J. Math. Sci. Univ. Tokyo
2 (1995), 441–498.
A construction of the fundamental
solution for Schrödinger equations
By Naoto Kumano-go
Abstract. In this paper, applying the skip method in Fujiwara
[3] to the theory of multi-products of Fourier integral operators in Kitada and H.Kumano-go [6], we give a construction of the fundamental
solution for the Cauchy problem of a pseudo-differential equation of
Schrödinger’s type. We regard this construction as a multi-product of
Fourier integral operators, and investigate the pointwise convergence of
the phase function and that of the symbol. Here we use neither the
solution of the Hamilton-Jacobi equation in [6] nor the action of the
classical orbit in [2],[4].
0.
Introduction
Let Lh be a Schrödinger operator defined by Lh ≡ i∂t + Kh (t, X, Dx ) on
[0, T ] with a parameter 0 < h < 1. For sufficiently small T0 (0 < T0 ≤ T ),
let Uh (t, s) (0 ≤ s ≤ t ≤ T0 ) be the fundamental solution for the operator
Lh such that
(0.1)
Lh Uh (t, s) = 0
on [s, T0 ]
Uh (s, s) = I (0 ≤ s ≤ T0 ) .
Noting that
(0.2)
Kh (t, X, Dx )u(x)
≡ eix·ξ Kh (t, x, ξ)û(ξ)d−ξ (u ∈ S, d−ξ ≡ (2π)−n dξ) ,
1991 Mathematics Subject Classification.
58G15.
441
Primary 35S05; Secondary 58D30, 47G30,
442
Naoto Kumano-go
one may ask whether
Uh (t, s)u(x) =
eix·ξ ei
t
s
Kh (τ,x,ξ)dτ
û(ξ)d−ξ ?
Of course, the answer is “ No ”. However, this simple idea is our starting
point. In what follows, we set
t
i st Kh (τ )dτ
(X, Dx )u(x) ≡ eix·ξ ei s Kh (τ,x,ξ)dτ û(ξ)d−ξ (u ∈ S) .
(0.3)
e
By the result of Kitada and H.Kumano-go [6], we have the following:
R 2n with a parameter
Let Hh (t, x, ξ) be a real-valued function on [0, T ]×R
R 2n
0 < h < 1, which has a continuous derivative ∂ξα Dxβ Hh (t, x, ξ) on [0, T ]×R
for any α, β, satisfying
(0.4)
|∂ξα Dxβ {hρ−δ Hh (t, hδ x, h−ρ ξ)}|
Cα,β x; ξ
2−|α+β| (|α + β| ≤ 1)
≤
Cα,β
(|α + β| ≥ 2) ,
on [0, T ] × R 2n . Here Cα,β is a constant independent of a parameter 0 <
h < 1.
Let Wh (t, x, ξ) be a complex-valued function on [0, T ] × R 2n with a parameter 0 < h < 1, which has a continuous derivative ∂ξα Dxβ Wh (t, x, ξ) on
[0, T ] × R 2n for any α, β, satisfying
(0.5)
|∂ξα Dxβ {Wh (t, hδ x, h−ρ ξ)}| ≤ Cα,β
on [0, T ] × R 2n . Here Cα,β
is a constant independent of a parameter 0 <
h < 1.
Set
Kh (t, x, ξ) ≡ Hh (t, x, ξ) + Wh (t, x, ξ)
(0.6)
Lh ≡ i∂t + Kh (t, X, Dx ) .
Then, using the solution φh (t, s)(x, ξ) of the Hamilton-Jacobi equation


 −∂t φh (t, s)(x, ξ) + Hh (t, x, ∇x φh (t, s)(x, ξ)) = 0
on [s, T0 ] × R 2n
(0.7)

 φh (s, s)(x, ξ) = x · ξ ,
Fundamental solution for Schrödinger equations
443
we can construct Uh (t, s) from the Fourier integral operator I(φh (t, s)) with
the phase function φh (t, s)(x, ξ) and the symbol 1, by the method of the
successive approximation and by H.Kumano-go-Taniguchi theorem. Especially, in the case Wh (t, x, ξ) ≡ 0, we have
Uh (t, s) =
(0.8)
lim
|∆t,s |→0
I(φh (t, t1 ))I(φh (t1 , t2 )) · · · I(φh (tν , s)) ,
where ∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν ≥ tν+1 ≡ s (≥ 0) is an arbitrary
division of interval [s,t] into subintervals, and |∆t,s | ≡ max |tj−1 − tj |.
1≤j≤ν+1
Now, in this case Wh (t, x, ξ) ≡ 0, by applying Fujiwara’s skip method in
[3] to Kitada and H.Kumano-go’s theory, we can construct Uh (t, s) directly
with (0.8). Here the following facts are important:
(A) φh (t, θ)#φh (θ, s) = φh (t, s) (T0 ≥ t ≥ θ ≥ s ≥ 0)
(B) There exist a constant Cl and rh (t, θ, s)(x, ξ) ∈ B0ρ,δ (h)
(T0 ≥ t ≥ θ ≥ s ≥ 0) such that
I(φh (t, θ))I(φh (θ, s)) = I(φh (t, s)) + rh (t, θ, s)(φh (t, s); X, Dx )
and
(0)
|rh (t, θ, s)|l
≤ Cl |t − θ||θ − s| .
The arguments about the solution of the Hamilton-Jacobi equation in
Kitada and H.Kumano-go [6], however, seem to be too technical. Then, can
we construct Uh (t, s) in a simple style without the solution of the HamiltonJacobi equation ? On the other hand, we can rewrite (0.8) with
Uh (t, s) =
lim
|∆t,s |→0
ei(φh (t,t1 )(x,ξ)−x·ξ) (X, Dx )ei(φh (t1 ,t2 )(x,ξ)−x·ξ) (X, Dx )
· · · ei(φh (tν ,s)(x,ξ)−x·ξ) (X, Dx ) .
Furthermore, φh (tj−1 , tj )(x, ξ) − x · ξ is approximated to
i
tj−1
tj−1
tj
Hh (τ, x, ξ)dτ .
H (τ )dτ
h
(X, Dx ) ? Our answer
Then, can we construct Uh (t, s) from e tj
is to get the following expression for general cases including Wh (t, x, ξ) ≡ 0:
(0.9)
Uh (t, s) =
lim
i
e
t
t1
Kh (τ )dτ
|∆t,s |→0
i stν Kh (τ )dτ
···e
i
(X, Dx )e
(X, Dx ) .
t
1
t2
Kh (τ )dτ
(X, Dx )
444
Naoto Kumano-go
Here we consider (A) and (B). If we replace φh (t, s)(x, ξ) by x · ξ +
t
s Hh (τ, x, ξ)dτ , then (A) does not hold any longer and (B) requires some
improvement. Therefore, in order to get (0.9), we have to make the best of
the theory of multi-products of phase functions in Kitada and H.Kumano-go
[6].
Main Theorems
The main theorems for a Schrödinger equation are Theorem 5.1, 5.3 and
5.4. For the details, see Section 5. Roughly speaking, Theorem 5.1, 5.3 and
5.4 are as follows:
R 2n with a parameter
Let Hh (t, x, ξ) be a real-valued function on [0, T ]×R
R 2n
0 < h < 1, which has a continuous derivative ∂ξα Dxβ Hh (t, x, ξ) on [0, T ]×R
for any α, β, satisfying
(0.10)
|∂ξα Dxβ {hρ−δ Hh (t, hδ x, h−ρ ξ)}|
Cα,β x; ξ
2−|α+β| (|α + β| ≤ 1)
≤
Cα,β
(|α + β| ≥ 2) ,
on [0, T ] × R 2n . Here Cα,β is a constant independent of a parameter 0 <
h < 1.
Let Wh (t, x, ξ) be a complex-valued function on [0, T ] × R 2n with a parameter 0 < h < 1, which has a continuous derivative ∂ξα Dxβ Wh (t, x, ξ) on
[0, T ] × R 2n for any α, β, satisfying

C0,0 x; ξ
(α = β = 0)
 |∂ α Dxβ {Wh (t, hδ x, h−ρ ξ)}| ≤
ξ
(|α + β| ≥ 1)
Cα,β
(0.11)

δ
−ρ
Im{Wh (t, h x, h ξ)} ≥ −C0,0 ,
is a constant independent of a parameter 0 <
on [0, T ] × R 2n . Here Cα,β
h < 1.
Set
Kh (t, x, ξ) ≡ Hh (t, x, ξ) + Wh (t, x, ξ)
(0.12)
Lh ≡ i∂t + Kh (t, X, Dx ) .
Furthermore, let T0 (0 < T0 ≤ T ) be sufficiently small.
Fundamental solution for Schrödinger equations
445
Theorem 5.1.
For any ∆t0 ,tν+1 : (T0 ≥) t0 ≥ t1 ≥ · · · ≥ tν+1 (≥ 0), there exists
a Fourier integral operator ph (∆t0 ,tν+1 )(φh (∆t0 ,tν+1 ); X, Dx ) with a phase
function φh (∆t0 ,tν+1 )(x, ξ) and a symbol ph (∆t0 ,tν+1 )(x, ξ) such that
(0.13)
ph (∆t0 ,tν+1 )(φh (∆t0 ,tν+1 ); X, Dx )
i
=e
t
0
t1
i
···e
Kh (τ )dτ
tν
tν+1
i
(X, Dx )e
Kh (τ )dτ
t
1
t2
Kh (τ )dτ
(X, Dx )
(X, Dx ) .
Furthermore, the set {φh (∆t0 ,tν+1 )}∆t0 ,tν+1 is bounded in Bδ−ρ,2
ρ,δ (h), and the
0
set {ph (∆t0 ,tν+1 )}∆t0 ,tν+1 is bounded in Bρ,δ (h).
Theorem 5.3.
For any (T0 ≥) t ≥ s (≥ 0), there exist a phase function φh (t, s)(x, ξ) and
a symbol ph (t, s)(x, ξ) such that φh (∆t,s ) uniformly converges to φh (t, s) in
Bδ−ρ,2
ρ,δ (h) as |∆t,s | tends to 0, and ph (∆t,s ) uniformly converges to ph (t, s)
0,1
in Bρ,δ
(h) as |∆t,s | tends to 0.
Furthermore, it follows that
(0.14)
eiφh (t,s)(x,ξ)−ix ·ξ ph (t, s)(x, ξ)
ν+1 ···
exp
i(xj−1 − xj ) · ξ j
= lim Os —
|∆t,s |→0
tj−1
+i
Kh (τ, xj−1 , ξ j )dτ
j=1
dx1 d−ξ 1 · · · dxν d−ξ ν ,
tj
where ∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν+1 ≡ s (≥ 0), x0 ≡ x, xν+1 ≡ x
and ξ ν+1 ≡ ξ.
Theorem 5.4.
The Fourier integral operator Uh (t, s) ≡ ph (t, s)(φh (t, s); X, Dx ) with the
phase function φh (t, s)(x, ξ) and the symbol ph (t, s)(x, ξ) in Theorem 5.3,
has the following form:
(0.15)
Uh (t, s) =
lim
i
e
t
t1
Kh (τ )dτ
|∆t,s |→0
i stν Kh (τ )dτ
···e
i
(X, Dx )e
(X, Dx ) ,
t
1
t2
Kh (τ )dτ
(X, Dx )
446
Naoto Kumano-go
where ∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν ≥ tν+1 ≡ s (≥ 0).
Furthermore, Uh (t, s) is the fundamental solution for the operator Lh .
An example of a Schrödinger operator Lh :
n h∂ h∂ 1 xj
xk
aj,k (t)
Lh ≡ i∂t +
h
i
i
j,k=1
h∂ xk
+ cj,k (t)xj xk
+ bj,k (t)xj
i
n h∂ xj
+ bj (t)xj + c(t, x)
aj (t)
+
i
Remark.
(0.17)
j=1
+ d(t, x).
Here aj,k (t), bj,k (t), cj,k (t), aj (t) and bj (t) are real-valued continuous functions on [0, T ]. c(t, x) is a real-valued function on [0, T ] × R 2n which has
a continuous and bounded derivative Dxβ c(t, x) on [0, T ] × R 2n for any β.
d(t, x) is a complex-valued function on [0, T ] × R 2n , which has a continuous
and bounded derivative Dxβ d(t, x) on [0, T ] × R 2n for any β = 0, and whose
imaginary part Im{d(t, x)} is lower bounded on [0, T ] × R 2n .
The Plan of This Paper
The plan of this paper is as follows.
m,λ
Section 1. We introduce the classes of symbols Bm
ρ,δ (h), Bρ,δ (h) and
m,λ
the families of pseudo-differential operators B m
ρ,δ (h), B ρ,δ (h) in Kitada and
H.Kumano-go [6].
Section 2. We define the class of phase functions Pρ,δ (τ, h, {κl }∞
l=0 )
and investigate the properties of the multi-products of phase functions in
more detail. It is important that the multi-product of phase functions
(φ1,h #φ2,h # · · · #φν+1,h )(x, ξ) is ‘ close ’ to x · ξ + ν+1
j=1 Jj,h (x, ξ) where
Jj,h (x, ξ) ≡ φj,h (x, ξ) − x · ξ. This section plays an essential role to prove
the pointwise convergence of the phase function in Section 5.
Section 3. We introduce the families of Fourier integral operators
m,λ
Bm
ρ,δ (φh ), B ρ,δ (φh ) in Kitada and H.Kumano-go [6].
Fundamental solution for Schrödinger equations
447
Section 4. We apply Fujiwara’s skip method in [3] to Kitada and
H.Kumano-go’s theory of multi-products of Fourier integral operators, and
investigate some properties. This section plays an essential role to prove
the convergence of the symbol in Section 5.
Section 5. We give a construction of the fundamental solution for the
Cauchy problem of a pseudo-differential equation of Schrödinger’s type. We
regard this construction as a multi-product of Fourier integral operators,
and investigate the pointwise convergence of the phase function and that of
the symbol.
Acknowledgements. I would like to express my sincere gratitude to Professor K. Kataoka, Professor H. Komatsu, Professor D. Fujiwara, Professor
K. Taniguchi, and Professor K. Yajima for helpful discussions and suggestions. It was very happy that I could learn Fujiwara’s skip method from
Professor D. Fujiwara, and H.Kumano-go-Taniguchi theorem from Professor K. Taniguchi. Moreover, Lemma 4.7 is a revised version of that in
Fujiwara [4].
Notation
For x = (x1 , . . . , xn ) ∈ R nx , ξ = (ξ1 , . . . , ξn ) ∈ R nξ and multi-indices of
non-negative integers α = (α1 , . . . , αn ), β = (β1 , . . . , βn ), we employ the
usual notation:
|α| = α1 + · · · + αn , |β| = β1 + · · · + βn ,
x · ξ = x1 ξ1 + · · · + xn ξn , x
= (1 + |x|2 )1/2 , x; ξ
= (1 + |x|2 + |ξ|2 )1/2 ,
∂ξj =
∂
∂
, Dxj = −i
, ∂ξα = ∂ξα11 · · · ∂ξαnn , Dxβ = Dxβ11 · · · Dxβnn .
∂ξj
∂xj
S denotes the Schwartz space of rapidly decreasing C ∞ -functions on R n .
R 2n ) denotes the set of C ∞ -functions on R 2n whose derivatives are all
B(R
R 2n ) denotes the set of C ∞ -functions g(x, ξ) on R 2n whose
bounded. B 1,∞ (R
R 2n )
derivatives ∂ξα Dxβ g(x, ξ) are bounded for |α+β| ≥ 1. For u ∈ S, f ∈ B(R
R 2n ), we define semi-norms |u|l,S , |f |l,B and |g|l,B1,∞ (l =
and g ∈ B1,∞ (R
0, 1, 2, . . . ), respectively by
|u|l,S ≡ max sup |x
k ∂xα u(x)| (l = 0, 1, 2, . . . ) ,
k+|α|≤l x
448
Naoto Kumano-go
|f |l,B ≡ max sup |∂ξα Dxβ f (x, ξ)| (l = 0, 1, 2, . . . ) ,
|α+β|≤l x,ξ
and
|g|l,B1,∞ ≡

sup |g(x, ξ)|/x; ξ
(l = 0)



