2,214 research outputs found
On the boundedness of wave operators for two-dimensional Schr\"odinger operators with threshold obstructions
Let be a Schr\"odinger operator on with
real-valued potential , and let . If has sufficient
pointwise decay, the wave operators are known to be bounded on for all if zero is not an eigenvalue or resonance. We show that if there is an
s-wave resonance or an eigenvalue only at zero, then the wave operators are
bounded on for . This result stands in
contrast to results in higher dimensions, where the presence of zero energy
obstructions is known to shrink the range of valid exponents .Comment: Revised according to referee's comments. 22 pages, to appear in J.
Funct. Ana
L^p boundedness of the wave operator for the one dimensional Schroedinger operator
Given a one dimensional perturbed Schroedinger operator H=-(d/dx)^2+V(x) we
consider the associated wave operators W_+, W_- defined as the strong L^2
limits as s-> \pm\infty of the operators e^{isH} e^{-isH_0} We prove that the
wave operators are bounded operators on L^p for all 1<p<\infty, provided
(1+|x|)^2 V(x) is integrable, or else (1+|x|)V(x) is integrable and 0 is not a
resonance. For p=\infty we obtain an estimate in terms of the Hilbert
transform. Some applications to dispersive estimates for equations with
variable rough coefficients are given.Comment: 26 page
Inverse Scattering at a Fixed Quasi-Energy for Potentials Periodic in Time
We prove that the scattering matrix at a fixed quasi--energy determines
uniquely a time--periodic potential that decays exponentially at infinity. We
consider potentials that for each fixed time belong to in space. The
exponent 3/2 is critical for the singularities of the potential in space. For
this singular class of potentials the result is new even in the
time--independent case, where it was only known for bounded exponentially
decreasing potentials.Comment: In this revised version I give a more detailed motivation of the
class of potentials that I consider and I have corrected some typo
Massive torsion modes, chiral gravity, and the Adler-Bell-Jackiw anomaly
Regularization of quantum field theories introduces a mass scale which breaks
axial rotational and scaling invariances. We demonstrate from first principles
that axial torsion and torsion trace modes have non-transverse vacuum
polarization tensors, and become massive as a result. The underlying reasons
are similar to those responsible for the Adler-Bell-Jackiw (ABJ) and scaling
anomalies. Since these are the only torsion components that can couple
minimally to spin 1/2 particles, the anomalous generation of masses for these
modes, naturally of the order of the regulator scale, may help to explain why
torsion and its associated effects, including CPT violation in chiral gravity,
have so far escaped detection. As a simpler manifestation of the reasons
underpinning the ABJ anomaly than triangle diagrams, the vacuum polarization
demonstration is also pedagogically useful. In addition it is shown that the
teleparallel limit of a Weyl fermion theory coupled only to the left-handed
spin connection leads to a counter term which is the Samuel-Jacobson-Smolin
action of chiral gravity in four dimensions.Comment: 7 pages, RevTeX fil
The phase diagram of the anisotropic Spin-1 Heisenberg Chain
We applied the Density Matrix Renormalization Group to the XXZ spin-1 quantum
chain. In studing this model we aim to clarify controversials about the point
where the massive Haldane phase appears.Comment: 2 pages (standart LaTex), 1 figure (PostScript) uuencode
KAM for the quantum harmonic oscillator
In this paper we prove an abstract KAM theorem for infinite dimensional
Hamiltonians systems. This result extends previous works of S.B. Kuksin and J.
P\"oschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an
application we show that some 1D nonlinear Schr\"odinger equations with
harmonic potential admits many quasi-periodic solutions. In a second
application we prove the reducibility of the 1D Schr\"odinger equations with
the harmonic potential and a quasi periodic in time potential.Comment: 54 pages. To appear in Comm. Math. Phy
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