Melrose, R., G. Uhlmann.
Generalized backscattering and the LaxPhillips transform
(pp. 355372)
A B S T R A C T S
PSEUDODIFFERENTIAL OPERATORS AND WEIGHTED NORMED SYMBOL SPACES
J. Sjöstrand
johannes@math.polytechnique.fr
2000 Mathematics Subject Classification: 35S05.
Key words:
Pseudodifferential operator, symbol, modulation space.
This work is the continuation of two earlier ones by the author
and stimulated by many more recent contributions. We develop a
very general calculus of pseudodifferential operators with microlocally
defined normed symbol spaces. The goal was to attain the natural degree
of generality in the case when the underlying metric on the cotangent
space is constant. We also give sufficient conditions for our
operators to belong to Schattenvon Neumann classes.
WEIGHTED DISPERSIVE ESTIMATES FOR SOLUTIONS OF THE
SCHRÖDINGER EQUATION
Fernando Cardoso
fernando@dmat.ufpe.br,
Claudio Cuevas
cch@dmat.ufpe.br,
Georgi Vodev
georgi.vodev@math.univnantes.fr
2000 Mathematics Subject Classification: 35L15, 35B40, 47F05.
Key words:
Potential, dispersive estimates.
We obtain < x >^{s}L^{1}→ < x >^{−s}L^{∞}
time decay estimates for the Schrödinger group
e^{it(−Δ+V)}, where
V Î L^{∞}(R^{n}), n ≥ 3, is a realvalued potential satisfying
V(x) = O(< x >^{−n+1/2−ε}),
ε > 0.
A SIMPLE EXAMPLE OF LOCALIZED PARAMETRIC RESONANCE FOR
THE WAVE EQUATION
Ferruccio Colombini
colombini@dm.unipi.it,
Jeffrey Rauch
rauch@umich.edu
2000 Mathematics Subject Classification: 35L05, 35P25, 47A40.
Key words:
Localized parametric resonance, growing
solutions.
We construct solutions growing exponentially in time of the
equation
u_{tt} − (a(t,x)u_{x})_{x} = 0, (t,x) ∈ R^{1+1}_{t,x}, 

with a > 0 periodic in t and constant outside a compact in x.
The function a is discontinuous representing a well which
oscillates in time. The novelty is that by a careful geometric
optics construction, all reflected and transmitted phases are linear
functions of (t,x) rendering the example elementary.
MICROLOCAL APPROACH TO TENSOR TOMOGRAPHY
AND BOUNDARY AND LENS RIGIDITY
Plamen Stefanov
stefanov@math.purdue.edu
2000 Mathematics Subject Classification:
53C24, 53C65, 53C21.
Key words:
Boundary rigidity, tensor tomography,
inverse problems.
This is a survey of the recent results by the author and Gunther
Uhlmann on the boundary rigidity problem and on the associated
tensor tomography problem.
ACCURATE WKB APPROXIMATION FOR A 1D PROBLEM WITH LOW
REGULARITY
Francis Nier
Francis.Nier@univrennes1.fr
2000 Mathematics Subject Classification:
34L40, 65L10, 65Z05, 81Q20.
Key words:
WKB method, low regularity.
This article is concerned with the analysis of the WKB expansion in
a classically forbidden region for a one dimensional boundary value
Schrodinger equation with a non smooth potential. The assumed
regularity of the potential is the one coming from a non linear
problem and seems to be the critical one for which a good
exponential decay estimate can be proved for the first remainder
term. The treatment of the boundary conditions brings also some
interesting subtleties which require a careful application of
Carleman's method.
GLOBAL WAVES WITH NONPOSITIVE ENERGY IN GENERAL
RELATIVITY
Alain Bachelot
bachelot@math.ubordeaux1.fr
2000 Mathematics Subject Classification:
35Lxx, 35Pxx, 81Uxx, 83Cxx.
Key words:
Global Cauchy problem, causality, superradiance, time machine,
blackhole, scattering.
The theory of the waves equations has a long history since M. Riesz
and J. Hadamard.
