49 research outputs found
A note on local behavior of eigenfunctions of the Schr\"odinger operator
We show that a real eigenfunction of the Schr\"odinger operator changes sign
near some point in under a suitable assumption on the potential.Comment: To appear in J. Math. Phys., 6 page
From resolvent estimates to unique continuation for the Schr\"odinger equation
In this paper we develop an abstract method to handle the problem of unique
continuation for the Schr\"odinger equation . In
general the problem is to find a class of potentials which allows the
unique continuation. The key point of our work is to make a direct link between
the problem and the weighted resolvent estimates
. We carry out it
in an abstract way, and thereby we do not need to deal with each of the
potential classes. To do so, we will make use of limiting absorption principle
and Kato -smoothing theorem in spectral theory, and employ some tools from
harmonic analysis. Once the resolvent estimate is set up for a potential class,
from our abstract theory the unique continuation would follow from the same
potential class. Also, it turns out that there can be no dented surface on the
boundary of the maximal open zero set of the solution . In this regard,
another main issue for us is to know which class of potentials allows the
resolvent estimate. We establish such a new class which contains previously
known ones, and will also apply it to the problem of well-posedness for the
equation.Comment: To appear in Trans. Amer. Math. Soc., 31 page
Unique continuation for fractional Schr\"odinger operators in three and higher dimensions
We prove the unique continuation property for the differential inequality
, where and , .Comment: To appear in Proc. Amer. Math. Soc., 5 page
On absolute continuity of the spectrum of periodic Schr\"odinger operators
In this paper we find a new condition on a real periodic potential for which
the self-adjoint Schr\"odinger operator may be defined by a quadratic form and
the spectrum of the operator is purely absolutely continuous. This is based on
resolvent estimates and spectral projection estimates in weighted spaces
on the torus, and an oscillatory integral theorem.Comment: To appear in Monatsh. Math., 17 page
Global unique continuation from a half space for the Schr\"odinger equation
We obtain a global unique continuation result for the differential inequality
in . This is the first
result on global unique continuation for the Schr\"odinger equation with
time-independent potentials in . Our method is based on
a new type of Carleman estimates for the operator on
. As a corollary of the result, we also obtain a new unique
continuation result for some parabolic equations.Comment: Revised version, to appear in Journal of Functional Analysi
On minimal support properties of solutions of Schr\"odinger equations
In this paper we obtain minimal support properties of solutions of
Schr\"odinger equations. We improve previously known conditions on the
potential for which the measure of the support of solutions cannot be too
small. We also use these properties to obtain some new results on unique
continuation for the Schr\"odinger operator.Comment: Revised version, to appear in Journal of Mathematical Analysis and
Application
On unique continuation for Schr\"odinger operators of fractional and higher orders
In this note we study the property of unique continuation for solutions of
, where is in a function class of
potentials including for
. In particular, when , our result gives a unique
continuation theorem for the fractional () Schr\"odinger operator
in the full range of values.Comment: Revised version, to appear in Mathematische Nachrichte
A note on the Schr\"odinger smoothing effect
The Kato-Yajima smoothing estimate is a smoothing weighted estimate
with a singular power weight for the Schr\"odinger propagator. The weight has
been generalized relatively recently to Morrey-Campanato weights. In this paper
we make this generalization more sharp in terms of the so-called Kerman-Sawyer
weights. Our result is based on a more sharpened Fourier restriction estimate
in a weighted space. Obtained results are also extended to the fractional
Schr\"odinger propagator.Comment: To appear in Math. Nachr., 8 page
A remark on unique continuation for the Cauchy-Riemann operator
In this note we obtain a unique continuation result for the differential
inequality , where
denotes the Cauchy-Riemann operator
and is a function in .Comment: To appear in Bull. Korean Math. Soc., 5 page
On a reverse H\"older inequality for Schr\"odinger operators
We obtain a reverse H\"older inequality for the eigenfuctions of the
Schr\"odinger operator with slowly decaying potentials. The class of potentials
includes singular potentials which decay like with
, especially the Coulomb potential.Comment: 7 pages, modified proo