$L^ p$ estimates for Schrödinger evolution equations
HTML articles powered by AMS MathViewer
 by M. Balabane and H. A. Emamirad PDF
 Trans. Amer. Math. Soc. 292 (1985), 357373 Request permission
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 $(1 + t{^{  c}})$, 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 ${L^p}({{\mathbf {R}}^n})$.References

M. Balabane and H. A. EmamiRad, Prépublications mathématiques n$^{o}$ 22, Université ParisNord, 1981.
 M. Balabane and H. A. Emamirad, Smooth distribution group and Schrödinger equation in $L^{p}(\textbf {R}^{n})$, J. Math. Anal. Appl. 70 (1979), no. 1, 61–71. MR 541059, DOI 10.1016/0022247X(79)900751 —, Pseudo differential parabolic systems in ${L^p}({{\mathbf {R}}^n})$, Contributions to NonLinear P.D.E., Research Notes in Mathematics, no. 89, Pitman, New York, 1983, pp. 1630.
 Claude Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport, Ann. Sci. École Norm. Sup. (4) 3 (1970), 185–233 (French). MR 274925, DOI 10.24033/asens.1190
 Philip Brenner, The Cauchy problem for systems in $L_{p}$ and $L_{p,\alpha }$, Ark. Mat. 11 (1973), 75–101. MR 324249, DOI 10.1007/BF02388508
 J. J. Duistermaat, Fourier integral operators, Courant Institute of Mathematical Sciences, New York University, New York, 1973. Translated from Dutch notes of a course given at Nijmegen University, February 1970 to December 1971. MR 0451313
 J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math. 27 (1974), 207–281. MR 405513, DOI 10.1002/cpa.3160270205
 Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
 J.L. Lions, Les semi groupes distributions, Portugal. Math. 19 (1960), 141–164 (French). MR 143045
 J.L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5–68 (French). MR 165343, DOI 10.1007/BF02684796
 Juan C. Peral, $L^{p}$ estimates for the wave equation, J. Functional Analysis 36 (1980), no. 1, 114–145. MR 568979, DOI 10.1016/00221236(80)90110X
 Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 E. T. Whittaker and J. N. Watson, A course of modern analysis, Cambridge Univ. Press, London, 1969.
Additional Information
 © Copyright 1985 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 292 (1985), 357373
 MSC: Primary 35K22; Secondary 35J10
 DOI: https://doi.org/10.1090/S00029947198508059683
 MathSciNet review: 805968