937 research outputs found
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
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
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
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
Local trace formulae and scaling asymptotics in Toeplitz quantization, II
In the spectral theory of positive elliptic operators, an important role is
played by certain smoothing kernels, related to the Fourier transform of the
trace of a wave operator, which may be heuristically interpreted as smoothed
spectral projectors asymptotically drifting to the right of the spectrum. In
the setting of Toeplitz quantization, we consider analogues of these, where the
wave operator is replaced by the Hardy space compression of a linearized
Hamiltonian flow, possibly composed with a family of zeroth order Toeplitz
operators. We study the local asymptotics of these smoothing kernels, and
specifically how they concentrate on the fixed loci of the linearized dynamics.Comment: Typos corrected. Slight expository change
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
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
The first coefficients of the asymptotic expansion of the Bergman kernel of the spin^c Dirac operator
We establish the existence of the asymptotic expansion of the Bergman kernel
associated to the spin-c Dirac operators acting on high tensor powers of line
bundles with non-degenerate mixed curvature (negative and positive eigenvalues)
by extending the paper " On the asymptotic expansion of Bergman kernel "
(math.DG/0404494) of Dai-Liu-Ma. We compute the second coefficient b_1 in the
asymptotic expansion using the method of our paper "Generalized Bergman kernels
on symplectic manifolds" (math.DG/0411559).Comment: 21 pages, to appear in Internat. J. Math. Precisions added in the
abstrac
Spectral and scattering theory for some abstract QFT Hamiltonians
We introduce an abstract class of bosonic QFT Hamiltonians and study their
spectral and scattering theories. These Hamiltonians are of the form
H=\d\G(\omega)+ V acting on the bosonic Fock space \G(\ch), where
is a massive one-particle Hamiltonian acting on and is a Wick
polynomial \Wick(w) for a kernel satisfying some decay properties at
infinity. We describe the essential spectrum of , prove a Mourre estimate
outside a set of thresholds and prove the existence of asymptotic fields. Our
main result is the {\em asymptotic completeness} of the scattering theory,
which means that the CCR representations given by the asymptotic fields are of
Fock type, with the asymptotic vacua equal to the bound states of . As a
consequence is unitarily equivalent to a collection of second quantized
Hamiltonians
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