called linearizable, this can
be achieved simply using algebraic manipulations. Here, we first present an
effective characterization of the spectral functions in terms of the given
initial and boundary data for the general case of non-linearizable boundary
conditions. This characterization is based on the analysis of the so-called
global relation and on the introduction of the so-called
Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the
spectral functions. We then concentrate on the physically significant case of
-periodic Dirichlet boundary data. After presenting certain heuristic
arguments which suggest that the Neumann boundary values become periodic as
, we show that for the case of the NLS with a sine-wave as
Dirichlet data, the asymptotics of the Neumann boundary values can be computed
explicitly at least up to third order in a perturbative expansion and indeed at
least up to this order are asymptotically periodic.Comment: 29 page
Quantum ergodicity of C* dynamical systems
This paper contains a very simple and general proof that eigenfunctions of
quantizations of classically ergodic systems become uniformly distributed in
phase space. This ergodicity property of eigenfunctions f is shown to follow
from a convexity inequality for the invariant states (Af,f). This proof of
ergodicity of eigenfunctions simplifies previous proofs (due to A.I.
Shnirelman, Colin de Verdiere and the author) and extends the result to the
much more general framework of C* dynamical systems.Comment: Only very minor differences with the published versio
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
Scattering Theory for Jacobi Operators with Steplike Quasi-Periodic Background
We develop direct and inverse scattering theory for Jacobi operators with
steplike quasi-periodic finite-gap background in the same isospectral class. We
derive the corresponding Gel'fand-Levitan-Marchenko equation and find minimal
scattering data which determine the perturbed operator uniquely. In addition,
we show how the transmission coefficients can be reconstructed from the
eigenvalues and one of the reflection coefficients.Comment: 14 page
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