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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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$L^ p$ estimates for Schrödinger evolution equations
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by M. Balabane and H. A. Emamirad PDF
Trans. Amer. Math. Soc. 292 (1985), 357-373 Request permission


We prove that for Cauchy data in ${L^1}({{\mathbf {R}}^n})$, the solution of a Schrödinger evolution equation with constant coefficients of order $2m$ is uniformly bounded for $t \ne 0$, with bound $(1 + |t{|^{ - c}})$, where $c$ is an integer, $c > n/2m - 1$. Moreover it belongs to ${L^q}({{\mathbf {R}}^n})$ if $q > q(m,n)$, with its ${L^p}$ norm bounded by $(|t{|^{c’}} + |t{|^{ - c}})$, where $c’$ is an integer, $c’ > n/q$. A maximal local decay result is proved. Interpolating between ${L^1}$ and ${L^2}$, we derive $({L^p},{L^q})$ estimates. On the other hand, we prove that for Cauchy data in ${L^p}({{\mathbf {R}}^n})$, such a Cauchy problem is well posed as a distribution in the $t$-variable with values in ${L^p}({{\mathbf {R}}^n})$, and we compute the order of the distribution. We apply these two results to the study of Schrödinger equations with potential in ${L^p}({{\mathbf {R}}^n})$. We give an estimate of the resolvent operator in that case, and prove an asymptotic boundedness for the solution when the Cauchy data belongs to a subspace of ${L^p}({{\mathbf {R}}^n})$.
    M. Balabane and H. A. Emami-Rad, Pré-publications mathématiques n$^{o}$ 22, Université Paris-Nord, 1981.
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 292 (1985), 357-373
  • MSC: Primary 35K22; Secondary 35J10
  • DOI:
  • MathSciNet review: 805968