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\(L^ p\) estimates for Schrödinger evolution equations. (English) Zbl 0588.35029
Let consider the Schrödinger evolution equation \[ (*)\quad \partial u/\partial t=(iP(D)+V(x))u,\quad u(0,x)=u_ 0(x)\in L^ p(R^ n) \] with P(\(\xi)\) an elliptic polynomial of order 2m with Im \(P_{2m}(\xi)=0.\)
For \(V=0\), \(u_ 0\in S(R^ n)\), P with constant coefficients and the hypotheses:
(H1) \(P(\xi)\)-real valued elliptic polynomial with principal part \(p(\xi)\) of degree 2m,
(H2) for \(u\in S^{n-1}\) (the unit sphere of \(R^ n)\) the restriction to \(S^{n-1}\) of \(\psi(\xi)=<u,\xi >p^{-1/m}(\xi)\) is nondegenerate at its critical points,
(H3) \(m\geq 1\) and \(n\geq 3\) or (H3’) \(m\geq 2\), \(n>3+2/m-1.\)
The authors prove the theorem:
a) If (H1), (H2) and (H3) are fulfilled, the solution u(t,.) of the Cauchy problem with Cauchy data in \(L^ 1(R^ n)\) belongs to \(L^{\infty}(R^ n)\) for \(t>0\) and \(\| u(t,.)\|_{L^{\infty}(R^ n)}\leq C_{\infty}(1+| t|^{- c})\| u_ 0\|_{L^ 1(R^ n)}.\)
b) If (H1), (H2) and (H3’) are fulfilled then \(u(t,.)\in L^ q(R^ n)\) for \(q(m,n)<q\leq \infty\) and \(\| u(t,.)\|_{L^ q(R^ n)}\leq C_ q(| t|^{c'}+| t|^{-c})\| u_ 0\|_{L^ 1(R^ n)}\) where \(C_{\infty}\), \(C_ q\) absolute constants, \(1/q(m,n)=(m-1)(n-3)/(2m-1)n-2/(2m-1)n\), \(c>n/2m-1\), \(c'>n/q.\)
On the other hand the authors prove that for Cauchy data in \(L^ q(R^ n)\) such a Cauchy problem is well posed as a distribution in t-variable with values in \(L^ p(R^ n)\) and a computation of the order of the solution is given too.
It deserves to underline the general results concerning \((L^ p,L^ q)\) estimates and smooth distribution groups \((1<p<\infty)\) respectively smooth distribution semigroups on a Banach space.
Finally the authors apply these results to give an estimate of the resolvent operator of (*) and an asymptotic boundedness for the solution when u belong to a subspace of \(L^ p(R^ n).\)
The bibliography lists 12 titles of which 3 are of the authors.
Reviewer: P.T.Crăciunaş

MSC:
35J10 Schrödinger operator, Schrödinger equation
35D10 Regularity of generalized solutions of PDE (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35B40 Asymptotic behavior of solutions to PDEs
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