It is impossible to cite all the important results in the area, but
we mention the authors related with our work:
J. Leray [34] and
Y. ChoquetBruhat [9] (Cauchy problem), P. Lax and R. Phillips
[33] (scattering theory for a compactly supported perturbation),
L. Hörmander [27]
and JM. Bony [7] (microlocal analysis). In all these domains,
V. Petkov has made fundamental contributions, mainly in microlocal
analysis,
scattering theory, dynamical zeta functions (see in particular the
monography [42]).
In this paper we present a survey of some recent results on the global
existence and the asymptotic behaviour of waves, when the conserved
energy is not definite positive. This unusual situation arises in
important cosmological models of the General Relativity where the
gravitational curvature is very strong. We consider the case of the
closed timelike curves (violation of the causality) [1], and
the charged blackholes (superradiance) [3].
ON THE CAUCHY PROBLEM FOR NON EFFECTIVELY HYPERBOLIC
OPERATORS, THE IVRIIPETKOVHÖRMANDER CONDITION AND THE GEVREY
WELL POSEDNESS
Tatsuo Nishitani
nishitani@math.sci.osakau.ac.jp
2000 Mathematics Subject Classification:
35L15, Secondary 35L30.
Key words:
Cauchy problem, non effectively
hyperbolic, Gevrey wellposedness, null bicharacteristic, Hamilton
map, elementary decomposition, positive trace.
In this paper we prove that for non effectively hyperbolic operators
with smooth double characteristics with the Hamilton map exhibiting
a Jordan block of size 4 on the double characteristic manifold the
Cauchy problem is well posed in the Gevrey 6 class if the strict
IvriiPetkovHörmander condition is satisfied.
DYNAMICAL RESONANCES AND SSF SINGULARITIES FOR A MAGNETIC
SCHRÖDINGER OPERATOR
María Angélica Astaburuaga
angelica@mat.puc.cl,
Philippe Briet
briet@cpt.univmrs.fr,
Vincent Bruneau
vbruneau@math.ubordeaux1.fr,
Claudio Fernández
cfernand@mat.puc.cl,
Georgi Raikov
graikov@mat.puc.cl,
2000 Mathematics Subject Classification:
35P25, 35J10, 47F05, 81Q10.
Key words:
Magnetic Schrödinger operators,
resonances, Mourre estimates, spectral shift function.
We consider the Hamiltonian H of a 3D spinless nonrelativistic
quantum particle subject to parallel constant magnetic and
nonconstant electric field. The operator H has infinitely many
eigenvalues of infinite multiplicity embedded in its continuous
spectrum. We perturb H by appropriate scalar potentials V and
investigate the transformation of these embedded eigenvalues into
resonances. First, we assume that the electric potentials are
dilationanalytic with respect to the variable along the magnetic
field, and obtain an asymptotic expansion of the resonances as the
coupling constant ϰ of the perturbation tends to zero.
Further, under the assumption that the Fermi Golden Rule holds true,
we deduce estimates for the time evolution of the resonance states
with and without analyticity assumptions; in the second case we
obtain these results as a corollary of suitable Mourre estimates and
a recent article of Cattaneo, Graf and Hunziker [11]. Next, we
describe sets of perturbations V for which the Fermi Golden Rule
is valid at each embedded eigenvalue of H; these sets turn out to
be dense in various suitable topologies. Finally, we assume that V
decays fast enough at infinity and is of definite sign, introduce
the Krein spectral shift function for the operator pair (H+V, H),
and study its singularities at the energies which coincide with
eigenvalues of infinite multiplicity of the unperturbed operator
H.
SPECTRA OF RUELLE TRANSFER OPERATORS FOR CONTACT FLOWS
Luchezar Stoyanov
stoyanov@maths.uwa.edu.au
2000 Mathematics Subject Classification:
Primary: 37D20, 37D40; Secondary: 37A25, 37D50.
Key words:
Axiom A
flow, basic set, Ruelle transfer operator, contact flow, Gibbs
measure.
In this survey article we discuss some recent results concerning strong
spectral estimates for Ruelle transfer operators for contact flows on basic sets
similar to these of Dolgopyat obtained in the case of Anosov flows with
C^{1}
stable and unstable foliations. Some applications of Dolgopyat's results
and the more recent ones are also described.