 x,ξ




sup |g(x, ξ)|/x; ξ
+
x,ξ
max
sup |∂ξα Dxβ g(x, ξ)|
1≤|α+β|≤l x,ξ
(l = 1, 2, . . . ) .
R 2n ) are Fréchet spaces with these semi-norms.
R 2n ) and B 1,∞ (R
Then, S, B(R
The Fourier transform û(ξ) ≡ F[u](ξ) and the inverse Fourier transform
F[v](x) are defined respectively by
û(ξ) ≡ F[u](ξ) ≡ e−ix·ξ u(x)dx (u ∈ S)
F[v](x) ≡ eix·ξ v(ξ)d−ξ (v ∈ S),
where d−ξ ≡ (2π)−n dξ. Os —
See [7].
1.
·dyd−η denotes the usual oscillatory integrals.
Pseudo-Differential Operators
In this section, we introduce the basic properties of the pseudodifferential operators defined by Kitada and H.Kumano-go in [6]. For the
details, see [6]. In what follows, for simplicity, let n0 ≡ 2[n/2 + 1].
Definition 1.1 ( A Family of Pseudo-Differential Operators B m
ρ,δ (h) ).
(1) We say that a family {ph }0<h<1 of C ∞ -functions ph ≡
ph (x, ξ, x , ξ , x ) belongs to the class {Bm
ρ,δ (h)}0<h<1 (m ∈ R , 0 ≤
δ ≤ ρ ≤ 1) of symbols, if ph satisfies
(1.1)
sup
sup
0<h<1 x,ξ,x ,ξ ,x
|∂ξα Dxβ ∂ξα Dxβ Dxβ ph (x, ξ, x , ξ , x )|
hm+ρ|α+α |−δ|β+β +β |
< ∞,
for any α, β, α , β , β . We write {ph }0<h<1 ∈ {Bm
ρ,δ (h)}0<h<1 , or
(h).
simply ph ∈ Bm
ρ,δ
Fundamental solution for Schrödinger equations
449
(2) For ph (x, ξ, x , ξ , x ) ∈ Bm
ρ,δ (h), we define a family {Ph }0<h<1 of
pseudo-differential operators Ph ≡ ph (X, Dx , X , Dx , X ) by
(1.2)
ph (X, Dx , X , Dx , X )u(x)
1 1
2 2
e−i(y ·η +y ·η ) ph (x, η 1 , x + y 1 , η 2 , x + y 1 + y 2 )
≡ Os —
× u(x + y 1 + y 2 )dy 2 d−η 2 dy 1 d−η 1 (u ∈ S) .
m
Bm
We write {Ph }0<h<1 ∈ {B
ρ,δ (h)}0<h<1 , or simply Ph ∈ B ρ,δ (h).
Remark.
(m)
(l = 0, 1, 2, . . . ) by
1◦ . For ph ∈ Bm
ρ,δ (h), we define semi-norms |ph |l
(1.3)
(m)
|ph |l
≡
max
sup
x,ξ,x ,ξ ,x
2◦ .
(1.4)
sup
|α+β+α +β +β |≤l 0<h<1
|∂ξα Dxβ ∂ξα Dxβ Dxβ ph (x, ξ, x , ξ , x )|
hm+ρ|α+α |−δ|β+β +β |
.
Then Bm
ρ,δ (h) is a Fréchet space with these semi-norms.
For ph (x, ξ, x , ξ , x ) ∈ Bm
ph by
ρ,δ (h), we define ph (x, ξ, x , ξ , x ) ≡ h−m ph (hδ x, h−ρ ξ, hδ x , h−ρ ξ , hδ x ) .
Then we have
ph ∈ B00,0 (h),
(1.5)
and
(1.6)
(0)
(m)
|
ph |l ( in B00,0 (h)) = |ph |l
( in Bm
ρ,δ (h)) .
3◦ . Symbols ph (x, ξ), ph (ξ, x ) ∈ Bm
ρ,δ (h) are often called single symbols.
◦
4 . For symbols ph (x, ξ), ph (ξ, x ), ph (x, ξ, x ) ∈ Bm
ρ,δ (h), we have the
representation formulae:
ph (X, Dx )u(x) = eix·ξ ph (x, ξ)û(ξ)d−ξ (u ∈ S) ,
(1.7)
F[ph (Dx , X )u](ξ) = e−ix ·ξ ph (ξ, x )u(x )dx (u ∈ S) ,
(1.8)
(1.9)
ph (X, Dx , X )u(x)
ei(x−x )·ξ ph (x, ξ, x )u(x )dx d−ξ (u ∈ S) .
= Os —
450
Naoto Kumano-go
Definition 1.2 ( A Family of Pseudo-Differential Operators B m,λ
ρ,δ (h) ).
(1) We say that a family {ph }0<h<1 of C ∞ -functions ph ≡ ph (x, ξ) belongs to the class {Bm,λ
ρ,δ (h)}0<h<1 (m, λ ∈ R , 0 ≤ δ ≤ ρ ≤ 1) of
−δ
symbols, if ph (x, ξ)h x; hρ ξ
−λ belongs to Bm
ρ,δ (h).
m,λ
We write {ph }0<h<1 ∈ {Bm,λ
ρ,δ (h)}0<h<1 , or simply ph ∈ Bρ,δ (h).
(2) For ph (x, ξ) ∈ Bm,λ
ρ,δ (h), we define a family {Ph }0<h<1 of pseudodifferential operators Ph ≡ ph (X, Dx ) by
ph (X, Dx )u(x) ≡
(1.10)
eix·ξ ph (x, ξ)û(ξ)d−ξ (u ∈ S) .
m,λ
B m,λ
We write {Ph }0<h<1 ∈ {B
ρ,δ (h)}0<h<1 , or simply Ph ∈ B ρ,δ (h).
Remark.
1◦ . For ph ∈ Bm,λ
ρ,δ (h), we define semi-norms |ph |l
(m,λ)
(1.11)
(m,λ)
|ph |l
≡ max
sup sup
(l = 0, 1, 2, . . . ) by
|∂ξα Dxβ {ph (x, ξ)h−δ x; hρ ξ
−λ }|
hm+ρ|α|−δ|β|
|α+β|≤l 0<h<1 x,ξ
Then Bm,λ
ρ,δ (h) is a Fréchet space with these semi-norms.
2◦ . For ph (x, ξ) ∈ Bm,λ
ph by
ρ,δ (h), we define ph (x, ξ) ≡ h−m ph (hδ x, h−ρ ξ) .
(1.12)
Then we have
ph ∈ B0,λ
0,0 (h) ,
(1.13)
and
(1.14)
(0,λ)
|
ph |l
(m,λ)
( in B0,λ
0,0 (h)) = |ph |l
( in Bm,λ
ρ,δ (h)) .
.
Fundamental solution for Schrödinger equations
451
Proposition 1.3 ( Continuity on S ).
(1) There exist a constant Mm,h,l (depending only on m, h) and integers
l , l such that
(1.15)
|ph (X, Dx , X , Dx , X )u|l,S ≤ Mm,h,l |ph |l |u|l ,S
(m)
(ph (x, ξ, x , ξ , x ) ∈ Bm
ρ,δ (h), u ∈ S) .
(2) There exist a constant Mm,λ,h,l (depending only on m, λ, h) and integers lλ , lλ (depending only on λ) such that
(m,λ)
|ph (X, Dx )u|l,S ≤ Mm,λ,h,l |ph |l
(1.16)
λ
|u|lλ ,S
(ph (x, ξ) ∈ Bm,λ
ρ,δ (h), u ∈ S) .
Proof. See [6]. Proposition 1.4 ( Multi-products ).
There exist constants A and Cl , satisfying the following:
m
For pj,h (x, ξ, x ) ∈ Bρ,δj (h) (ν = 1, 2, . . . , j = 1, 2, . . . , ν + 1), define
qν+1,h (x, ξ, x ) by
(1.17)
qν+1,h (x, ξ, x ) ≡ Os —
···
exp(−i
ν
yj · ηj )
j=1
×
ν
pj,h (x + ȳ j−1 , ξ + η j , x + ȳ j )
j=1
· pν+1,h (x + ȳ ν , ξ, x )dy ν d−η ν · · · dy 1 d−η 1
(ȳ 0 ≡ 0, ȳ j ≡ y 1 + · · · + y j , j = 1, 2, . . . , ν) .
Then it follows that
(1.18)
(1.19)
qν+1,h (x, ξ, x ) ∈ Bρ,δν+1 (h) ,
m̄
qν+1,h (X, Dx , X )
= p1,h (X, Dx , X )p2,h (X, Dx , X ) · · · pν+1,h (X, Dx , X ) ,
452
Naoto Kumano-go
and
(1.20)
(m̄
)
|qν+1,h |l ν+1
≤ Cl A
ν+1
ν+1
max
l1 +l2 +···+lν+1 ≤l
(m )
j
|pj,h |lj +3n
,
0
j=1
where m̄ν+1 ≡ m1 + m2 + · · · + mν+1 .
Proof. See [6]. Proposition 1.5 ( Simplified Symbols ).
There exists a constant Cl satisfying the following:
For ph (x, ξ, x , ξ , x ) ∈ Bm
ρ,δ (h), define ph,L (x, ξ, x ) and ph,R (x, ξ, x ),
respectively by
−iy·η
ph,L (x, ξ, x ) ≡ Os —
e
ph (x, ξ + η, x + y, ξ, x )dyd−η
−iy·η
(1.21)
e
ph (x, ξ, x + y, ξ − η, x )dyd−η .
ph,R (x, ξ, x ) ≡ Os —
Then it follows that
(1.22)
(1.23)
ph,L (x, ξ, x ), ph,R (x, ξ, x ) ∈ Bm
ρ,δ (h) ,
ph (X, Dx , X , Dx , X ) = ph,L (X, Dx , X ) = ph,R (X, Dx , X ) ,
and
(1.24)
(m)
|ph,L |l
(m)
, |ph,R |l
(m)
≤ Cl |ph |l+2n0 .
Proof. See [6]. Proposition 1.6 ( The Inverse of (I − ph (X, Dx , X )) ).
Let the constant A be the same as that in Proposition 1.4. There exists
a constant Cl satisfying the following:
(0)
If ph (x, ξ, x ) ∈ B0ρ,δ (h) satisfies |ph |3n0 ≤ 1/(2A), there exists
qh (x, ξ, x ) ∈ B0ρ,δ (h) such that
(1.25)
I = qh (X, Dx , X )(I − ph (X, Dx , X ))
= (I − ph (X, Dx , X ))qh (X, Dx , X ) ,
Fundamental solution for Schrödinger equations
453
and
(0)
|qh |l
(1.26)
(0)
≤ Cl {max(1, |ph |l+3n0 )}l .
Proof. See [6]. Proposition 1.7 ( L 2 -Boundedness ).
There exists a constant C such that
(1.27)
ph (X, Dx , X )uL 2 ≤ Chm |ph |3n0 uL 2
(m)
(ph (x, ξ, x ) ∈ Bm
ρ,δ (h), u ∈ S) .
Proof. See [6] and [7]. Definition 1.8 ( Hilbert Spaces, H 0,h , H 2,h ).
(1) We define the Hilbert space H 0,h as the completion of S with respect
to the norm
u0,h ≡ hρ−δ uL 2 .
(1.28)
(2) We define the Hilbert space H 2,h as the completion of S with respect
to the norm
(1.29)
u2,h ≡
(h
−δ
1/2
α
ρ
β
x) (h Dx )
u(x)L2 2
.
|α+β|≤2
2.
Phase Functions
In this section, we define phase functions and investigate the properties
of multi-products of phase functions in more detail. This section plays an
essential role to prove the pointwise convergence of the phase function in
Section 5. The definition of phase functions below is a little different from
that in Kitada and H.Kumano-go [6]. This definition is essential in this
paper.
454
Naoto Kumano-go
Definition 2.1 ( A Family of Phase Functions Pρ,δ (τ, h, {κl }∞
l=0 ) ).
For τ ≥ 0 and a sequence of positive constants {κl }∞
(κ
=
1, κl ≤
l=0 0
∞
κl+1 ), we say that a family {φh }0<h<1 of real-valued C -functions φh (x, ξ)
belongs to the class {Pρ,δ (τ, h, {κl }∞
l=0 )}0<h<1 of phase functions, if
Jh (x, ξ) ≡ φh (x, ξ) − x · ξ satisfies
(2.1)
sup
0<h<1
sup
|α+β|≤1
+
|∂ξα Dxβ {hρ−δ Jh (hδ x, h−ρ ξ)}|
x; ξ
2−|α+β|
x,ξ
2≤|α+β|≤2+l
sup |∂ξα Dxβ {hρ−δ Jh (hδ x, h−ρ ξ)}|
x,ξ
≤ κl τ ,
for any l = 0, 1, 2, . . . .
We write {φh }0<h<1 ∈ {Pρ,δ (τ, h, {κl }∞
l=0 )}0<h<1 , or simply φh ∈
∞
Pρ,δ (τ, h, {κl }l=0 ). Furthermore, for simplicity, we sometimes omit {κl }∞
l=0
and write simply Pρ,δ (τ, h).
Remark. For φh ∈ Pρ,δ (τ, h, {κl }∞
l=0 ), we define φh and Jh respectively
by
(2.2)
φh (x, ξ) ≡ hρ−δ φh (hδ x, h−ρ ξ)
h (x, ξ) ≡ hρ−δ Jh (hδ x, h−ρ ξ) .
J
Then we have
(2.3)
h ∈ P0,0 (τ, h, {κ}∞ ) .
φ
l=0
Proposition 2.2.
∞
τj < 1/2.
Let φj,h ∈ Pρ,δ (τj , h)(j = 1, 2, . . . ) and
j=1
Then, for any ν = 1, 2, 3, . . . , 0 < h < 1 and (x, ξ) ∈ R 2n , the solution
j
j
{Xν,h
, Ξν,h
}νj=1 (x, ξ) (∈ R 2νn ) of the equation
(2.4)
 j
j−1
j
, Ξν,h
)
X = ∇ξ φj,h (Xν,h

 ν,h
j
j
j+1
= ∇x φj+1,h (Xν,h
, Ξν,h
)
Ξν,h


0 ≡ x, Ξ ν+1 ≡ ξ) ,
(j = 1, . . . , ν, Xν,h
ν,h
Fundamental solution for Schrödinger equations
j ,Ξ
j }ν (x, ξ) (∈ R 2νn ) of the equation
and the solution {X
ν,h
ν,h j=1
 j
j,h (X
j−1 , Ξ
j )
X = ∇ξ φ
ν,h
ν,h

 ν,h
j
j
j+1 )
Ξν,h = ∇x φj+1,h (Xν,h , Ξ
ν,h


0
ν+1 ≡ ξ) ,
(j = 1, . . . , ν, Xν,h ≡ x, Ξ
ν,h
(2.5)
exist uniquely. Furthermore, it follows that
j
j (x, ξ) ∈ C ∞ (R
R 2n ) (j = 0, 1, . . . , ν)
Xν,h (x, ξ), X
ν,h
(2.6)
j
j (x, ξ) ∈ C ∞ (R
R 2n ) (j = 1, 2, . . . , ν + 1) ,
(x, ξ), Ξ
Ξν,h
ν,h
and
(2.7)
j (x, ξ) = h−δ X j (hδ x, h−ρ ξ) (j = 0, 1, . . . , ν)
X
ν,h
ν,h
j (x, ξ) = hρ Ξ j (hδ x, h−ρ ξ) (j = 1, 2, . . . , ν + 1) .
Ξ
ν,h
ν,h
Proof. See [6]. Definition 2.3 ( Multi-products of Phase Functions ).
∞
τj < 1/2 .
Let φj,h ∈ Pρ,δ (τj , h)(j = 1, 2, . . . ) and
j=1
For ν = 1, 2, . . . , we define the #-(ν + 1) products
(2.8)
Φν+1,h ≡ φ1,h #φ2,h # · · · #φν+1,h
1,h #φ
2,h # · · · #φ
ν+1,h ,
ν+1,h ≡ φ
Φ
respectively by
(2.9)

ν+1
ν
j−1
j
j
j


 Φν+1,h (x, ξ) ≡ j=1 φj,h (Xν,h , Ξν,h ) − j=1 Xν,h · Ξν,h
ν+1
ν


j,h (X
ν+1,h (x, ξ) ≡ φ
j ·Ξ
j−1 , Ξ
j ) − X
j .
Φ
ν,h
ν,h
ν,h
ν,h
j=1
j=1
Remark. Set
Jj,h (x, ξ) ≡ φj,h (x, ξ) − x · ξ (j = 1, 2, . . . , ν + 1)
(2.10)
j,h (x, ξ) − x · ξ (j = 1, 2, . . . , ν + 1) ,
j,h (x, ξ) ≡ φ
J
455
456
Naoto Kumano-go
and
(2.11)
J ν+1,h (x, ξ) ≡ Φν+1,h (x, ξ) − x · ξ
ν+1,h (x, ξ) − x · ξ .
J ν+1,h (x, ξ) ≡ Φ
Then we get
(2.12)

ν+1
ν+1,h (x, ξ) = J
j,h (X
j−1 , Ξ
j )

J

ν,h
ν,h


j=1




ν
j −X
j−1 ) · (Ξ
j − ξ)
(X
−
ν,h
ν,h
ν,h

j=1




ν+1


j,h )(X
j−1 , Ξ
j ).
 ∇(x,ξ)J ν+1,h (x, ξ) =
(∇(x,ξ) J
ν,h
j=1
ν,h
Furthermore, using (2.7), we have
(2.13)
ν+1,h (x, ξ) = hρ−δ Φν+1,h (hδ x, h−ρ ξ) .
Φ
Proposition 2.4.
(1) Let φj,h ∈ Pρ,δ (τj , h)(j = 1, 2, . . . ) and
∞
τj ≤ 1/4.
j=1
Then it follows that
(2.14)
 j
j−1 |, |Ξ
j−1 − Ξ
j | ≤ 2τj x; ξ
|Xν,h − X

ν,h
ν,h
ν,h




j−1
−1
 2 x; ξ
≤ x + θ(X
j


ν,h − x); ξ + θ(Ξν,h − ξ)
≤ 2x; ξ




j − ξ))| ≤ |f |0,B
j−1 − x), ξ + θ(Ξ
 |f (x + θ(X
ν,h
ν,h

j−1 − x), ξ + θ(Ξ
j − ξ))| ≤ 2|g| 1,∞ x; ξ
|g(x + θ(X

0,B

ν,h
ν,h






ν = 1, 2, . . . , j = 1, 2, . . . , ν + 1, |θ| ≤ 1, 0 < h < 1,




.



