Lower Bounds for Order of Decay or of Growthin Time for Solutions to Linear and Non-linear Schrödinger Equations

Tohru Ozawa, Nakao Hayashi

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

We Study lower bounds of decay (or of growth) order in time for solutions to the Cauchy problem for the Schrodinger equation: where f is a linear or non-linear complex-valued function. Under some conditions on f and φ, it is shown that every nontrivial solution u has the estimate for sufficiently large k>0 and for any q∈[2, ∞]. In the previous work [12] of the first named author, we imposed on the assumption that u is asymptotically free. In this article, however, we shall show the assumption is, in fact, irrelevant to the results.

Original languageEnglish
Pages (from-to)847-859
Number of pages13
JournalPublications of the Research Institute for Mathematical Sciences
Volume25
Issue number6
DOIs
Publication statusPublished - 1989
Externally publishedYes

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Nonlinear Equations
Decay
Lower bound
Order of Growth
Schrodinger Equation
Nontrivial Solution
Cauchy Problem
Estimate

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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AB - We Study lower bounds of decay (or of growth) order in time for solutions to the Cauchy problem for the Schrodinger equation: where f is a linear or non-linear complex-valued function. Under some conditions on f and φ, it is shown that every nontrivial solution u has the estimate for sufficiently large k>0 and for any q∈[2, ∞]. In the previous work [12] of the first named author, we imposed on the assumption that u is asymptotically free. In this article, however, we shall show the assumption is, in fact, irrelevant to the results.

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