Download analytically smootidng effect for schrodinger equations

ANALYTICALLY SMOOTIDNG EFFECT
FOR SCHRODINGER EQUATIONS
KUNIHIKO KAJITANI
Institute of Mathematics
University of Tsukuba
305 Tsukuba Ibaraki Japan
§ 1 . Introduction
We shall prove that if the initial data decay exponentially, then the solutions of the
Cauchy problem for Schrodinger equation are analytic with respect to space variables.
Let us consider the following Cauchy problem for Schrodinger equation
%
(1)
t
u(t, x) -
�
.6.u(t, x)
=
O, t E R, x E Rn,
u(O, x) = uo (x), x E Rn ,
(2)
where
.6.
is Laplacian in Rn . Then we can prove
Theorem 1 . Assume that for 80 > 0 ec5o< x > uo belongs to L2 (�) . Then there are G >
O, p > 0 and 8 > 0 such that the solution u(t,x) of (1) and (2) satisfies
(3)
where
<
x >= (1
+ xf + . . . + X� ) 1/2.
We should remark that A. Jensen in [3] and Hayashi,Nakarnitsu & Tsutsurni in [2]
showed that if < x > k uo(x) E L2 (�), the solution u of ( 1 ) and (2) belongs to Hfoc for
t i= 0, Hayashi & Saitoh in [1] proved that if e6< x > 2 uo for some 8 > 0 is in L2(Rn), the
solution u is analytic in x for t i= 0 and Kato & Taniguti in [6] show that if Uo satisfies
II (x V') j uo ll :S Gj + l j!2 for j=O,l,2, , , . ,then the solution is analytic in x for t i= O. Moreover
by use of the methods in [4.5] we can treat Schrodinger equations of more general form.
§2 Fourier integral operators
We begin to introduce functional spaces on � . For p 2 0 we define Hp={u E
Ltoc ; eP<€>u(O E L2 } and for p :S 0, Hp =the dual space of H_p . For p and 8 E R we denote
by Hp,c5 a functional space spanned by the functions u(x) in Ltoc(Rn) such that ec5(x)u(x) E
Hp , where u(O stands for a Fourier transform of u and < x > = (1 + xf + . + x� ) 1/2 .
"
Lemma 2.1. Let Ipi < 1 and 181 < 1 and u E Hp,o.
(i) eo <x > eP< D >u belongs to L2(Rn), where eP<D >u(x) stands for the inverse Fourier
tranform of eP<€ >u(O ·
(ii)If p > 0, u satisfies
(2.1 )
for Va
E Nn
and x
E Rn,
where II . II Hp. 6 denotes the norm o n Hp,o.
Let cp(x, O = X · E - ifX . E < E > - I , where x · E = L, xjEj . We denote by I..,
as
follows,
(2.2)
for u
E
Hp,o and also define II) by
E
Hp,o ,
(2.3)
for v
Lemma 2.2. Let cp = x . E - ifX . E < E > - 1 . Then h.., and I!{.., map comtinuously from
Hp,o to Hp,o' if 8' < 8 - I f I and 181, W I and If I < l.
Lemma 2.3. Let cp = x · E - ifx · E < E >- 1 and lEI small enough. Then 1.., I!!.., = J(D) and
I!!..,I.., = J' (D), where J(O = det(I - iE'VTJ (T] < T] >-1)-1 ITJ=.v(€ ) , where W is a solution
of the equation W - iEW < W > - 1 = E and J'(O = det( -I + iE'V € (E < E > -1 » - 1 .
Noting that there exists the inverse matrix of I ± iE'V €E < E > - 1
we obtain J(D)-l and J' (D)- l . Thus we proved
if
lEI is small enough,
Lemma 2.4. Let cp = x . E - iEx . E < E > - 1 and lEI small enough. Then I.., has the
inverse 1:;; 1 = J' (D) -l I..,I!!.., = I!!.., I.., J(D)- l which maps continuously from Hp,o to Hp,o'
if 8' < 8 - l E I and 181, W I < l .
§3. Proof of Theorem
In order to prove Theorem 1 it suffices to show the following theorem
Theorem 3.1. Let cp = X · E -ifX . E < E > - 1 Assume I..,uo
u of (1)-(2) satisfies that I:;;l U(t, . ) is in H2<t,o .
E
L 2 (�». Then the solution
This theorem and Lemma 2 . 4 assure that the solution u (t,x) belongs to H2<t,o' for
8' > IE . If we choose E such that Et > 0 we obtain the estimate ( l . 1) by use of (ii) of Lemma
2. l .
Proof of Theorem 3 . 1 ; Put v = I:;; l U. Then noting that u = I..,v satisfies
l
8xj u = 1.., ( 8xj ( 1 - iE < D > - l ) V ) ,
we get the following equation for v,
35 1
v (O, x)
Since -i6 ( 1
- i( < D
=
I; luO(x).
> - 1 ) 2= -i6 - 2(6
:
<
D > - 1 +i(26
l e 2 t <D > - 1 v (t , ' ) lli2
t I - < l:.
which implies Theorem 3 . 1 .
[1J
[2J
[3J
[4J
[5J
[6J
Hayashi N.
=
0, 'it
<
D > -2, we can show,
E R,
References
&
Saitoh S . , Analyticity and smoothing effect for Schrodinger equation" Ann. lnst. Henri
Poincare . Math.
52 ( 1990) , 163-173.
Hayashi S , Nakarnitsu K
&
Tsutsumi M ,
On solutions of the initial value problem for the nonlinear
Schrodinger equations in one space dimension, Math.Z.
192 ( 1986), 637-650.
Jensen A, Commutator method and a Smoothing property of the Schrodinger evolution group, Math.
Z.
191 (1986), 53-59.
Kajitani K
459-473.
Kajitani K
&
Baba A, The Cauchy problem for Schrodinger type equations, Bull. Sci.math.
&
Yamaguti K,
Ann. S.N.S. Pisa.
Kato K
1-1
&
1 1 9 (1995),
On global real analytic solutions of the degenemte Kirchhoff equations,
21 ( 1994), 279-297.
Taniguti K, Gevrey regularizing effect for nonlinear Schrodinger equations, preprint
TENNOUDAI TSUKUBA IBARAKI 305 JAPAN
address: kajitanilDmath.tsulruba . ac . jp
E-mail
352
(1995).