ν+1 ≡ X
ν , Ξ
0 ≡ Ξ
1
R 2n ), X
R 2n ), g(x, ξ) ∈ B1,∞ (R
f (x, ξ) ∈ B(R
ν,h
ν,h
ν,h
ν,h
(2) There exists a constant c0 satisfying the following:
∞
τj ≤ 1/4.
Let φj,h ∈ Pρ,δ (τj , h)(j = 1, 2, . . . ) and
j=1
Fundamental solution for Schrödinger equations
457
Then it follows that
 →
→
j −X
j−1 )|, |∇x (Ξ
j−1 − Ξ
j )| ≤ c τj

∇
(
X
|
x

0
ν,h
ν,h
ν,h
ν,h



→
 → j
j−1 )|, |∇ξ (Ξ
j−1 − Ξ
j )| ≤ c τj
|∇ξ (Xν,h − X
0
ν,h
ν,h
ν,h


(ν = 1, 2, . . . , j = 1, 2, . . . , ν + 1, 0 < h < 1,




ν+1 ≡ X
ν , Ξ
0 ≡ Ξ
1 ).
X
ν,h
ν,h
ν,h
ν,h
(2.15)
(3) For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exist constants cl (depending only on κ1 , κ2 , . . . , κl−1 , κl ) and cl−1 , c
l−1 (depending only
on κ1 , κ2 , . . . , κl−1 ), satisfying the following:
∞
)
(j
=
1,
2,
.
.
.
)
and
τj ≤ 1/4.
Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
l=0
j=1
Then it follows that
(2.16)

→
→
j − X
j−1 )|, |∂ α Dxβ ∇x (Ξ
j−1 − Ξ
j )| ≤ c
 |∂ α Dxβ ∇x (X


ξ
ξ
ν,h
ν,h
ν,h
ν,h
|α+β| τj



→
→


j − X
j−1 )|, |∂ α Dxβ ∇ξ (Ξ
j−1 − Ξ
j )| ≤ c

|∂ξα Dxβ ∇ξ (X

ξ
ν,h
ν,h
ν,h
ν,h
|α+β| τj




 |∂ α Dβ f (x + θ(X
j−1 − x), ξ + θ(Ξ
j − ξ))| ≤ c
|f |
x
ξ
ν,h
ν,h
|α+β|−1
|α+β|,B

β

j−1 − x), ξ + θ(Ξ
j − ξ))| ≤ c

|∂ξα Dx g(x + θ(X

ν,h
ν,h
|α+β|−1 |g||α+β|,B1,∞







ν = 1, 2, . . . , j = 1, 2, . . . , ν + 1, |θ| ≤ 1, |α + β| ≥ 1, 0 < h < 1,




.


 f (x, ξ) ∈ B(R
ν+1 ≡ X
ν , Ξ
0 ≡ Ξ
1
R 2n ), X
R 2n ), g(x, ξ) ∈ B1,∞ (R
ν,h
ν,h
ν,h
ν,h
Proof. By induction. See [6].
Proposition 2.5.
For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exist a constant c0 > 1(inde∞
pendent of {κl }l=0 ) and a sequence of positive constants {κl }∞
l=0 (κ0 = 1,
κl ≤ κl+1 , κl ≤ κl ) (depending only on κ1 , κ2 , . . . , κl ), satisfying the following:
∞
(1) Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
)(j
=
1,
2,
.
.
.
)
and
τj ≤ 1/4.
l=0
j=1
Then, for any ν = 1, 2, . . . , it follows that
(2.17)
Φν+1,h ≡ (φ1,h #φ2,h # · · · #φν+1,h ) ∈ Pρ,δ (c0 τ̄ν+1 , h, {κl }∞
l=0 )
1,h #φ
2,h # · · · #φ
ν+1,h ) ∈ P0,0 (c0 τ̄ν+1 , h, {κ }∞ ) ,
ν+1,h ≡ (φ
Φ
l l=0
458
Naoto Kumano-go
where τ̄ν+1 ≡ τ1 + τ2 + · · · + τν+1 .
(2) Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
l=0 )(j = 1, 2, . . . ) and
∞
τj ≤ 1/(4c0 ).
j=1
Then, for any ν = 1, 2, . . . , it follows that
Φν+2,h ≡ (φ1,h #φ2,h # · · · #φν+1,h #φν+2,h )
(2.18)
= (φ1,h #φ2,h # · · · #φν+1,h )#φν+2,h
= φ1,h #(φ2,h # · · · #φν+1,h #φν+2,h ) .
Furthermore, if {Xh , Ξh }(x, ξ) is the solution of the equations
(2.19)
Xh = ∇ξ Φν+1,h (x, Ξh )
Ξh = ∇x φν+2,h (Xh , ξ) ,
then it follows that
j
(x, ξ)
Xν+1,h
(2.20)
=
j
Xν,h
(x, Ξh ) (j = 1, 2, . . . , ν)
Xh (x, ξ) (j = ν + 1) ,
and
(2.21)
j
(x, ξ)
Ξν+1,h
=
j
Ξν,h
(x, Ξh ) (j = 1, 2, . . . , ν)
Ξh (x, ξ) (j = ν + 1) .
Proof. See [6]. Lemma 2.6.
(1) Let φj,h ∈ Pρ,δ (τj , h)(j = 1, 2, . . . ) and
∞
τj ≤ 1/4.
j=1
Then it follows that
(2.22)
j − ξ) + θ η
±1
j−1 − x) + θ y; ξ + θ(Ξ
x + θ(X
ν,h
ν,h
≤ 4y; η
x; ξ
±1
(ν = 1, 2, . . . , j = 1, 2, . . . , ν + 1,
|θ|, |θ |, |θ | ≤ 1, 0 < h < 1) .
Fundamental solution for Schrödinger equations
459
(2) For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), λ ∈ R , α and β , there exists
a constant Cλ,α ,β ,l−1 (depending only on λ, α , β , κ1 , κ2 , . . . , κl−1 )
satisfying the following:
∞
Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
τj ≤ 1/4.
l=0 ) (j = 1, 2, . . . ) and
j=1
Then it follows that
j−1 − x) + θ y; ξ + θ(Ξ
j − ξ) + θ η
λ |
(2.23) |∂ξα Dxβ ∂ηα Dyβ x + θ(X
ν,h
ν,h
≤ Cλ,α ,β ,|α+β|−1 y; η
|λ| x; ξ
λ
(ν = 1, 2, . . . , j = 1, 2, . . . , ν + 1,
|θ|, |θ |, |θ | ≤ 1, 0 < h < 1) .
Proof. By induction. Note Proposition 2.4. 3
j=1
For φj,h ∈ Pρ,δ (τj , h)(j = 1, 3) and φ2,ϑ,h ∈ Pρ,δ (τ2 , h)(0 ≤ ϑ ≤ 1) with
j ,Ξ
j }2 (x, ξ) be the solution of the equations
τj < 1/2, let {X
2,ϑ,h
2,ϑ,h j=1
(2.24)
 1
1,h (X
0 ,Ξ
1 ),
X2,ϑ,h = ∇ξ φ
2,ϑ,h
2,ϑ,h




1
1
2,ϑ,h (X
2

= ∇x φ
Ξ

2,ϑ,h , Ξ2,ϑ,h )

 2,ϑ,h
1
2
2
X
2,ϑ,h = ∇ξ φ2,ϑ,h (X2,ϑ,h , Ξ2,ϑ,h ),




2
3
2
Ξ

2,ϑ,h = ∇x φ3,h (X2,ϑ,h , Ξ2,ϑ,h )



0
3
(X
2,ϑ,h ≡ x, Ξ2,ϑ,h ≡ ξ, 0 ≤ ϑ ≤ 1) .
2,ϑ,h (x, ξ) be continuously differentiable on 0 ≤ ϑ ≤ 1.
Furthermore, let φ
Set
(2.25)
2,ϑ,h (x, ξ) − x · ξ
J2,ϑ,h (x, ξ) ≡ φ
1,h #φ
2,ϑ,h #φ
3,h )(x, ξ) − x · ξ .
J 3,ϑ,h (x, ξ) ≡ (φ
Then we get the following lemma.
460
Naoto Kumano-go
Lemma 2.7.
∞
For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ) and {χl }l=0 (0 ≤ χl ≤ χl+1 ), there
exist a constant Cl and a sequence of positive constants {χl }∞
l=0 (0 ≤ χl ≤
χl+1 , χl ≤ χl ) (depending only on κ1 , κ2 , . . . , κl and χ0 , χ1 , . . . , χl ), satisfying the following:
∞
Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
l=0 )(j = 1, 3), φ2,ϑ,h ∈ Pρ,δ (τ2 , h, {κl }l=0 )(0 ≤
3
2,ϑ,h (x, ξ) satisfy
ϑ ≤ 1) and
τj ≤ 1/4, and let ∂ϑ J
j=1
(2.26)

2
2

 |∂ϑ J2,ϑ,h (x, ξ)| ≤ χ0 (τ2 ) x; ξ
2,ϑ,h (x, ξ)| ≤ χ|α+β| (τ2 )2 x; ξ
|∂ξα Dxβ ∇(x,ξ) ∂ϑ J


(0 < h < 1, 0 ≤ ϑ ≤ 1) .
Then it follows that
(2.27)
j |, |∂ α Dxβ ∂ϑ Ξ
j | ≤ C|α+β| (τ2 )2 x; ξ
|∂ξα Dxβ ∂ϑ X
ξ
2,ϑ,h
2,ϑ,h
(j = 1, 2, 0 < h < 1, 0 ≤ ϑ ≤ 1) ,
and
(2.28)

(x, ξ)| ≤ χ0 (τ2 )2 x; ξ
2
|∂ J

 ϑ 3,ϑ,h
|∂ξα Dxβ ∇(x,ξ) ∂ϑJ 3,ϑ,h (x, ξ)| ≤ χ|α+β| (τ2 )2 x; ξ


(0 < h < 1, 0 ≤ ϑ ≤ 1) .
Proof. By induction. Note Proposition 2.4 and Lemma 2.6. Let φj,h ∈ Pρ,δ (τj , h)(j = 1, 2, . . . ) and
∞
τj < 1/2.
j=1
For ν = 1, 2, . . . and 0 ≤ ϑ ≤ 1, set

ν+1

(x,
ξ)
≡
ϑΦ
(x,
ξ)
+
(1
−
ϑ)
x
·
ξ
+
J
(x,
ξ)
Φ

ν+1,ϑ,h
ν+1,h
j,h

j=1
(2.29)
ν+1


j,h (x, ξ) ,
ν+1,h (x, ξ) + (1 − ϑ) x · ξ + J
ν+1,ϑ,h (x, ξ) ≡ ϑΦ
Φ
j=1
and
(2.30)
J ν+1,ϑ,h (x, ξ) ≡ Φν+1,ϑ,h (x, ξ) − x · ξ
ν+1,ϑ,h (x, ξ) − x · ξ .
J ν+1,ϑ,h (x, ξ) ≡ Φ
Fundamental solution for Schrödinger equations
461
Then we have
(2.31)

Φν+1,1,h (x, ξ) = Φν+1,h (x, ξ),




ν+1



(x,
ξ)
=
x
·
ξ
+
Jj,h (x, ξ)
Φ

ν+1,0,h

j=1









ν+1,h (x, ξ),
ν+1,1,h (x, ξ) = Φ
Φ
ν+1
j,h (x, ξ) ,
ν+1,0,h (x, ξ) = x · ξ + J
Φ
j=1
and
(2.32)
ν+1,ϑ,h (x, ξ) = hρ−δ Φν+1,ϑ,h (hδ x, h−ρ ξ) .
Φ
Furthermore, we get the following lemma.
Lemma 2.8.
For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exist a constant c0 > 1(inde ∞
pendent of {κl }∞
l=0 ) and sequences of positive constants {κl }l=0 (κ0 = 1, κl ≤
κl+1 , κl ≤ κl ), {χl }∞
l=0 (0 ≤ χl ≤ χl+1 )(depending only on κ1 , κ2 , . . . , κl ),
satisfying the following:
∞
)(j
=
1,
2,
.
.
.
)
and
τj ≤ 1/4.
Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
l=0
j=1
Then it follows that
(2.33)
Φν+1,ϑ,h ∈ Pρ,δ (c0 τ̄ν+1 , h, {κl }∞
l=0 )
ν+1,ϑ,h ∈ P0,0 (c0 τ̄ν+1 , h, {κ }∞ ) ,
Φ
l l=0
and
(2.34)

2
2

 |∂ϑJ ν+1,ϑ,h (x, ξ)| ≤ χ0 (τ̄ν+1 ) x; ξ
|∂ξα Dxβ ∇(x,ξ) ∂ϑJ ν+1,ϑ,h (x, ξ)| ≤ χ|α+β| (τ̄ν+1 )2 x; ξ


(ν = 1, 2, . . . , 0 ≤ ϑ ≤ 1, 0 < h < 1) ,
where τ̄ν+1 ≡ τ1 + τ2 + · · · + τν+1 .
462
Naoto Kumano-go
Proof. From (2.12), we have
(2.35)
∂ϑJ ν+1,ϑ,h (x, ξ) = −
+
ν+1 1
→
0
j=1
ν
j −X
j−1 ) · (Ξ
j − ξ)
(X
ν,h
ν,h
ν,h
j=1
j,h )(x + θ(X
j−1 − x), ξ + θ(Ξ
j − ξ))dθ
(∇(x,ξ) J
ν,h
ν,h
j−1
Xν,h − x
·
j − ξ ,
Ξ
ν,h
and
(2.36) ∇(x,ξ) ∂ϑJ ν+1,ϑ,h (x, ξ)
ν+1 1
→
j,h )(x + θ(X
j−1 − x), ξ + θ(Ξ
j − ξ))dθ
=
(∇(x,ξ) ∇(x,ξ) J
ν,h
ν,h
j=1
0
j−1
Xν,h − x
·
j − ξ .
Ξ
ν,h
Using Proposition 2.4, we get (2.34). 3.
Fourier Integral Operators
In this section, we introduce the basic properties of the Fourier integral
operators defined by Kitada and H.Kumano-go in [6]. For the details, see
[6]. In what follows, for simplicity, let n0 ≡ 2[n/2 + 1].
m
∗
Definition 3.1 ( Fourier Integral Operators B m
ρ,δ (φh ), B ρ,δ (φh ) ).
(1) For 0 ≤ τ < 1, φh ∈ Pρ,δ (τ, h) and ph (x, ξ) ∈ Bm
ρ,δ (h), we define a
family of Fourier integral operators ph (φh ; X, Dx ) by
(3.1)
ph (φh ; X, Dx )u(x) =
eiφh (x,ξ) ph (x, ξ)û(ξ)d−ξ (u ∈ S).
Bm
We write {ph (φh ; X, Dx )}0<h<1 ∈ {B
ρ,δ (φh )}0<h<1 , or simply
m
ph (φh ; X, Dx ) ∈ B ρ,δ (φh ). We sometimes use this definition for
0 ≤ τ < ∞. ( See Theorem 4.1.)
Fundamental solution for Schrödinger equations
463
(2) For 0 ≤ τ < 1, φh ∈ Pρ,δ (τ, h) and qh (ξ, x ) ∈ Bm
ρ,δ (h), we define a
family of conjugate Fourier integral operators qh (φ∗h ; Dx , X ) by
∗
(3.2)
F[qh (φ ; Dx , X )u](ξ) = e−iφh (x ,ξ) qh (ξ, x )u(x )dx (u ∈ S).
∗
Bm
We write {qh (φ∗h ; Dx , X )}0<h<1 ∈ {B
ρ,δ (φh )}0<h<1 , or simply
m
∗
∗
qh (φh ; Dx , X ) ∈ B ρ,δ (φh ).
Definition 3.2 ( Fourier Integral Operators B m,λ
ρ,δ (φh ) ).
For 0 ≤ τ < 1, φh ∈ Pρ,δ (τ, h) and ph (x, ξ) ∈ Bm,λ
ρ,δ (h), we define a family
of Fourier integral operators ph (φh ; X, Dx ) by
(3.3)
ph (φh ; X, Dx )u(x) = eiφh (x,ξ) ph (x, ξ)û(ξ)d−ξ (u ∈ S).
We
write
{ph (φh ; X, Dx )}0<h<1
B m,λ
ρ,δ (φh ).
ph (φh ; X, Dx ) ∈
( See Theorem 4.2.)
B m,λ
{B
ρ,δ (φh )}0<h<1 ,
∈
or
simply
We sometimes use this definition for 0 ≤ τ < ∞.
In the following proposition, it is a revised point that Mm,h,l , Mm,λ,h,l , l
and l are independent of the choice of phase functions as well as the choice
of symbols.
Proposition 3.3 ( Continuity on S ).
(1) For any 0 ≤ τ0 < 1 and {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exist
a constant Mm,h,l (depending only on m, h, τ0 , {κl }∞
l=0 (κ0 = 1, κl ≤
κl+1 )) and integers l , l such that

