Analytically Smoothing Effect for Schrödinger Type Equations with Variable Coefficients

Kajitani, Kunihiko; Wakabayashi, Seiichiro

doi:10.1007/978-1-4757-3214-6_10

Cited by 11 publications

(13 citation statements)

References 7 publications

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“…Some related results are obtained for the linear and nonlinear Schrö-dinger equations. For the linear variable coefficients case, see Craig-KappelerStrauss [7], Kajitani-Wakabayashi [13] (see also [25]) and for the nonlinear case, Chihara [3]. They are giving global weighted uniform estimates for solutions with arbitrary order of derivatives in space variable.…”

Section: Then There Exists a Unique Solution V ∈ C((−t T ) H S )∩Xmentioning

confidence: 99%

Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

Kato¹,

Ogawa²

2000

Math Ann

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We show that a solution of the Cauchy problem for the KdV equation,has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity atthe solution is analytic in both space and time variable. The above condition allows us to take as initial data the Dirac δ measure or the Cauchy principal value of 1/x. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [20] and a systematic use of the dilation generator 3t∂ t + x∂ x .

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Section: Then There Exists a Unique Solution V ∈ C((−t T ) H S )∩Xmentioning

confidence: 99%

Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

Kato¹,

Ogawa²

2000

Math Ann

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“…Instead, a local smoothing effect has been used to study the local smoothness of solutions to Schrödinger equations. The smoothing effects for the Schrödinger equation has been a very rich source of recent investigation; see e.g., [1,2,3,4,5,6,7,9,10,11,12,13,14,17,19,20,21,22,23,24,26,27,28,29,30]. In particular, Craig, Kappeler, and Strauss [1] showed that this effect may be considered as a microlocal phenomenon, and this observation inspired a series of investigations, both in the C ∞ case (in particular, [5,26]) and in the analytic case [21,22,23].…”

Section: Introductionmentioning

confidence: 96%

Analytic smoothing effect for the Schrödinger equation with long‐range perturbation

Martinez

Nakamura

Sordoni

2005

Comm Pure Appl Math

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We study the microlocal analytic singularity of solutions to the Schrödinger equation with analytic coefficients. Using microlocal weight estimates developed for estimating phase space tunneling, we prove microlocal smoothing estimates that generalize results by Robbiano and Zuily. We show that the exponential decay of the initial state in a cone in the phase space implies microlocal analytic regularity of the solution at a positive time. We suppose the Schrödinger operator is a long-range-type perturbation of the Laplacian, and we employ positive commutator-type estimates to prove the smoothing property.

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“…Ito and Nakamura [9] also extended these results to Schrödinger operators on scattering manifolds using the same idea and a construction of classical mechanical scattering theory on scattering manifolds. Before Hassel and Wunsch's work [5], many investigations have been made to study the possible smoothness of e −itH f , giving rise to a wide series of results, both in the C ∞ case and in the analytic case; see, e.g., [1][2][3][4]6,7,[10][11][12][13][14][15]20,22,[26][27][28][32][33][34][35][36]. In particular, the microlocal study of this phenomenon was started with [1], and has probably reached its most refined degree of sophistication in [32], where the notion of quadratic scattering wave front set is introduced in the C ∞ case.…”

Section: Introductionmentioning

confidence: 99%

Analytic wave front set for solutions to Schrödinger equations

Martinez

Nakamura

Sordoni

2009

Advances in Mathematics

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This paper is a continuation of [A. Martinez, S. Nakamura, V. Sordoni, Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math. LIX (2006) 1330-1351],where an analytic smoothing effect was proved for long-range type perturbations of the Laplacian H 0 on R n . In this paper, we consider short-range type perturbations H of the Laplacian on R n , and we characterize the analytic wave front set of the solution to the Schrödinger equation: e −itH f , in terms of that of the free solution: e −itH 0 f , for t < 0 in the forward non-trapping region. The same result holds for t > 0 in the backward non-trapping region. This result is an analytic analogue of results by Hassel and Wunsch [

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Analytically Smoothing Effect for Schrödinger Type Equations with Variable Coefficients

Cited by 11 publications

References 7 publications

Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

Analytic smoothing effect for the Schrödinger equation with long‐range perturbation

Analytic wave front set for solutions to Schrödinger equations

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