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Transactions of the American Mathematical Society

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



$ L\sp p$ estimates for Schrödinger evolution equations

Authors: M. Balabane and H. A. Emamirad
Journal: Trans. Amer. Math. Soc. 292 (1985), 357-373
MSC: Primary 35K22; Secondary 35J10
MathSciNet review: 805968
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Abstract: 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 + \vert t{\vert^{ - 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 $ (\vert t{\vert^{c'}} + \vert t{\vert^{ - 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})$.

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Keywords: $ {L^p}$ estimates, Schrödinger evolution equation, distribution semigroup, local decay, asymptotic boundedness, functional calculus in $ {L^p}$
Article copyright: © Copyright 1985 American Mathematical Society