(m)
|ph (φh ; X, Dx )u|l,S ≤ Mm,h,l |ph |l |u|l ,S




(m)

|qh (φ∗h ; Dx , X )u|l,S ≤ Mm,h,l |qh |l |u|l ,S
(3.4)

(0 ≤ τ ≤ τ0 , φh ∈ Pρ,δ (τ, h, {κl }∞

l=0 ),



ph (x, ξ), qh (ξ, x ) ∈ Bm
ρ,δ (h), u ∈ S) .
(2) For any 0 ≤ τ0 < 1 and {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exist a constant Mm,λ,h,l (depending only on m, λ, h, τ0 , {κl }∞
l=0 (κ0 =
1, κl ≤ κl+1 )) and integers lλ , lλ (depending only on λ) such that
(3.5)
(m,λ)
|ph (φh ; X, Dx )u|l,S ≤ Mm,λ,h,l |ph |l
λ
|u|lλ ,S
m,λ
(0 ≤ τ ≤ τ0 , φh ∈ Pρ,δ (τ, h, {κl }∞
l=0 ), ph (x, ξ) ∈ Bρ,δ (h), u ∈ S) .
464
Naoto Kumano-go
Proof. See [6]. m
∗
∗ B m (φ ) ⊂ B m+m (h)).
Bm
Bm
Proposition 3.4 (B
ρ,δ (φh ), B ρ,δ (φh )◦B
ρ,δ (φh )◦B
ρ,δ h
ρ,δ
For any 0 ≤ τ0 < 1 and {κl }∞
(κ
=
1,
κ
≤
κ
),
there
exists a
l
l+1
l=0 0
constant Cl+2n0 (depending only on τ0 , κ1 , κ2 , . . . , κl+2n0 ) satisfying the following:
m
Let 0 ≤ τ ≤ τ0 , φh ∈ Pρ,δ (τ, h, {κl }∞
l=0 ), ph (x, ξ) ∈ B ρ,δ (h) and
m+m
(h) such
qh (ξ, x ) ∈ B m
ρ,δ (h). Then there exist rh (x, ξ), rh (ξ, x ) ∈ B ρ,δ
that
ph (φh ; X, Dx )qh (φ∗h ; Dx , X ) = rh (X, Dx )
(3.6)
qh (φ∗h ; Dx , X )ph (φh ; X, Dx ) = rh (Dx , X ) ,
and
(3.7)
(m+m )
|rh |l
(m+m )
, |rh |l
(m)
(m )
≤ Cl+2n0 |ph |l+2n0 |qh |l+2n0 .
Proof. See [6]. m
m+m
Bm
Bm
Proposition 3.5 ( B m
(φh ) ).
ρ,δ (φh ), B ρ,δ (φh )◦B
ρ,δ (h)◦B
ρ,δ (h) ⊂ B ρ,δ
∞
For any 0 ≤ τ0 < 1 and {κl }l=0 (κ0 = 1, κl ≤ κl+1 ), there exists a
constant Cl+2n0 −1 (depending only on τ0 , κ1 , κ2 , . . . , κl+2n0 −1 ) satisfying the
following:
m
Let 0 ≤ τ ≤ τ0 , φh ∈ Pρ,δ (τ, h, {κl }∞
l=0 ), ph (x, ξ) ∈ B ρ,δ (h) and qh (x, ξ) ∈
m+m
(h) such that
Bm
ρ,δ (h). Then there exist rh (x, ξ), rh (x, ξ) ∈ B ρ,δ
(3.8)
ph (X, Dx )qh (φh ; X, Dx ) = rh (φh ; X, Dx )
qh (φh ; X, Dx )ph (X, Dx ) = rh (φh ; X, Dx ) ,
and
(3.9)
(m+m )
|rh |l
(m+m )
, |rh |l
Proof. See [6]. (m)
(m )
≤ Cl+2n0 −1 |ph |l+2n0 |qh |l+2n0 .
Fundamental solution for Schrödinger equations
465
m
∗
∗ B m (h) ⊂ B m+m (φ∗ ) ).
Bm
Proposition 3.6 ( B m
ρ,δ (φh ), B ρ,δ (φh )◦B
ρ,δ (h)◦B
ρ,δ
h
ρ,δ
(κ
=
1,
κ
≤
κ
),
there
exists
a
For any 0 ≤ τ0 < 1 and {κl }∞
0
l
l+1
l=0
constant Cl+2n0 −1 (depending only on τ0 , κ1 , κ2 , . . . , κl+2n0 −1 ) satisfying the
following:
m
Let 0 ≤ τ ≤ τ0 , φh ∈ Pρ,δ (τ, h, {κl }∞
l=0 ), ph (ξ, x ) ∈ B ρ,δ (h) and
m+m
(h) such
qh (ξ, x ) ∈ B m
ρ,δ (h). Then there exist rh (ξ, x ), rh (ξ, x ) ∈ B ρ,δ
that
(3.10)
ph (Dx , X )qh (φ∗h ; Dx , X ) = rh (φ∗h ; Dx , X )
qh (φ∗h ; Dx , X )ph (Dx , X ) = rh (φ∗h ; Dx , X ) ,
and
(3.11)
(m+m )
|rh |l
(m+m )
, |rh |l
(m)
(m )
≤ Cl+2n0 −1 |ph |l+2n0 |qh |l+2n0 .
Proof. See [6]. Proposition 3.7 ( L 2 -Boundedness ).
For any 0 ≤ τ0 < 1 and {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exists a
constant C5n0 (depending only on τ0 , κ1 , κ2 , . . . , κ5n0 ) such that
(3.12)

(m)
ph (φh ; X, Dx )uL2 ≤ C5n0 hm |ph |5n0 uL2





(m)
qh (φ∗h ; Dx , X )uL 2 ≤ C5n0 hm |qh |5n0 uL 2


(0 ≤ τ ≤ τ0 , φh ∈ Pρ,δ (τ, h, {κl }∞

l=0 ),


m
ph (x, ξ), qh (ξ, x ) ∈ Bρ,δ (h), u ∈ S) .
Proof. See [6]. Proposition 3.8 ( Inverse of I(φh ), I(φ∗h ) ).
For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exist constants 0 < τ0 <
1(depending only on κ1 , κ2 , . . . , κ5n0 ) and Cl+7n0 (depending only on
κ1 , κ2 , . . . , κl+7n0 ), satisfying the following:
Let φh ∈ Pρ,δ (τ, h, {κl }∞
l=0 ) and 0 ≤ τ ≤ τ0 .
466
Naoto Kumano-go
Then there exist rh (ξ, x ), rh (x, ξ) ∈ B0ρ,δ (h) such that
(3.13)
I(φh )rh (φ∗h ; Dx , X ) = rh (φ∗h ; Dx , X )I(φh ) = I
I(φ∗h )rh (φh ; X, Dx ) = rh (φh ; X, Dx )I(φ∗h ) = I ,
and
|rh |l , |rh |l
(0)
(3.14)
(0)
≤ Cl+7n0 .
Proof. See [6]. 4.
Multi-Products of Fourier Integral Operators
In this section, we apply Fujiwara’s skip method in [3] to Kitada and
H.Kumano-go’s theory of multi-products of Fourier integral operators. This
section plays an essential role to prove the pointwise convergence of the
symbol in Section 5.
m2
m1 +m2
1
Theorem 4.1 ( B m
(φ1,h #φ2,h ) ).
ρ,δ (φ1,h ) ◦ B ρ,δ (φ2,h ) ⊂ B ρ,δ
∞
For any {κl }l=0 (κ0 = 1, κl ≤ κl+1 ), there exist constants C3l+2n+1 (de
pending only on κ1 , κ2 , . . . , κ3l+2n+1 ) and C3l+2n+3
(depending only on
κ1 , κ2 , . . . , κ3l+2n+3 ), satisfying the following:
mj
Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
l=0 ), pj,h (x, ξ) ∈ Bρ,δ (h)(j = 1, 2), τ1 + τ2 ≤
1/4, and {Xh , Ξh }(x, ξ) be the solution of the equations
(4.1)
Xh = ∇ξ φ1,h (x, Ξh )
Ξh = ∇x φ2,h (Xh , ξ) .
1 +m2
Then there exist qh (x, ξ), rh (x, ξ) ∈ Bm
(h) such that
ρ,δ
(4.2)
qh (x, ξ) = p1,h (x, Ξh (x, ξ))p2,h (Xh (x, ξ), ξ) + rh (x, ξ) ,
(4.3)
p1,h (φ1,h ; X, Dx )p2,h (φ2,h ; X, Dx ) = qh (φ1,h #φ2,h ; X, Dx ) ,
(4.4)
(m1 +m2 )
|qh |l
(m )
(m )
1
2
≤ C3l+2n+1 |p1,h |3l+2n+1
|p2,h |3l+2n+1
,
Fundamental solution for Schrödinger equations
467
and
(4.5)
(m1 +m2 )
|rh |l
1
1
≤ C3l+2n+3
(τ1 |p1,h |3l+2n+3
+ max |∂ξj p1,h |3l+2n+3
)
(m )
(m +ρ)
1≤j≤n
×
(m2 )
(τ2 |p2,h |3l+2n+3
(m −δ)
2
+ max |Dxj p2,h |3l+2n+3
).
1≤j≤n
Proof.
(I) For u ∈ S, we can write
(4.6)
p1,h (φ1,h ; X, Dx )p2,h (φ2,h ; X, Dx )u(x)
= ei(φ1,h #φ2,h )(x,ξ) qh (x, ξ)û(ξ)d−ξ ,
where
(4.7) ψh (x, ξ , x , ξ) ≡ φ1,h (x, ξ ) − x · ξ + φ2,h (x , ξ) − (φ1,h #φ2,h )(x, ξ) ,
and
(4.8)
Set
(4.9)
qh (x, ξ) ≡ Os —
eiψh (x,ξ ,x ,ξ) p1,h (x, ξ )p2,h (x , ξ)dx d−ξ .

q (x, ξ) ≡ h−(m1 +m2 ) qh (hδ x, h−ρ ξ)

 h
pj,h (x, ξ) ≡ h−mj pj,h (hδ x, h−ρ ξ) ∈ B00,0 (h) (j = 1, 2)


ψh (x, ξ , x , ξ) ≡ hρ−δ ψh (hδ x, h−ρ ξ , hδ x , h−ρ ξ) .
By a change of variables: (x, ξ, x , ξ ) → (hδ x, h−ρ ξ, hδ x , h−ρ ξ ), we have
−2σ −2nσ
(4.10)
Os —
eih ψh (x,ξ ,x ,ξ)
qh (x, ξ) = h
·
p1,h (x, ξ )
p2,h (x , ξ)dx d−ξ ,
where σ ≡ (ρ − δ)/2. Furthermore, set
 −δ
δ
−ρ

 Xh (x, ξ) ≡ h Xh (h x, h ξ)
h (x, ξ) ≡ hρ Ξh (hδ x, h−ρ ξ)
Ξ
(4.11)


Jj,h (x, ξ) ≡ hρ−δ φj,h (hδ x, h−ρ ξ) − x · ξ (j = 1, 2) .
468
Naoto Kumano-go
h (x, ξ) + hσ y, ξ = Ξ
h (x, ξ) + hσ η, we have
By a change of variables: x = X
qh (x, ξ) = Os —
(4.12)
h (x, ξ) + hσ η)
eiϕh (y,η;x,ξ)
p1,h (x, Ξ
h (x, ξ) + hσ y, ξ)dyd−η ,
·
p2,h (X
where
h (x, Ξ
h (x, ξ) + hσ η, X
h (x, ξ) + hσ y, ξ) .
ϕ
h (y, η; x, ξ) ≡ h−2σ ψ
(4.13)
Setting
(4.14)
h (η; x, ξ) ≡
A
h (y; x, ξ) ≡
B
1
0
1
→
1,h )(x, Ξ
h (x, ξ) + θhσ η)dθη · η
(1 − θ)(∇ξ ∇ξ J
→
σ
0 (1 − θ)(∇x ∇x J2,h )(Xh (x, ξ) + θh y, ξ)dθy · y ,
we can write
(4.15)
h (η; x, ξ) + B
h (y; x, ξ) .
ϕ
h (y, η; x, ξ) = −y · η + A
(II) For 0 ≤ ϑ ≤ 1, set
h (ϑη; x, ξ) + B
h (y; x, ξ) .
ϕ
ϑ,h (y, η; x, ξ) ≡ −y · η + A
(4.16)
Then we have
(4.17)
h (y, η; x, ξ)
ϕ
1,h (y, η; x, ξ) = ϕ
h (y; x, ξ) .
ϕ
0,h (y, η; x, ξ) = −y · η + B
Furthermore, we get
(4.18)
h (η; x, ξ) =
∇η A
h (y; x, ξ) =
∇y B
1
→
σ
0 (∇ξ ∇ξ J1,h )(x, Ξh (x, ξ) + θh η)dθη
1
0
→
2,h )(X
h (x, ξ) + θhσ y, ξ)dθy ,
(∇x ∇x J
and
(4.19)
h )(ϑη; x, ξ)
ϑ,h (y, η; x, ξ) = −y + ϑ(∇η A
∇η ϕ
h )(y; x, ξ) .
ϑ,h (y, η; x, ξ) = −η + (∇y B
∇y ϕ
Fundamental solution for Schrödinger equations
469
By Proposition 2.4, there exists a constant Cl (depending only on κ1 , κ2 , . . . ,
κl ) such that
(4.20)
h )(ϑη; x, ξ)| ≤ C |∂ξα Dxβ ∂ηα (∇η A
|α+β+α | τ1 η
(0 ≤ ϑ ≤ 1)
h )(y; x, ξ)| ≤ C |∂ξα Dxβ Dyβ (∇y B
|α+β+β | τ2 y
,
and
(4.21)
 −1
ϑ,h (y, η; x, ξ)
≤ 3y; η
3 y; η
≤ ∇(y,η) ϕ