SPECTRAL SHIFT FUNCTION FOR THE PERTURBATIONS OF
SCHRÖDINGER OPERATORS AT HIGH ENERGY
Rachid Assel
rachid.assel@fsm.rnu.tn,
Mouez Dimassi
dimassi@math.univparis13.fr,
2000 Mathematics Subject Classification:
35P20, 35J10, 35Q40.
Key words:
High energy, asymptotic expansions,
spectral shift function.
We give a complete pointwise asymptotic expansion for the Spectral
Shift Function for Schrödinger operators that are perturbations of
the Laplacian on R^{n} with slowly decaying potentials.
RESOLVENT AND SCATTERING MATRIX AT THE MAXIMUM OF THE
POTENTIAL
Ivana Alexandrova
alexandrovai@ecu.edu,
JeanFrançois Bony
bony@math.ubordeaux1.fr,
Thierry Ramond
thierry.ramond@math.upsud.fr
2000 Mathematics Subject Classification:
35P25, 81U20, 35S30, 47A10, 35B38.
Key words:
Scattering matrix, resolvent, spectral function, Schrödinger
equation, Fourier integral operator, critical energy.
We study the microlocal structure of the resolvent of the
semiclassical Schrödinger operator with short range potential at
an energy which is a unique nondegenerate global maximum of the
potential. We prove that it is a semiclassical Fourier integral
operator quantizing the incoming and outgoing Lagrangian
submanifolds associated to the fixed hyperbolic point. We then
discuss two applications of this result to describing the structure
of the spectral function and the scattering matrix of the
Schrödinger operator at the critical energy.
ON AN ODE RELEVANT FOR THE GENERAL THEORY OF THE
HYPERBOLIC CAUCHY PROBLEM
Enrico Bernardi
enrico.bernardi@unibo.it,
Antonio Bove
bove@bo.infn.it
2000 Mathematics Subject Classification:
34E20, 35L80, 35L15.
Key words:
Cauchy problem, zeroes of entire
functions, asymptotic expansions, Hamilton systems.
In this paper we study an ODE in the complex plane. This is a key step
in the search of new necessary conditions for the well posedness of
the Cauchy Problem for hyperbolic operators with double
characteristics.
SUPERSYMMETRY AND GHOSTS IN QUANTUM MECHANICS
Didier Robert
didier.robert@univnantes.fr
2000 Mathematics Subject Classification:
81Q60, 35Q40.
Key words:
Supersymmetric quantum mechanics, Hamiltonian and Lagrangian
mechanics, bosons and fermions.
A standard supersymmetric quantum system is defined by a
Hamiltonian [^H] = ½([^Q]^{*}[^Q] +[^Q][^Q]^{*}), where the supercharge [^Q] satisfies [^Q]^{2} = 0,
[^Q] commutes with [^H].
So we have [^H] ≥ 0 and the quantum spectrum of [^H] is non negative.
On the other hand PaisUlhenbeck proposed in 1950 a model in
quantumfield theory where the d'Alembert operator
^{[¯]} = [(∂^{2})/( ∂t^{2})] − Δ_{x} is replaced by fourth order operator ^{[¯]}(^{[¯]} + m^{2}), in order to eliminate the divergences occuring in quantum field theory.
But then the Hamiltonian of the system, obtained by second
quantization, has large negative energies called ``ghosts" by
physicists. We report here on a joint work with A. Smilga
(SUBATECH, Nantes) where we consider a similar problem for some
models in quantum mechanics which are invariant under
supersymmetric transformations. We show in particular that
``ghosts" are still present.
GENERALIZED BACKSCATTERING AND THE LAXPHILLIPS
TRANSFORM
Richard Melrose
rbm@math.mit.edu,
Gunther Uhlmann
gunther@math.washington.edu,
2000 Mathematics Subject Classification:
35P25, 35R30, 58J50.
Key words:
Backscattering, Radon transform,
Fredholm family, holomorphy, potential scattering, inversion.
Using the freespace translation representation (modified
Radon transform) of Lax and Phillips in odd dimensions, it is shown that
the generalized backscattering transform (so outgoing angle
w = Sq in terms of the incoming angle with S orthogonal and IdS
invertible) may be further restricted to give an entire, globally
Fredholm, operator on appropriate Sobolev spaces of potentials with
compact support. As a corollary we show that the modified backscattering
map is a local isomorphism near elements of a generic set of
potentials.
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