36 research outputs found
SPHERICAL HARMONICS AND HARDY'S INEQUALITIES (Mathematical aspects of quantum fields and related topics)
We consider the derivative operators for radial direction and spherical direction. We also investigate the operator which takes the spherical average for functions. We reconfirm those properties with particular attention to orthogonality. As an application, the Hardy type inequality is presented with spherical derivatives in the framework of equalities. This clarifies the difference between contribution by radial and spherical derivatives in the improved Hardy inequality as well as nonexistence of nontrivial extremizers without compactness arguments
Scattering Theory for the Dirac Equation with a Nonlocal Term
Consider a scattering problem for the Dirac equation with a nonlocal term including he Hartree type. We show the existence of scattering operators for small initial data n the subcritical and critical Sobolev spaces
The well-posedness of the stochastic nonlinear Schr\"odinger equations in
The Cauchy problem for the stochastic nonlinear Schr\"odinger equation with a
multiplicable noise is considered where the nonlinear term is of a power type
and its coefficients are complex numbers. In particular, it is extremely
important to consider the complex coefficients in the noise which cover
non-conservative case, because they include measurement effects in quantum
physics. The main purpose of this paper is to construct classical solutions in
for the problem in question. The time local well-posedness
in and was investigated in the papers
[7,8]. In this paper we study the well-posedness in by
making use of the rescaling approach as a main tool for dealing with the
multiple noise, where we need to take advantage of a slight modification of the
deterministic Strichartz estimate to fit into requirements under the setting of
. The other difficulty lies on the discussion on smoothness
of functions in the nonlinear term, where the proof of time local
well-posedness for the case of -solutions does not go similarly as in the
cases of -solutions or -solutions, because of the complexity in the
computation of the nonlinear term with lower exponent . The techniques
of Kato [18,19] work well on this difficulty even for the stochastic equations.
We use the stochastic Strichartz estimate [4,16,17] as well to deal with white
noise which did not appear in the proof for -solutions or -solutions.
We also discuss time-global solutions in .Comment: 31 pages, no figures. arXiv admin note: text overlap with
arXiv:1404.5039 by other author
Global solutions of stochastic nonlinear Schr\"odinger system with quadratic interaction
The time-global existence of solutions to a system of stochastic
Schr\"odinger equations with multiplicative noise and the quadratic nonlinear
terms are discussed in this paper. The same system in the deterministic
treatment was studied in [18] where the mass and energy are conserved. In our
stochastic situation, those are not conserved and which causes several
difficulties in the arguments of composing a-priori estimate.Comment: 28 pages, no figure
A supersolutions perspective on hypercontractivity
The purpose of this article is to expose an algebraic closure property of
supersolutions to certain diffusion equations. This closure property quickly
gives rise to a monotone quantity which generates a hypercontractivity
inequality. Our abstract argument applies to a general Markov semigroup whose
generator is a diffusion and satisfies a curvature condition.Comment: 7 page
Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation
We prove endpoint Strichartz estimates for the Klein-Gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions. As an application, global wellposedness of the nonlinear Dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable
Remarks on the Rellich inequality
We study the Rellich inequalities in the framework of equalities. We present equalities which imply the Rellich inequalities by dropping remainders. This provides a simple and direct understanding of the Rellich inequalities as well as the nonexistence of nontrivial extremisers. © 2016 Springer-Verlag Berlin HeidelbergEmbargo Period 12 month