 |∂ α Dxβ ∂ηα Dyβ ∇(y,η) ϕ
ϑ,h (y, η; x, ξ)| ≤ C|α+β+α
+β | y; η
ξ

2

ϑ,h (y, η; x, ξ)| ≤ C|α+β+α
|∂ξα Dxβ ∂ηα Dyβ ϕ
+β | y; η



(0 ≤ ϑ ≤ 1) .
h + hσ (ϑη))
h +
(III) Setting f (ϑ; y, η; x, ξ) ≡ eiϕϑ,h (y,η;x,ξ)
p1,h (x, Ξ
p2,h (X
1
hσ y, ξ) and using the formula: f (1) = f (0) + 0 f (ϑ)dϑ, we have
(4.22)
h)
p1,h (x, Ξ
qh (x, ξ) = h + hσ y, ξ)dyd−η
· Os —
e−iy·η eiBh (y;x,ξ)
p2,h (X
1
f (ϑ; y, η; x, ξ)dϑdyd−η
+ Os —
0
h )
h , ξ) + p2,h (X
rh (x, ξ) ,
=
p1,h (x, Ξ
where
(4.23)
rh (x, ξ) ≡
1
rϑ,h (x, ξ)dϑ ,
0
and
(4.24)
(e−iy·η η)
rϑ,h (x, ξ) ≡ Os —
h )(ϑη)
h + hσ (ϑη))
· i(∇η A
p1,h (x, Ξ
h + hσ (ϑη))
p1,h )(x, Ξ
+ hσ (∇ξ h + hσ y, ξ)dyd−η .
p2,h (X
× eiAh (ϑη) eiBh (y)
470
Naoto Kumano-go
Integrating by parts, we get
rϑ,h (x, ξ) = Os —
(4.25)
eiϕϑ,h (y,η;x,ξ) aϑ,h (y, η; x, ξ)dyd−η ,
where
h )(ϑη)
h + hσ (ϑη))
p1,h (x, Ξ
aϑ,h (y, η; x, ξ) ≡ i(∇η A
h + hσ (ϑη))
p1,h )(x, Ξ
+ hσ (∇ξ h )(y)
h + hσ y, ξ)
p2,h (X
· (∇y B
h + hσ y, ξ) .
p2,h )(X
+ hσ (−i∇x
(4.26)
(IV) Set
aϑ,h,α,β (y, η; x, ξ)
eiϕϑ,h (y,η;x,ξ) α β
iϕ
ϑ,h (y,η;x,ξ)
aϑ,h (y, η; x, ξ) .
≡ ∂ξ Dx e
(4.27)
By (4.20), (4.21) and Proposition 2.4, there exists a constant Cl (depending
only on κ1 , κ2 , . . . , κl ) such that
2|α+β|+2
(4.28) |∂ξγ Dxµ ∂ηα Dyβ aϑ,h,α,β (y, η; x, ξ)| ≤ C|α+β+α
+β +γ+µ| y; η
(0)
(0)
p1,h ||α+β+α +γ+µ| + max |∂ξj p1,h ||α+β+α +γ+µ| )
× (τ1 |
1≤j≤n
(0)
(0)
p2,h ||α+β+β +γ+µ| + max |Dxj p2,h ||α+β+β +γ+µ| )
× (τ2 |
1≤j≤n
(0 ≤ ϑ ≤ 1) .
Hence, for l = 2|α + β| + 2n + 3, we can write
(4.29)
∂ξα Dxβ rϑ,h (x, ξ)
=
= Os —
eiϕϑ,h (y,η;x,ξ) aϑ,h,α,β (y, η; x, ξ)dyd−η
eiϕϑ,h (1 + i∇(y,η) · ∇(y,η) ϕ
ϑ,h )l
· ∇(y,η) ϕ
ϑ,h −2l aϑ,h,α,β (y, η; x, ξ)dyd−η ,
Fundamental solution for Schrödinger equations
471
and there exists a constant C3l+2n+3
(depending only on κ1 , κ2 , . . . ,
κ3l+2n+3 ) such that
(0)
(4.30) |
rϑ,h |l
≤ C3l+2n+3
(τ1 |
p1,h |3l+2n+3 + max |∂ξj p1,h |3l+2n+3 )
(0)
(0)
1≤j≤n
(0)
(0)
p2,h |3l+2n+3 + max |Dxj p2,h |3l+2n+3 ) (0 ≤ ϑ ≤ 1) .
× (τ2 |
1≤j≤n
From (4.23), we have
(4.31)
(0)
|
rh |l
≤ C3l+2n+3
(τ1 |
p1,h |3l+2n+3 + max |∂ξj p1,h |3l+2n+3 )
(0)
(0)
1≤j≤n
(0)
(0)
p2,h |3l+2n+3 + max |Dxj p2,h |3l+2n+3 ) .
× (τ2 |
1≤j≤n
Furthermore, we can get the estimate for qh (x, ξ) in a way similar to this. 1 ,λ1
2 ,λ2
1 +m2 ,λ1 +λ2
Bm
Bm
(φ1,h )◦B
(φ2,h ) ⊂ B m
(φ1,h #φ2,h )).
Theorem 4.2 (B
ρ,δ
ρ,δ
ρ,δ
∞
For any λ1 , λ2 ∈ R and {κl }l=0 (κ0 = 1, κl ≤ κl+1 ), there exists a constant Cλ1 ,λ2 ,lλ1 ,λ2 (depending only on λ1 , λ2 , κ1 , κ2 , . . . , κlλ1 ,λ2 ) satisfying the
following:
mj ,λj
Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
(h)(j = 1, 2) and τ1 +
l=0 ), pj,h (x, ξ) ∈ Bρ,δ
1 +m2 ,λ1 +λ2
τ2 ≤ 1/4. Then there exists qh (x, ξ) ∈ Bm
(h) such that
ρ,δ
(4.32)
p1,h (φ1,h ; X, Dx )p2,h (φ2,h ; X, Dx ) = qh (φ1,h #φ2,h ; X, Dx ) ,
and
(4.33)
(m1 +m2 ,λ1 +λ2 )
|qh |l
(m1 ,λ1 )
≤ Cλ1 ,λ2 ,lλ1 ,λ2 |p1,h |lλ
1 ,λ2
(m2 ,λ2 )
|p2,h |lλ
1 ,λ2
,
where lλ1 ,λ2 ≡ 3l + 2n + 1 + [|λ1 | + |λ2 |].
Proof.
(I) is the same as that in the proof of Theorem 4.1 except for the following
modification:
(4.34)
pj,h (x, ξ) ≡ h−mj pj,h (hδ x, h−ρ ξ) ∈ B0,0 j (h) (j = 1, 2) .
0,λ
472
Naoto Kumano-go
(II) By Proposition 2.4, there exists a constant Cl (depending only on κ1 ,
κ2 , . . . , κl ) such that
 −1
3 y; η
≤ ∇(y,η) ϕ
h (y, η; x, ξ)
≤ 3y; η



β
β
|∂ξα Dx ∂ηα Dy ∇(y,η) ϕ
h (y, η; x, ξ)| ≤ C|α+β+α
(4.35)
+β | y; η


 α β α β 2
h (y, η; x, ξ)| ≤ C|α+β+α
|∂ξ Dx ∂η Dy ϕ
+β | y; η
.
(III) Set
pj,h (x, ξ) ≡ pj,h (x, ξ)x; ξ
−λj ∈ B00,0 (h) (j = 1, 2) .
(4.36)
Then we have
qh (x, ξ) = Os —
(4.37)
eiϕh (y,η;x,ξ) ah (y, η; x, ξ)dyd−η ,
where
h + hσ η)
h + hσ y, ξ)
ah (y, η; x, ξ) ≡ p1,h (x, Ξ
p2,h (X
(4.38)
h + hσ η
λ1 X
h + hσ y; ξ
λ2 .
× x; Ξ
(IV) Set
(4.39)
eiϕh (y,η;x,ξ) ah,α,β (y, η; x, ξ) ≡ ∂ξα Dxβ eiϕh (y,η;x,ξ) ah (y, η; x, ξ) .
By (4.35), Proposition 2.4 and Lemma 2.6, there exists a constant Cλ1 ,λ2 ,l
(depending only on λ1 , λ2 , κ1 , κ2 , . . . , κl ) such that
(4.40)
ah,α,β (y, η; x, ξ)|
|∂ξγ Dxµ ∂ηα Dyβ ≤ Cλ1 ,λ2 ,|α+β+α +β +γ+µ| y; η
|λ1 |+|λ2 |+2|α+β| x; ξ
λ1 +λ2
× |
p1,h ||α+β+α +γ+µ| |
p2,h ||α+β+β +γ+µ| .
(0)
(0)
Hence, for l = 2|α + β| + 2n + 1 + [|λ1 | + |λ2 |], we can write
α β
(4.41) ∂ξ Dx qh (x, ξ) = Os —
eiϕh (y,η;x,ξ) ah,α,β (y, η; x, ξ)dyd−η
=
eiϕh (1 + i∇(y,η) · ∇(y,η) ϕ
h )l
· ∇(y,η) ϕ
h −2l ah,α,β (y, η; x, ξ)dyd−η
Fundamental solution for Schrödinger equations
473
and there exists a constant Cλ1 ,λ2 ,lλ1 ,λ2 (depending only on λ1 , λ2 , κ1 , κ2 , . . . ,
κlλ1 ,λ2 ) such that
(0,λ1 +λ2 )
|
qh |l
(4.42)
≤ Cλ1 ,λ2 ,lλ1 ,λ2 |
p1,h |lλ
(0)
|
p2,h |lλ
(0)
1 ,λ2
1 ,λ2
.
Therefore, we can get (4.33). Theorem 4.3 ( H.Kumano-go-Taniguchi theorem ).
For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exist constants 0 < τ0 <
1/(4c0 ) (depending only on κ1 , κ2 , . . . , κ5n0 ) and C3l+23n0 (depending only
on κ1 , κ2 , . . . , κ3l+23n0 ) satisfying the following:
mj
Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
l=0 ), pj,h (x, ξ) ∈ Bρ,δ (h)(j = 1, 2, . . . ) and
∞
τj ≤ τ0 .
j=1
m̄
Then, for any ν = 1, 2, . . . , there exists rν+1,h (x, ξ) ∈ Bρ,δν+1 (h) such
that
(4.43) rν+1,h (φ1,h #φ2,h # · · · #φν+1,h ; X, Dx )
= p1,h (φ1,h ; X, Dx )p2,h (φ2,h ; X, Dx ) · · · pν+1,h (φν+1,h ; X, Dx ) ,
and
(4.44)
(m̄ν+1 )
|rν+1,h |l
≤ (C3l+23n0 )ν+1
ν+1
(m )
j
|pj,h |l+11n
,
0
j=1
where m̄ν+1 ≡ m1 + m2 + · · · + mν+1 .
Proof.
(I) By Proposition 2.5(1), there exist a constant c0 > 1(independent
∞
of {κl }∞
l=0 ) and a sequence of positive constants {κl }l=0 (κ0 = 1, κl ≤
κl+1 , κl ≤ κl )(depending only on κ1 , κ2 , . . . , κl ), satisfying the following:
∞
For any φj,h ∈ Pρ,δ (τj , h, {κl }∞
)(j
=
1,
2,
.
.
.
)
with
τj ≤ 1/4, it
l=0
j=1
follows that
(4.45)
Φj,h = (φ1,h #φ2,h # · · · #φj,h ) ∈ Pρ,δ (c0 τ¯j , h, {κl }) .
474
Naoto Kumano-go
(II) By Proposition 3.8, there exists a constant 0 < τ0 < 1/(4c0 )(depending only on κ1 , κ2 , . . . , κ5n0 ) satisfying the following:
∞
For any φj,h ∈ Pρ,δ (τj , h, {κl }∞
τj ≤ τ0 /c0 (≡
l=0 )(j = 1, 2, . . . ) with
τ0 ), there exist sj,h (ξ, x ), tj,h (ξ, x ) ∈ B0ρ,δ (h) such that
(4.46)
j=1
I(φj,h )sj,h (φ∗j,h ; Dx , X ) = I
tj,h (Φ∗j,h ; Dx , X )I(Φj,h ) = I .
(III) For any j = 1, 2, . . . , set
I(Φj−1,h )I(φj,h )
(4.47) qj,h (X, Dx ) ≡
∗
∗
◦
sj,h (φj,h ; Dx , X )pj,h (φj,h ; X, Dx ) tj,h (Φj,h ; Dx , X ) ,
where Φ0,h ≡ x · ξ. Then we have
(4.48)
rν+1,h (φ1,h #φ2,h # · · · #φν+1,h ; X, Dx )
= p1,h (φ1,h ; X, Dx )p2,h (φ2,h ; X, Dx ) · · · pν+1,h (φν+1,h ; X, Dx )
= q1,h (X, Dx )q2,h (X, Dx ) · · · qν+1,h (X, Dx )I(Φν+1,h ) .
By Proposition 3.5, 1.4, 3.4, 3.6 and Theorem 4.1, we get (4.44). Definition 4.4.
For τ ≥ 0 and a sequence of positive constants {κl }∞
l=0 (κ0 = 1, κl ≤
κl+1 ), we say that a family {ωh }0<h<1 of complex-valued C ∞ -functions
ωh (x, ξ) belongs to the class {Ωρ,δ (τ, h, {κl }∞
h (x, ξ) ≡
l=0 )}0<h<1 , if ω
δ
−ρ
ωh (h x, h ξ) satisfies

ωh (x, ξ)} ≥ −κ0 τ
inf inf Im{

 0<h<1 x,ξ
(4.49)
h (x,ξ)|

 sup
sup |ωx;ξ
+
0<h<1
x,ξ
sup |∂ξα Dxβ ω
h (x, ξ)|
1≤|α+β|≤1+l x,ξ
≤ κl τ ,
for any l = 0, 1, 2, . . . .
We write {ωh }0<h<1 ∈ {Ωρ,δ (τ, h, {κl }∞
l=0 )}0<h<1 , or simply ωh ∈
Ωρ,δ (τ, h, {κl }∞
).
l=0
Fundamental solution for Schrödinger equations
475
Proposition 4.5.
For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exist a constant c0 > 1(inde∞
pendent of {κl }l=0 ) and a sequence of positive constants {κl }∞
l=0 (κ0 = 1,
κl ≤ κl+1 , κl ≤ κl ) (depending only on κ1 , κ2 , . . . , κl ), satisfying the following:
∞
Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
l=0 ) and ωj,h ∈ Ωρ,δ (τj , h, {κl }l=0 ) (j = 1,
∞
τj ≤ 1/4.
2, . . . ) with
j=1
Then, for any ν = 1, 2, . . . , it follows that

∞
 Φν+1,h ≡ (φ1,h #φ2,h # · · · #φν+1,h ) ∈ Pρ,δ (c0 τ̄ν+1 , h, {κl }l=0 )
ν+1
(4.50)
j−1
j

ωj,h (Xν,h
, Ξν,h
) ∈ Ωρ,δ (c0 τ̄ν+1 , h, {κl }∞
l=0 ) ,
j=1
where τ̄ν+1 ≡ τ1 + τ2 + · · · + τν+1 .
Proof. By Proposition 2.4 and 2.5. Theorem 4.6 ( Fujiwara’s skip method ).
For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exist constants 0 < τ0 <
1/(4c0 ) (depending only on κ1 , κ2 , . . . , κ5n0 ) and C3l+36n0 , C3l+36n0 (depending only on κ1 , κ2 , . . . , κ3l+36n0 ) satisfying the following:
∞
For φj,h ∈ Pρ,δ (τj , h, {κl }∞
l=0 ) and ωj,h ∈ Ωρ,δ (τj , h, {κl }l=0 )(j = 1, 2, . . . )
∞
τj ≤ τ0 , set
with
j=1
(4.51)

 φ1,2,...,ν+1,h (x, ξ) ≡ (φ1,h #φ2,h # · · · #φν+1,h )(x, ξ)
ν+1
j−1
j
 ω1,2,...,ν+1,h (x, ξ) ≡
ωj,h (Xν,h
, Ξν,h
),
j=1
and
(4.52)

 pj,h (x, ξ) ≡ exp iωj,h (x, ξ) ∈ B0ρ,δ (h)
 p#
(x,
ξ)
≡
exp
iω
(x,
ξ)
∈ B0ρ,δ (h) .
1,2,...,ν+1,h
1,2,...,ν+1,h
Then, for any ν = 1, 2, . . . , there exists Υ1,2,...,ν+1,h (x, ξ) ∈ B0ρ,δ (h) such that
(4.53)
p1,h (φ1,h ; X, Dx )p2,h (φ2,h ; X, Dx ) · · · pν+1,h (φν+1,h ; X, Dx )
= p#
1,2,...,ν+1,h (φ1,2,...,ν+1,h ; X, Dx )
+ Υ1,2,...,ν+1,h (φ1,2,...,ν+1,h ; X, Dx ) ,
476
(4.54)
Naoto Kumano-go
(0)
|Υ1,2,...,ν+1,h |l
≤ C3l+36n
(τ̄ν+1 )2 ,
0
and
(4.55)
(0)
|(p#
1,2,...,ν+1,h + Υ1,2,...,ν+1,h )|l
≤ C3l+36n
.
0
Proof.
(I) By Proposition 2.5 and Proposition 4.5, there exist a constant c0 > 1
and a sequence of positive constants {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 , κl ≤ κl )
(depending only on κ1 , κ2 , . . . , κl ) such that
(4.56)

φ1,2,...,ν,h ∈ Pρ,δ (c0 τ̄ν , h, {κl }∞
l=0 ),



∞
φ
ν+1,h ∈ Pρ,δ (c0 τν+1 , h, {κl }l=0 )

ω1,2,...,ν,h ∈ Ωρ,δ (c0 τ̄ν , h, {κl }∞

l=0 ),


ων+1,h ∈ Ωρ,δ (c0 τν+1 , h, {κl }∞
l=0 ) .
Hence, there exist a constant Cl > 1(depending only on κ1 , κ2 , . . . , κl ) such
that
 #
(0)
(0)
|p1,2,...,ν,h |l ≤ Cl , |pν+1,h |l ≤ Cl




(ρ)
(−δ)
|∂ξk p#
≤ Cl τν+1
1,2,...,ν,h |l ≤ Cl τ̄ν , |Dxk pν+1,h |l
(4.57)

ν


 (ν = 1, 2, . . . , k = 1, 2, . . . , n, p# ≡ pj,h , τ̄ν ≡
τj ) .
j,h
j=1
By Theorem 4.1 and Proposition 2.5(2), there exist a constant C3l+2n+3
0
(depending only on κ1 , κ2 , . . . , κ3l+2n+3 ) and r1,2,...,ν+1,h (x, ξ) ∈ Bρ,δ (h) such
that
(4.58)
p#
1,2,...,ν,h (φ1,2,...,ν,h ; X, Dx )pν+1,h (φν+1,h ; X, Dx )
= p#
1,2,...,ν+1,h (φ1,2,...,ν+1,h ; X, Dx )
+ r1,2,...,ν+1,h (φ1,2,...,ν+1,h ; X, Dx ) ,
and
(4.59)
(0)
|r1,2,...,ν+1,h |l
≤ C3l+2n+3
(τ̄ν )τν+1 .
Fundamental solution for Schrödinger equations
477
For simplicity, set
#
P1,2,...,ν+1,h ≡ p#
1,2,...,ν+1,h (φ1,2,...,ν+1,h ; X, Dx )
(4.60)
R1,2,...,ν+1,h ≡ r1,2,...,ν+1,h (φ1,2,...,ν+1,h ; X, Dx ) .
Furthermore, replacing {φ1,h , φ2,h , . . . , φν+1,h } and {ω1,h , ω2,h , . . . , ων+1,h }
by {φj+1,h , φj+2,h , . . . , φj+ν+1,h } and {ωj+1,h , ωj+2,h , . . . , ωj+ν+1,h }, set
#
Pj+1,j+2,...,j+ν+1,h
and Rj+1,j+2,...,j+ν+1,h .
(II) We use Fujiwara’s skip method: Using (4.58) inductively, we can
get
(4.61)
p1,h (φ1,h ; X, Dx )p2,h (φ2,h ; X, Dx ) · · · pν+1,h (φν+1,h ; X, Dx )
= p#
1,2,...,ν+1,h (φ1,2,...,ν+1,h ; X, Dx )
+ Υ1,2,...,ν+1,h (φ1,2,...,ν+1,h ; X, Dx ) ,
where
(4.62)
Υ1,2,...,ν+1,h (φ1,2,...,ν+1,h ; X, Dx )
Rj0 +1,j0 +2,...,j1 ,h Rj1 +1,j1 +2,...,j2 ,h
≡
· · · Rjs−1 +1,js−1 +2,...,js ,h Pj#s +1,js +2,...,ν+1,h ,
stands for the summation with respect to the sequences of integers
(j1 , j2 , . . . , js−1 , js ) with the property
(4.63) 0 ≡ j0 < j1 − 1 < j1 < j2 − 1 < · · · < js−1 < js − 1 < js ≤ ν + 1 ,
and, in the special case of js = ν + 1, we set Pj#s +1,js +2,...,ν+1,h ≡ I.
Now, let C3l+23n0 is the same constant as that in Theorem 4.3.
By (4.57), (4.59), (4.62), (4.63) and Theorem 4.3, we have
(4.64)
(0)
|Υ1,2,...,ν+1,h |l
(C3l+23n0 )s+1 Cl+11n
≤
0
·
s k=1
≤
j
k −1
C3(l+11n0 )+2n+3 (
τk )τjk
k =jk−1 +1
C3l+23n0 Cl+11n
0
ν+1
j=1
≤ C3l+36n
(τ̄ν+1 )2 .
0
1+
(τ̄ν+1 )τj
C3l+23n0 C3l+36n
0
−1
478
Naoto Kumano-go
Here C3l+36n
is a constant depending only on κ1 , κ2 , . . . , κ3l+36n0 . 0
The following lemma is a revised version of that in Fujiwara [4].
Lemma 4.7.
∞
For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ) and {χl }l=0 (0 ≤ χl ≤ χl+1 ),
there exists a constant C3l+4n+5 (depending only on κ1 , κ2 , . . . , κ3l+4n+5 and
χ0 , χ1 , . . . , χ3l+4n+5 ) satisfying the following:
0
Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
l=0 ), pj,h (x, ξ) ∈ Bρ,δ (h) (j = 1, 3), φ2,ϑ,h ∈
3
0
τj ≤ 1/(4c0 ).
Pρ,δ (τ2 , h, {κl }∞
l=0 ), p2,ϑ,h (x, ξ) ∈ Bρ,δ (h) (0 ≤ ϑ ≤ 1) and
j=1
2,ϑ,h
(0 ≤ ϑ ≤ 1) and let φ
Furthermore, let ∂ϑ p2,ϑ,h (x, ξ) ∈
satisfy (2.26).
Then, there exists qϑ,h (x, ξ) ∈ B0ρ,δ (h) such that
0,1
Bρ,δ
(h)
(4.65)
qϑ,h (φ1,h #φ2,ϑ,h #φ3,h ; X, Dx )
= p1,h (φ1,h ; X, Dx )p2,ϑ,h (φ2,ϑ,h ; X, Dx )p3,h (φ3,h ; X, Dx ) ,
∂ϑ qϑ,h (x, ξ) ∈ B0,1
ρ,δ (h) ,
(4.66)
and
(4.67)
(0,1)
|∂ϑ qϑ,h |l
(0)
(0)
≤ C3l+4n+5 |p1,h |3l+4n+5 |p3,h |3l+4n+5
(0)
(0,1)
× (τ2 )2 |p2,h |3l+4n+5 + |∂ϑ p2,ϑ,h |3l+4n+5
(0 ≤ ϑ ≤ 1) .
Proof.
(I) For u ∈ S, we can write
(4.68) qϑ,h (x, ξ) = h
−4nσ
Os —
−2σ ψ
ϑ,h (x,ξ 1 ,x1 ,ξ 2 ,x2 ,ξ)
eih
p2,ϑ,h (x1 , ξ 2 )
p3,h (x2 , ξ)dx1 d−ξ 1 dx2 d−ξ 2 ,
×
p1,h (x, ξ 1 )
Fundamental solution for Schrödinger equations
where
(4.69)
ϑ,h (x, ξ 1 , x1 , ξ 2 , x2 , ξ) ≡ φ
1,h (x, ξ 1 ) − x1 · ξ 1 + φ
2,ϑ,h (x1 , ξ 2 )
ψ
3,h (x2 , ξ)
− x2 · ξ 2 + φ
2,ϑ,h #φ
3,h )(x, ξ) ,
1,h #φ
− (φ
and σ ≡ (ρ − δ)/2.
j }2 (x, ξ) be the solution of the equations (2.24).
j ,Ξ
Let {X
2,ϑ,h
2,ϑ,h j=1
By a change of variables:
(4.70)
1 (x, ξ) + hσ y 1 , ξ 1 = Ξ
1 (x, ξ) + hσ η 1
x1 = X
2,ϑ,h
2,ϑ,h
2 (x, ξ) + hσ y 2 , ξ 2 = Ξ
2 (x, ξ) + hσ η 2 ,
x2 = X
2,ϑ,h
2,ϑ,h
we can write
qϑ,h (x, ξ) = Os —
(4.71) eiϕϑ,h (y
1 ,η 1 ,y 2 ,η 2 ;x,ξ)
·
aϑ,h (y 1 , η 1 , y 2 , η 2 ; x, ξ)dy 1 d−η 1 dy 2 d−η 2 ,
where
σ 1
1
p1,h (x, Ξ
aϑ,h (y 1 , η 1 , y 2 , η 2 ; x, ξ) ≡ 2,ϑ,h + h η )
(4.72)
σ 1 2
σ 2
1
×
p2,ϑ,h (X
2,ϑ,h + h y , Ξ2,ϑ,h + h η )
σ 2
2
×
p3,h (X
2,ϑ,h + h y , ξ) ,
and
(4.73) ϕ
ϑ,h (y 1 , η 1 , y 2 , η 2 ; x, ξ) ≡ h−2σ ψϑ,h (x, ξ 1 , x1 , ξ 2 , x2 , ξ)
= −y 1 · η 1 − y 2 · η 2 + y 1 · η 2
1
→
σ 1
1
1
1
+
(1 − θ)(∇ξ ∇ξ J1,h )(x, Ξ
2,ϑ,h + θh η )dθη · η
0
1
+
0
→
σ 1 2
σ 2
1
1
1
(1 − θ)(∇x ∇x J2,ϑ,h )(X
2,ϑ,h + θh y , Ξ2,ϑ,h + θh η )dθy · y
+2
0
1
→
σ 1 2
σ 2
1
2
1
(1 − θ)(∇x ∇ξ J2,ϑ,h )(X
2,ϑ,h + θh y , Ξ2,ϑ,h + θh η )dθy · η
479
480
Naoto Kumano-go
1
+
0
1
+
0
→
σ 1 2
σ 2
2
2
1
(1 − θ)(∇ξ ∇ξ J2,ϑ,h )(X
2,ϑ,h + θh y , Ξ2,ϑ,h + θh η )dθη · η
→
σ 2
2
2
2
(1 − θ)(∇x ∇x J3,h )(X
2,ϑ,h + θh y , ξ)dθy · y .
Furthermore, we have
(4.74)
→

1 + 1 (∇ ∇ J
σ 1
1
1
∇
=
−y
1ϕ

ϑ,h
ξ ξ 1,h )(x, Ξ2,ϑ,h + θh η )dθη
η
0





∇y 1 ϕ
ϑ,h = −η 1 + η 2




1 →

σ 1 2
σ 2
1
1

+

0 (∇x ∇x J2,ϑ,h )(X2,ϑ,h + θh y , Ξ2,ϑ,h + θh η )dθy




→

σ 1 2
σ 2
2
2,ϑ,h )(X
1
 + 1 (∇ξ ∇x J
2,ϑ,h + θh y , Ξ2,ϑ,h + θh η )dθη
0

∇η 2 ϕ
ϑ,h = y 1 − y 2




1 →

σ 1 2
σ 2
1

2,ϑ,h )(X
1

+ 0 (∇x ∇ξ J

2,ϑ,h + θh y , Ξ2,ϑ,h + θh η )dθy



1 →

σ 1 2
σ 2
2

2,ϑ,h )(X
1
+ 0 (∇ξ ∇ξ J

2,ϑ,h + θh y , Ξ2,ϑ,h + θh η )dθη




1 →
σ 1
2
3,h )(X
2
∇y 2 ϕ
ϑ,h = −η 2 + 0 (∇x ∇x J
2,ϑ,h + θh y , ξ)dθy .
(II) By (4.72), (4.73), (4.74) and Proposition 2.4, there exists a constant
C (independent of {κl }∞
l=0 ) and Cl (depending only on κ1 , κ2 , . . . , κl ) such
that
 −1 1 1 2 2
C y ; η ; y ; η ≤ ∇(y1 ,η1 ,y2 ,η2 ) ϕ
ϑ,h ≤ Cy 1 ; η 1 ; y 2 ; η 2 


 α β α1 β 1 α 2 β 2


ϑ,h |
|∂ Dx ∂η1 Dy1 ∂η2 Dy2 ∇(y1 ,η1 ,y2 ,η2 ) ϕ


 ξ
1 1 2 2
≤ C|α+β+α
(4.75)
1 +β 1 +α2 +β 2 | y ; η ; y ; η 

1
2

1
2


|∂ξα Dxβ ∂ηα1 Dyβ1 ∂ηα2 Dyβ2 ϕ
ϑ,h |




1 1 2 2 2
≤ C|α+β+α
1 +β 1 +α2 +β 2 | y ; η ; y ; η ,
and
(4.76)
1
1
2
2
|∂ξα Dxβ ∂ηα1 Dyβ1 ∂ηα2 Dyβ2 aϑ,h |
≤ C|α+β+α
p1,h ||α+β+α1 |
1 +β 1 +α2 +β 2 |−1 |
(0)
(0)
(0)
× |
p2,ϑ,h ||α+β+β 1 +α2 | |
p3,h ||α+β+β 2 | (0 ≤ ϑ ≤ 1) ,
Fundamental solution for Schrödinger equations
481
where y; η; y ; η ≡ (1 + |y|2 + |η|2 + |y |2 + |η |2 )1/2 .
Furthermore, by (4.72), (4.73), Lemma 2.6, 2.7, and Proposition 2.4,
there exists a constant Cl (depending only on κ1 , κ2 , . . . , κl and χ0 , χ1 , . . . ,
χl ) such that
(4.77)
1
1
2
2
ϑ,h |
|∂ξα Dxβ ∂ηα1 Dyβ1 ∂ηα2 Dyβ2 ∂ϑ ϕ
2 1 1 2 2 3
≤ C|α+β+α
1 +β 1 +α2 +β 2 |+1 (τ2 ) y ; η ; y ; η x; ξ
,
and
(4.78)
1
1
2
2
|∂ξα Dxβ ∂ηα1 Dyβ1 ∂ηα2 Dyβ2 ∂ϑ aϑ,h |
1 2
≤ C|α+β+α
1 +β 1 +α2 +β 2 | y ; η x; ξ
(0)
(0)
· |
p1,h ||α+β+α1 |+1 |
p3,h ||α+β+β 2 |+1
(0)
(0,1)
× (τ2 )2 |
p2,ϑ,h ||α+β+β 1 +α2 |+1 + |∂ϑ
p2,ϑ,h ||α+β+β 1 +α2 |
(0 ≤ ϑ ≤ 1) ,
Set
(4.79)
bϑ,h (y 1 , η 1 , y 2 , η 2 ; x, ξ) ≡ ∂ϑ aϑ,h (y 1 , η 1 , y 2 , η 2 ; x, ξ)
+ i∂ϑ ϕ
ϑ,h (y 1 , η 1 , y 2 , η 2 ; x, ξ)
aϑ,h (y 1 , η 1 , y 2 , η 2 ; x, ξ) .
Then we have
(4.80) ∂ϑ
qϑ,h (x, ξ) = Os —
eiϕϑ,h (y
1 ,η 1 ,y 2 ,η 2 ;x,ξ)
· bϑ,h (y 1 , η 1 , y 2 , η 2 ; x, ξ)dy 1 d−η 1 dy 2 d−η 2 .
(III) Set
(4.81)
eiϕϑ,h (y
bϑ,h,α,β (y 1 , η 1 , y 2 , η 2 ; x, ξ)
1 ,η 1 ,y 2 ,η 2 ;x,ξ)
1 1 2 2
≡ ∂ξα Dxβ eiϕϑ,h (y ,η ,y ,η ;x,ξ)bϑ,h (y 1 , η 1 , y 2 , η 2 ; x, ξ) .
482
Naoto Kumano-go
By (4.75)-(4.79), there exists a constant Cl (depending only on κ1 , κ2 , . . . ,
κl and χ0 , χ1 , . . . , χl ) such that
1
2
1
2
(4.82) |∂ξγ Dxµ ∂ηα1 Dyβ1 ∂ηα2 Dyβ2 bϑ,h,α,β (y 1 , η 1 , y 2 , η 2 ; x, ξ)|
1 1 2 2 2|α+β|+3
x; ξ
≤ C|α+β+α
1 +β 1 +α2 +β 2 +γ+µ|+1 y ; η ; y ; η (0)
(0)
× |
p1,h ||α+β+α1 +γ+µ|+1 |
p3,h ||α+β+β 2 +γ+µ|+1
(0)
(0,1)
× (τ2 )2 |
p2,ϑ,h ||α+β+β 1 +α2 +γ+µ|+1 + |∂ϑ
p2,ϑ,h ||α+β+β 1 +α2 +γ+µ|
(0 ≤ ϑ ≤ 1) .
Hence, for l = 2|α + β| + 4n + 4, we can write
(4.83) ∂ξα Dxβ ∂ϑ
qϑ,h (x, ξ)
1 1 2 2
eiϕϑ,h (y ,η ,y ,η ;x,ξ)
= Os —
· bϑ,h,α,β (y 1 , η 1 , y 2 , η 2 ; x, ξ)dy 1 d−η 1 dy 2 d−η 2
=
eiϕϑ,h (1 + i∇(y1 ,η1 ,y2 ,η2 ) · ∇(y1 ,η1 ,y2 ,η2 ) ϕ
ϑ,h )l
× ∇(y1 ,η1 ,y2 ,η2 ) ϕ
ϑ,h −2l bϑ,h,α,β (y 1 , η 1 , y 2 , η 2 ; x, ξ)dy 1 d−η 1 dy 2 d−η 2
and there exists a constant C3l+4n+5 (depending only on κ1 , κ2 , . . . ,
κ3l+4n+5 ) such that
(0,1)
(4.84) |∂ϑ
qϑ,h |l
(0)
(0)
≤ C3l+4n+5 |
p1,h |3l+4n+5 |
p3,h |3l+4n+5
(0)
(0,1)
p2,ϑ,h |3l+4n+5 + |∂ϑ
p2,ϑ,h |3l+4n+5 )
× ((τ2 )2 |
(0 ≤ ϑ ≤ 1) .
Therefore, we get (4.67). In Theorem 4.6, for ν = 1, 2, . . . and 0 ≤ ϑ ≤ 1, set
(4.85)
p1,2,...,ν+1,ϑ,h (x, ξ)
ν+1 j−1
j
ϑωj,h (Xν,h , Ξν,h ) + (1 − ϑ)ωj,h (x, ξ)
≡ exp i
j=1
+ ϑΥ1,2,...,ν+1,h (x, ξ) .
Fundamental solution for Schrödinger equations
483
Then we have

#
 p1,2,...,ν+1,1,h (x, ξ) = (p1,2,...,ν+1,h + Υ1,2,...,ν+1,h )(x, ξ)
ν+1
 p1,2,...,ν+1,0,h (x, ξ) = exp i
ωj,h (x, ξ) .
(4.86)
j=1
Furthermore, we get the following lemma.
Lemma 4.8.
For any {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), there exist constants 0 < τ0 <
1/(4c0 ) (depending only on κ1 , κ2 , . . . , κ5n0 ) and C3l+36n0 , C3l+36n0 (depending only on κ1 , κ2 , . . . , κ3l+36n0 ) satisfying the following:
∞
Let φj,h ∈ Pρ,δ (τj , h, {κl }∞
l=0 ) and ωj,h ∈ Ωρ,δ (τj , h, {κl }l=0 )(j = 1, 2, . . . )
∞
τj ≤ τ0 .
with
j=1
Then, it follows that
(4.87)
p1,2,...,ν+1,ϑ,h (x, ξ) ∈ B0ρ,δ (h)
0,1
(h) (0 ≤ ϑ ≤ 1) ,
∂ϑ p1,2,...,ν+1,ϑ,h (x, ξ) ∈ Bρ,δ
and
(4.88)
≤ C3l+36n
0
(0)
|p1,2,...,ν+1,ϑ,h |l
(0,1)
|∂ϑ p1,2,...,ν+1,ϑ,h |l
≤ C3l+36n
(τ̄ν+1 )2 (0 ≤ ϑ ≤ 1) .
0
Proof. Note that
(4.89)
ν+1 j−1 , Ξ
j ) − ω
(x,
ξ)
ω
j,h (X
j,h
ν,h
ν,h
j=1
ν+1 1
=
→
j−1 − x), ξ + θ(Ξ
j − ξ))dθ
(∇(x,ξ) ω
j,h )(x + θ(X
ν,h
ν,h
j=1 0
j−1
Xν,h − x
.
·
j − ξ
Ξ
ν,h
By Proposition 2.4, 4.5 and Theorem 4.6, we get (4.88). 484
5.
Naoto Kumano-go
A Construction of the Fundamental Solution
In this section, using the theory developed in Section 1, 2, 3 and 4, we
give a construction of the fundamental solution for the Cauchy problem of
a pseudo-differential equation of Schrödinger’s type. We regard this construction as a multi-product of Fourier integral operators, and investigate
the pointwise convergence of the phase function and that of the symbol.
R 2n with a parameter
Let Hh (t, x, ξ) be a real-valued function on [0, T ]×R
β
α
R 2n
0 < h < 1, which has a continuous derivative ∂ξ Dx Hh (t, x, ξ) on [0, T ]×R
for any α, β, satisfying
(5.1)
|∂ξα Dxβ {hρ−δ Hh (t, hδ x, h−ρ ξ)}|
Cα,β x; ξ
2−|α+β| (|α + β| ≤ 1)
≤
Cα,β
(|α + β| ≥ 2) ,
on [0, T ] × R 2n . Here Cα,β is a constant independent of a parameter 0 <
h < 1.
Let Wh (t, x, ξ) be a complex-valued function on [0, T ] × R 2n with a parameter 0 < h < 1, which has a continuous derivative ∂ξα Dxβ Wh (t, x, ξ) on
[0, T ] × R 2n for any α, β, satisfying

C0,0 x; ξ
(α = β = 0)
 |∂ α Dxβ {Wh (t, hδ x, h−ρ ξ)}| ≤
ξ
(|α + β| ≥ 1)
Cα,β
(5.2)

δ
−ρ
Im{Wh (t, h x, h ξ)} ≥ −C0,0 ,
on [0, T ] × R 2n . Here Cα,β
is a constant independent of a parameter 0 <
h < 1.
Set
Kh (t, x, ξ) ≡ Hh (t, x, ξ) + Wh (t, x, ξ)
(5.3)
Lh ≡ i∂t + Kh (t, X, Dx ) .
For sufficiently small T0 (0 < T0 ≤ T ), we consider the fundamental solution
Uh (t, s) (0 ≤ s ≤ t ≤ T0 ) for the operator Lh such that
(5.4)
Lh Uh (t, s) = 0 on [s, T0 ]
Uh (s, s) = I
(0 ≤ s ≤ T0 ) .
Fundamental solution for Schrödinger equations
485
Our purpose is to construct Uh (t, s) by
(5.5)
i
Uh (t, s) =
lim e
t
t1
Kh (τ )dτ
|∆t,s |→0
· · · ei
tν
s
i
(X, Dx )e
Kh (τ )dτ
t
1
t2
Kh (τ )dτ
(X, Dx )
(X, Dx ) ,
where ∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν ≥ tν+1 ≡ s (≥ 0) is an arbitrary
division of interval [s,t] into subintervals, |∆t,s | ≡ max |tj−1 − tj |, and
1≤j≤ν+1
i
e
(5.6)
tj−1
tj
Kh (τ )dτ
≡
(X, Dx )u(x)
i
eix·ξ e
tj−1
tj
Kh (τ,x,ξ)dτ
û(ξ)d−ξ (u ∈ S) .
Now, set
(5.7)
t
φh (t, s)(x, ξ) ≡ x · ξ + s Hh (τ, x, ξ)dτ
t
ωh (t, s)(x, ξ) ≡ s Wh (τ, x, ξ)dτ (0 ≤ s ≤ t ≤ T ) .
and
(5.8)
ph (t, s)(x, ξ) ≡ exp iωh (t, s)(x, ξ) (0 ≤ s ≤ t ≤ T ) .
Then we can write
(5.9)
ei
t
s
Kh (τ )dτ
(X, Dx )u(x) ≡ ph (t, s)(φh (t, s); X, Dx )u(x) .
t
Therefore, we regard the operator ei s Kh (τ )dτ (X, Dx ) as the Fourier integral
operator ph (t, s)(φh (t, s); X, Dx ) with the phase function φh (t, s)(x, ξ) and
the
symbol ph (t, s)(x, ξ),
and investigate the properties of
i st Kh (τ )dτ
e
(X, Dx ).
Theorem 5.1.
Under the assumptions (5.1), (5.2) and (5.3), there exist sufficiently
small T0 (0 < T0 ≤ T ), constants C1,l , c and a sequences of positive con
stants {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ), satisfying the following:
486
Naoto Kumano-go
For any ∆t0 ,tν+1 : (T0 ≥) t0 ≥ t1 ≥ · · · ≥ tν+1 (≥ 0), there exist a
phase function φh (∆t0 ,tν+1 )(x, ξ) ∈ Pρ,δ (c (t0 − tν+1 ), h, {κl }∞
l=0 ) and a sym0
bol ph (∆t0 ,tν+1 )(x, ξ) ∈ Bρ,δ (h) such that
(5.10)
i
ph (∆t0 ,tν+1 )(φh (∆t0 ,tν+1 ); X, Dx ) = e
i
·e
t
1
t2
Kh (τ )dτ
i
(X, Dx ) · · · e
tν
tν+1
t
0
t1
Kh (τ )dτ
Kh (τ )dτ
(X, Dx )
(X, Dx ) ,
and
(0)
|ph (∆t0 ,tν+1 )|l
(5.11)
≤ C1,l .
Furthermore, the following equation holds:
(5.12)
eiφh (∆t0 ,tν+1 )(x,ξ)−ix ·ξ ph (∆t0 ,tν+1 )(x, ξ)
ν+1 = Os —
···
exp
i(xj−1 − xj ) · ξ j
j=1
tj−1
+i
j−1
Kh (τ, x
j
, ξ )dτ
dx1 d−ξ 1 · · · dxν d−ξ ν ,
tj
where x0 ≡ x, xν+1 ≡ x and ξ ν+1 ≡ ξ.
Proof.
(I) By (5.1), (5.2) and (5.7), there exist a constant c > 0 and a sequence
of positive constants {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 ) such that
(5.13)
φh (t, s)(x, ξ) ∈ Pρ,δ (c(t − s), h, {κl }∞
l=0 )
ωh (t, s)(x, ξ) ∈ Ωρ,δ (c(t − s), h, {κl }∞
l=0 ) .
Let c0 > 1 be the same constant as that in Proposition 4.5, T ≡
min{T, 1/(4c0 c)} and c ≡ c0 c. For any ∆t0 ,tν+1 : (T ≥) t0 ≥ t1 ≥ · · · ≥
j
j
, Ξν,h
}νj=1 (x, ξ) be the solution of the equations
tν+1 (≥ 0), let {Xν,h
(5.14)
 j
j−1
j
X = ∇ξ φh (tj−1 , tj )(Xν,h
, Ξν,h
)

 ν,h
j
j
j+1
= ∇x φh (tj , tj+1 )(Xν,h
, Ξν,h
)
Ξν,h


0 ≡ x, Ξ ν+1 ≡ ξ) ,
(j = 1, . . . , ν, Xν,h
ν,h
Fundamental solution for Schrödinger equations
487
and set

φh (∆t0 ,tν+1 )(x, ξ)




≡ (φh (t0 , t1 )#φh (t1 , t2 )# · · · #φh (tν , tν+1 ))(x, ξ)
ν+1


j−1
j

ωh (tj−1 , tj )(Xν,h
, Ξν,h
).
 ωh (∆t0 ,tν+1 )(x, ξ) ≡
(5.15)
j=1
Then, by Proposition 2.5 and 4.5, there exists a sequence of positive con
stants {κl }∞
l=0 (κ0 = 1, κl ≤ κl+1 , κl ≤ κl ) such that

∞

 φh (∆t0 ,tν+1 )(x, ξ) ∈ Pρ,δ (c (t − s), h, {κl }l=0 )
ωh (∆t0 ,tν+1 )(x, ξ) ∈ Ωρ,δ (c (t − s), h, {κl }∞
l=0 )


(ν = 0, 1, 2, . . . , T ≥ t0 ≥ t1 ≥ · · · ≥ tν+1 ≥ 0) .
(5.16)
Furthermore, set
(5.17)
(∆
)(x,
ξ)
≡
exp
iω
(∆
)(x,
ξ)
∈ B0ρ,δ (h) .
p#
t
,t
t
,t
h
0 ν+1
0 ν+1
h
such that
Then, by (5.16), there exists a constant C1,l
(0)
|p#
h (∆t0 ,tν+1 )|l
(5.18)
≤ C1,l
(ν = 0, 1, 2, . . . , T ≥ t0 ≥ t1 ≥ · · · ≥ tν+1 ≥ 0) .
(II) We choose sufficiently small T0 (0 < T0 < T ) such that Theorem 4.6
satisfying the following:
holds. By Theorem 4.6, there exists a constant C1,l
For any ∆t0 ,tν+1 : (T0 ≥) t0 ≥ t1 ≥ · · · ≥ tν+1 (≥ 0), there exists
Υh (∆t0 ,tν+1 )(x, ξ) ∈ B0ρ,δ (h) such that
(5.19)
ph (t0 , t1 )(φh (t0 , t1 ); X, Dx )ph (t1 , t2 )(φh (t1 , t2 ); X, Dx )
=
· · · ph (tν , tν+1 )(φh (tν , tν+1 ); X, Dx )
#
ph (∆t0 ,tν+1 )(φh (∆t0 ,tν+1 ); X, Dx )
+ Υh (∆t0 ,tν+1 )(φh (∆t0 ,tν+1 ); X, Dx ) ,
and
(5.20)
(0)
|Υh (∆t0 ,tν+1 )|l
≤ C1,l
(t0 − tν+1 )2 .
488
Naoto Kumano-go
(III) Set
ph (∆t0 ,tν+1 ) = p#
h (∆t0 ,tν+1 ) + Υh (∆t0 ,tν+1 ) .
(5.21)
Then we get (5.11). Lemma 5.2.
(1) There exists a sequence of positive constants {χl }∞
l=0 (0 ≤ χl ≤ χl+1 )
satisfying the following:
For any ∆t0 ,tν+1 : (T0 ≥) t0 ≥ t1 ≥ · · · ≥ tν+1 (≥ 0), there exists
φϑ,h (∆t0 ,tν+1 ) ∈ P0,0 (c (t0 − tν+1 ), h, {κl }∞
l=0 ) (0 ≤ ϑ ≤ 1) such that
h (∆t ,t )(x, ξ)
φ1,h (∆t0 ,tν+1 )(x, ξ) = φ
0 ν+1
φ0,h (∆t ,t )(x, ξ) = φh (t0 , tν+1 )(x, ξ) ,
(5.22)
0 ν+1
and

(∆
)(x, ξ)| ≤ χ0 (t0 − tν+1 )2 x; ξ
2
|∂ J

 ϑ ϑ,h t0 ,tν+1
ϑ,h (∆t ,t )(x, ξ)| ≤ χ|α+β| (t0 − tν+1 )2 x; ξ
(5.23)
|∂ξα Dxβ ∇(x,ξ) ∂ϑ J
0 ν+1


(0 < h < 1, 0 ≤ ϑ ≤ 1) ,
ϑ,h (∆t ,t )(x, ξ) − x · ξ.
ϑ,h (∆t ,t )(x, ξ) ≡ φ
where J
0 ν+1
0 ν+1
(2) There exist constants C1,l , C2,l , satisfying the following:
For any ∆t0 ,tν+1 : (T0 ≥) t0 ≥ t1 ≥ · · · ≥ tν+1 (≥ 0), there exists
pϑ,h (∆t0 ,tν+1 )(x, ξ) ∈ B0ρ,δ (h) (0 ≤ ϑ ≤ 1) such that
(5.24)
p1,h (∆t0 ,tν+1 )(x, ξ) = ph (∆t0 ,tν+1 )(x, ξ)
p0,h (∆t0 ,tν+1 )(x, ξ) = ph (t0 , tν+1 )(x, ξ) ,
0,1
∂ϑ pϑ,h (∆t0 ,tν+1 )(x, ξ) ∈ Bρ,δ
(h) (0 ≤ ϑ ≤ 1) ,
(5.25)
and
(5.26)
(0)
≤ C1,l
|pϑ,h (∆t0 ,tν+1 )|l
(0,1)
|∂ϑ pϑ,h (∆t0 ,tν+1 )|l
≤ C2,l (t0 − tν+1 )2 (0 ≤ ϑ ≤ 1) .
Fundamental solution for Schrödinger equations
489
Proof. Set
h (∆t ,t )(x, ξ) + (1 − ϑ)φ
h (t0 , tν+1 )(x, ξ) ,
ϑ,h (∆t ,t )(x, ξ) ≡ ϑφ
(5.27) φ
0 ν+1
0 ν+1
and
(5.28) pϑ,h (∆t0 ,tν+1 )(x, ξ)
≡ exp i ϑωh (∆t0 ,tν+1 )(x, ξ) + (1 − ϑ)ωh (t0 , tν+1 )(x, ξ)
+ ϑΥh (∆t0 ,tν+1 )(x, ξ) .
By Lemma 2.8 and Lemma 4.8, we get (1) and (2). Theorem 5.3.
(1) There exists a constant C3,l satisfying the following:
For any (T0 ≥) t ≥ s (≥ 0), let ∆t,s be an arbitrary division of
interval [s, t] into subintervals, and ∆t,s be an arbitrary refinement
of ∆t,s .
Then it follows that
(5.29)
|φh (∆t,s ) − φh (∆t,s )|l
(δ−ρ,2)
≤ C3,l |∆t,s |(t − s) .
Furthermore, there exists a phase function φh (t, s) ∈ Pρ,δ (c (t −
s), h, {κl }∞
l=0 ) such that
(5.30)
(δ−ρ,2)
|φh (∆t,s ) − φh (t, s)|l
≤ C3,l |∆t,s |(t − s) .
(2) There exists a constant C4,l satisfying the following:
For any (T0 ≥) t ≥ s (≥ 0), let ∆t,s be an arbitrary division of
interval [s, t] into subintervals, and ∆t,s be an arbitrary refinement
of ∆t,s .
Then it follows that
(5.31)
|ph (∆t,s ) − ph (∆t,s )|l
(0,1)
≤ C4,l |∆t,s |(t − s) .
490
Naoto Kumano-go
Furthermore, there exists a symbol ph (t, s) ∈ B0ρ,δ (h) such that
(0,1)
|ph (∆t,s ) − ph (t, s)|l
(5.32)
≤ C4,l |∆t,s |(t − s) .
and
(0)
|ph (t, s)|l
(5.33)
≤ C1,l .
(3) Furthermore, the following equation holds:
(5.34)
eiφh (t,s)(x,ξ)−ix ·ξ ph (t, s)(x, ξ)
ν+1 ···
exp
i(xj−1 − xj ) · ξ j
= lim Os —
|∆t,s |→0
tj−1
+i
Kh (τ, xj−1 , ξ j )dτ
j=1
dx1 d−ξ 1 · · · dxν d−ξ ν ,
tj
where ∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν+1 ≡ s (≥ 0), x0 ≡ x,
xν+1 ≡ x and ξ ν+1 ≡ ξ.
Proof.
(I) Set
(5.35)
∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν+1 ≡ s (≥ 0) ,
and
(5.36)
∆t,s

t0 ≡ t0,0 ≥ t0,1 ≥ · · · ≥ t0,k0 ≡ t1






 t1 ≡ t1,0 ≥ t1,1 ≥ · · · ≥ t1,k1 ≡ t2
.................................
:


 tν ≡ tν,0 ≥ tν,1 ≥ · · · ≥ tν,k ≡ tν+1

ν



(kj > 0, j = 0, 1, 2, . . . , ν) .
Fundamental solution for Schrödinger equations
491
Then we have
h (∆ ) − φ
h (∆t,s )
(5.37) φ
t,s
ν+1 h (∆
=
φ
t0 ,tj−1 )#φh (∆tj−1 ,tj )#φh (∆tj ,tν+1 )
j=1
h (∆
h (tj−1 , tj )#φ
h (∆t ,t )
−φ
)#
φ
t0 ,tj−1
j ν+1
=
ν+1 j=1
1
0
ϑ,h (∆
h (∆t ,t ) dϑ .
h (∆
∂ϑ φ
)#
φ
)#
φ
t0 ,tj−1
tj−1 ,tj
j ν+1
By Lemma 2.7, Lemma 5.2 and Theorem 5.1, there exists a constant χl
such that
h (∆ ) − φ
h (∆t,s )}|
|∂ξα Dxβ {φ
t,s
ν+1 1
≤
χ|α+β|−1 (tj−1 − tj )2 x; ξ
2 dϑ
(5.38)
j=1
0
≤ χ|α+β|−1 |∆t,s |(t − s)x; ξ
2 .
Hence we get (1).
(II) Define qj,ϑ,h (∆t,s , ∆t,s )(x, ξ) ∈ B0ρ,δ (h) by
(5.39) qj,ϑ,h (∆t,s , ∆t,s )(φh (∆t0 ,tj−1 )#φϑ,h (∆tj−1 ,tj )#φh (∆tj ,tν+1 ); X, Dx )
≡ ph (∆t0 ,tj−1 )(φh (∆t0 ,tj−1 ); X, Dx )
◦ pϑ,h (∆tj−1 ,tj )(φϑ,h (∆tj−1 ,tj ); X, Dx )
◦ ph (∆tj ,tν+1 )(φh (∆tj ,tν+1 ); X, Dx ) .
Then we have
(5.40)
qj,1,h (∆t,s , ∆t,s )(x, ξ) = ph (∆t0 ,tj , ∆tj+1 ,tν+1 )(x, ξ)
qj,0,h (∆t,s , ∆t,s )(x, ξ) = ph (∆t0 ,tj−1 , ∆tj ,tν+1 )(x, ξ) .
492
Naoto Kumano-go
Hence, it follows that
(5.41)
ph (∆t,s ) − ph (∆t,s )
ν+1 =
ph (∆t0 ,tj , ∆tj+1 ,tν+1 ) − ph (∆t0 ,tj−1 , ∆tj ,tν+1 )
j=1
=
ν+1 1
j=1
0
∂ϑ qj,ϑ,h (∆t,s , ∆t,s ) dϑ .
By Lemma 4.7, Lemma 5.2 and Theorem 5.1, we get (2).
(III) From (1) and (2), (3) is clear. Theorem 5.4.
The Fourier integral operator Uh (t, s) ≡ ph (t, s)(φh (t, s); X, Dx ) with the
phase function φh (t, s)(x, ξ) and the symbol ph (t, s)(x, ξ) in Theorem 5.3,
satisfies the following:
(1) The operators Uh (t, s) : L 2 → L 2 , H 2,h → H 2,h are uniformly
bounded for 0 ≤ s ≤ t ≤ T0 and 0 < h < 1.
(2) The operators Kh (t, X, Dx )Uh (t, s), ∂t Uh (t, s) : H 2,h → H 0,h are
uniformly bounded for 0 ≤ s ≤ t ≤ T0 and 0 < h < 1.
(3) The following relation holds:
(5.42)
Uh (t, s) =
i
lim e
t
t1
Kh (τ )dτ
|∆t,s |→0
· · · ei
tν
s
i
(X, Dx )e
Kh (τ )dτ
t
1
t2
Kh (τ )dτ
(X, Dx )
(X, Dx ) ,
where ∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν ≥ tν+1 ≡ s (≥ 0) is
an arbitrary division of interval [s,t] into subintervals, and |∆t,s | ≡
max |tj−1 − tj |.
1≤j≤ν+1
(4) Uh (t, s) is the fundamental solution for the operator Lh such that
(5.43)
Lh Uh (t, s) = 0
on [s, T0 ]
Uh (s, s) = I (0 ≤ s ≤ T0 ) .
Fundamental solution for Schrödinger equations
493
Proof.
(0) By Proposition 3.7, Theorem 4.2 and Theorem 5.3, (1) and (2) are
clear. We have only to check (3) and (4) for u ∈ S.
(I) By Proposition 3.3 and Theorem 5.1, there exist a constant M1,h,l
and an integer l1 such that
|ph (∆t0 ,tν+1 )(φh (∆t0 ,tν+1 ); X, Dx )u|l,S ≤ M1,h,l |u|l1 ,S
∆t0 ,tν+1 : (T0 ≥) t0 ≥ t1 ≥ · · · ≥ tν+1 (≥ 0) .
(5.44)
Furthermore, we can write
(5.45)
ph (∆t0 ,tν+1 )(φh (∆t0 ,tν+1 ); X, Dx )
− ph (t0 , tν+1 )(φh (t0 , tν+1 ); X, Dx ) u(x)
1 iφϑ,h (∆t0 ,tν+1 )(x,ξ)
−
∂ϑ
pϑ,h (∆t0 ,tν+1 )(x, ξ)û(ξ)dξ dϑ
e
=
0
1
=
qϑ,h (∆t0 ,tν+1 )(φϑ,h (∆t0 ,tν+1 ); X, Dx )u(x)dϑ ,
0
where
(5.46) qϑ,h (∆t0 ,tν+1 )(x, ξ) ≡∂ϑ pϑ,h (∆t0 ,tν+1 )(x, ξ)
+ i∂ϑ φϑ,h (∆t0 ,tν+1 )(x, ξ)pϑ,h (∆t0 ,tν+1 )(x, ξ) .
Hence, by Proposition 3.3, Theorem 5.1 and Lemma 5.2, there exist a constant M2,h,l and an integer l2 such that
(5.47)
ph (∆t0 ,tν+1 )(φh (∆t0 ,tν+1 ); X, Dx )
− ph (t0 , tν+1 )(φh (t0 , tν+1 ); X, Dx ) u
≤ M2,h,l (t0 − tν+1 ) |u|l2 ,S
l,S
2
∆t0 ,tν+1 : (T0 ≥) t0 ≥ t1 ≥ · · · ≥ tν+1 (≥ 0) .
494
Naoto Kumano-go
Similarly, there exist a constant M3,h,l and an integer l3 such that
(5.48)
i∂t + K(t, X, Dx ) ph (t, t1 )(φh (t, t1 ); X, Dx )u
l,S
≤ M3,h,l (t − t1 )|u|l3 ,S
(T0 ≥ t ≥ t1 ≥ 0) .
(II) Let ∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν+1 ≡ s (≥ 0) and let ∆t,s be
any refinement of ∆t,s . Then we can write
(5.49)
ph (∆t,s )(φh (∆t,s ); X, Dx ) − ph (∆t,s )(φh (∆t,s ); X, Dx ) u
=
ν+1
ph (∆t0 ,tj−1 )(φh (∆t0 ,tj−1 ); X, Dx )
j=1
◦ ph (∆tj−1 ,tj )(φh (∆tj−1 ,tj ); X, Dx )
− ph (tj−1 , tj )(φh (tj−1 , tj ); X, Dx )
◦ ph (∆tj ,tν+1 )(φh (∆tj ,tν+1 ); X, Dx )u .
Hence, using (5.44) and (5.47), there exist a constant M4,h,l and an integer
l4 such that
(5.50) ph (∆t,s )(φh (∆t,s ); X, Dx ) − ph (∆t,s )(φh (∆t,s ); X, Dx ) u
l,S
≤ M4,h,l |∆t,s |(t − s)|u|l4 ,S
∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν+1 ≡ s (≥ 0) .
Therefore, there exists Uh (t, s)u ∈ S such that
Uh (t, s)u =
(5.51)
lim
|∆t,s |→0
ph (∆t,s )(φh (∆t,s ); X, Dx )u
( uniformly on [s, T0 ]) .
Similarly, using (5.44), (5.47) and (5.48), there exist a constant M5,h,l and
an integer l5 such that
(5.52) ∂t ph (∆t,s )(φh (∆t,s ); X, Dx ) − ∂t ph (∆t,s )(φh (∆t,s ); X, Dx ) u
l,S
≤ M5,h,l |∆t,s ||u|l5 ,S
∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν+1 ≡ s (≥ 0) .
Fundamental solution for Schrödinger equations
495
Therefore, there exists Vh (t, s)u ∈ S such that
(5.53)
Vh (t, s)u =
lim
|∆t,s |→0
∂t ph (∆t,s )(φh (∆t,s ); X, Dx )u
( uniformly on [s, T0 ]) .
Hence, Uh (t, s)u is differentiable and ∂t Uh (t, s)u = Vh (t, s)u.
Furthermore, using (5.44) and (5.48), there exist a constant M6,h,l and
an integer l6 such that
(5.54)
i∂t + Kh (t, X, Dx ) ph (∆t,s )(φh (∆t,s ); X, Dx )u
l,S
≤ M6,h,l (t − t1 )|u|l6 ,S
∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν+1 ≡ s (≥ 0) .
Therefore, we have
(5.55)
lim
|∆t,s |→0
i∂t + Kh (t, X, Dx ) ph (∆t,s )(φh (∆t,s ); X, Dx )u = 0
As a consequence, we get
(5.56)
Lh Uh (t, s)u = 0
on [s, T0 ]
Uh (s, s)u = u (0 ≤ s ≤ T0 ) .
(III) By Theorem 5.1 and Theorem 5.3, there exist a constant M7,h,l and
an integer l7 such that
(5.57) ph (∆t,s )(φh (∆t,s ); X, Dx ) − ph (t, s)(φh (t, s); X, Dx ) u
l,S
≤ M7,h,l |∆t,s |(t − s)|u|l7 ,S
∆t,s : (T0 ≥) t ≡ t0 ≥ t1 ≥ · · · ≥ tν+1 ≡ s (≥ 0) .
Hence, we have
(5.58)
Uh (t, s) = ph (t, s)(φh (t, s); X, Dx ) .
(VI) Finally, we prove the uniqueness of Uh (t, s):
496
Naoto Kumano-go
Let uh (t, s) ∈ C 1 ([s, T0 ]; S) be a solution of the Cauchy problem such
that
(5.59)
Lh uh (t, s) = 0
on [s, T0 ]
uh (s, s) = 0 .
Set
(5.60)
Kh (t, ξ, x ) ≡ Kh (t, x , ξ)
Lh ≡ i∂t + Kh (t, Dx , X ) .
Then there exists Wh (t, x, ξ) such that
Lh = i∂t + Hh (t, X, Dx ) + Wh (t, X, Dx )
(5.61)
and
(5.62)

C0,0 x; ξ
 |∂ α Dxβ {W (t, hδ x, h−ρ ξ)}| ≤
ξ
h
Cα,β

,
Im{Wh (t, hδ x, h−ρ ξ)} ≤ C0,0
(α = β = 0)
(|α + β| ≥ 1)
is a constant independent of 0 < h < 1.
on [0, T0 ] × R 2n . Here Cα,β
Hence, for any 0 ≤ θ ≤ T0 , there exists a fundamental solution U (t, θ)
(0 ≤ t ≤ θ) for the operator Lh such that
Lh Uh (t, θ) = 0
(5.63)
on [0, θ]
Uh (θ, θ) = I .
For any v(t) ∈ C 0 ([s, T0 ]; S), we set
(5.64)
zh (t) ≡ −i
t
T0
Uh (t, θ)v(θ)dθ (s ≤ t ≤ T0 ) .
Then we have
(5.65)
Lh zh (t) = v(t) on [s, T0 ]
zh (T0 ) = 0 .
Fundamental solution for Schrödinger equations
497
Hence, using (5.59) and (5.65), we have
(5.66)
T0
(uh (t, s), v(t))dt
s
=
s
T0
(uh (t, s), Lh zh (t))dt
t=T0 +
= − i(uh (t, s), zh (t))
t=s
T0
(Lh uh (t, s), zh (t))dt
s
= 0 (v(t) ∈ C 0 ([s, T0 ]; S)) .
Therefore, we get uh (t, s) = 0 on [s, T0 ]. References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Feynman, R. P., Space-time approach to non relativistic quantum mechanics,
Rev. of Modern Phys. 20 (1948), 367–387.
Fujiwara, D., Remarks on convergence of some Feynman path integrals, Duke
Math J. 47 (1980), 559–600.
Fujiwara, D., The stationary phase method with an estimate of the remainder
term on a space of large dimension, Nagoya Math. J. 124 (1991), 61–97.
Fujiwara, D., Some Feynman Path Integrals As Oscillatory Integrals Over A
Sobolev Manifold, preprint (1993).
Kitada, H., Fourier integral operators with weighted symbols and micro-local
resolvent estimates, J. Math. Soc. Japan 39 (1987), 455–476.
Kitada, H. and H. Kumano-go, A family of Fourier integral operators and the
fundamental solution for a Schrödinger equation, Osaka J. Math. 18 (1981),
291–360.
Kumano-go, H., PSEUDO-DIFFERENTIAL OPERATORS, MIT press,
Cambridge, Massachusetts and London, England, 1983.
Kumano-go, H. and K. Taniguchi, Fourier integral operators of multi-phase
and the fundamental solution for a hyperbolic system, Funkcial. Ekvac. 22
(1979), 161–196.
Taniguchi, K., Multi-products of Fourier integral operators and the fundamental solution for a hyperbolic system with involutive characteristics, Osaka J.
Math. 21 (1984), 169–224.
(Received December 22, 1994)
498
Naoto Kumano-go
Graduate School of Mathematical Sciences
University of Tokyo
3-8-1, Komaba
Meguro-ku, Tokyo 153
Japan