A WEIGHTED DISPERSIVE ESTIMATE FOR SCHRÖDINGER
OPERATORS IN DIMENSION TWO
M. BURAK ERDOĞAN AND WILLIAM R. GREEN
Abstract. Let H = −∆ + V , where V is a real valued potential on R2 satisfying |V (x)| .
hxi−3− . We prove that if zero is a regular point of the spectrum of H = −∆ + V , then
kw−1 eitH Pac f kL∞ (R2 ) .
1
kwf kL1 (R2 ) ,
|t| log2 (|t|)
|t| > 2,
with w(x) = log2 (2 + |x|). This decay rate was obtained by Murata in the setting of
weighted L2 spaces with polynomially growing weights.
1. Introduction
The free Schrödinger evolution on Rd ,
e−it∆ f (x) = Cd
1
td/2
Z
e−i|x−y|
2 /4t
f (y)dy,
Rd
satisfies the dispersive estimate
ke−it∆ f k∞ .
1
kf k1 .
|t|d/2
In recent years many authors (see, e.g., [20, 28, 13, 11, 12, 29, 14, 37, 9, 5, 6], and the survey
article [31]) worked on the problem of extending this bound to the perturbed Schrödinger
operator H = −∆ + V , where V is a real-valued potential with sufficient decay at infinity
and some smoothness for d > 3. Since the perturbed operator may have negative point
spectrum one needs to consider eitH Pac (H), where Pac (H) is the orthogonal projection
onto the absolutely continuous subspace of L2 (R2 ). One also assumes that zero is a regular
point of the spectrum of H. This is equivalent to the boundedness of the resolvent,
RV± (λ2 ) = RV (λ2 ± i0) = (H − (λ2 ± i0))−1 ,
as an operator between certain weighted L2 spaces as λ → 0.
It is easy to see that t−d/2 decay rate at infinity is optimal for the free evolution. In
dimensions d ≥ 3 one can not hope to have a faster decay rate for the perturbed operator.
In fact, it is known that (see, e.g., [26, 17, 24, 15, 16, 8, 36, 10, 3, 7]) the decay rate as
Date: January 31, 2012.
1
2
M. B. ERDOĞAN, W. R. GREEN
t → ∞ is in general slower if zero is not regular point of the spectrum. In dimensions d = 1
and d = 2, zero is not a regular point of the spectrum of the Laplacian since the constant
function is a resonance. Therefore, for the perturbed operator −∆ + V , one may expect
to have a faster dispersive decay at infinity if zero is regular. Indeed, in [24, Theorem 7.6],
Murata proved that if zero is a regular point of the spectrum, then for |t| > 2
3
kw1−1 eitH Pac (H)f kL2 (R1 ) . |t|− 2 kw1 f kL2 (R1 ) ,
kw2−1 eitH Pac (H)f kL2 (R2 ) . |t|−1 (log |t|)−2 kw2 f kL2 (R2 ) .
Here w1 and w2 are weight functions growing at a polynomial rate at infinity. It is also
assumed that the potential decays at a polynomial rate at infinity (for d = 2, it suffices to
1
assume that w2 (x) = hxi−3− and |V (x)| . hxi−6− where hxi := (1 + |x|2 ) 2 ). This type of
estimates are very useful in the study of nonlinear asymptotic stability of (multi) solitons
in lower dimensions since the dispersive decay rate in time is integrable at infinity (see, e.g.,
[30, 21, 22]). Also see [4, 32, 25, 34] for other applications of weighted dispersive estimates
to nonlinear PDEs.
In [31], Schlag extended Murata’s result for d = 1 to the L1 → L∞ setting. He proved
that if zero is regular, then
3
kw−1 eitH Pac (H)f kL∞ (R) . |t|− 2 kwf k1 ,
|t| > 2,
with w(x) = hxi provided khxi4 V k1 < ∞.
In this paper, we study the two dimensional case. Our main result is the following
Theorem 1.1. Let V (x) . hxi−2β for some β > 32 . If zero is a regular point of the spectrum
of H = −∆ + V , then we have
kw−1 eitH Pac f kL∞ (R2 ) .
1
kwf kL1 (R2 ) ,
|t| log2 (|t|)
|t| > 2,
where w(x) = log2 (2 + |x|).
We note that the requirement for the weight function and the potential is much weaker
than was assumed in [24]. We think similar bounds hold in the case of matrix Schrödinger
operators, which we plan to address in a subsequent paper.
There are not many results on L1 → L∞ estimates in the two dimensional case. In [29],
Schlag proved that
keitH Pac kL1 (R2 )→L∞ (R2 ) . |t|−1
WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
3
under the decay assumption |V | . hxi−3− and the assumption that zero is a regular point of
the spectrum. For the case when zero is not regular, see [7]. Yajima, [35], established that
the wave operators are bounded on Lp (R2 ) for 1 < p < ∞ if zero is regular. The hypotheses
on the potential V were relaxed slightly in [19]. High frequency dispersive estimates, similar
to those obtained in [29] were obtained by Moulin, [23], under an integrability condition on
the potential.
We also note that standard spectral theoretic results for H apply. Under our assumptions
we have that the spectrum of H can be expressed as the absolutely continuous spectrum,
the interval [0, ∞), and finitely many eigenvalues of finite multiplicity on (−∞, 0]. See [27]
for spectral theory and [33] for Birman-Schwinger type bounds.
As usual, Theorem 1.1 follows from (see, e.g., [13, 29, 7])
∞
Z
(1)
sup
L≥1
0
2
eitλ λχ(λ/L)[RV+ (λ2 ) − RV− (λ2 )](x, y)dλ .
w(x)w(y)
,
t log2 (t)
t > 2.
Here χ is an even smooth function supported in [−λ1 , λ1 ] and χ(x) = 1 for |x| < λ1 /2, and
λ1 is a sufficiently small number which is fixed throughout the paper.
In this paper we prove that
Theorem 1.2. Under the assumptions of Theorem 1.1, we have for t > 2
Z
(2)
e
sup
L≥1
∞
0
itλ2
λχ(λ/L)[RV+ (λ2 )
−
RV− (λ2 )](x, y)dλ
p
3
3
w(x)w(y) hxi 2 hyi 2
.
+
,
t1+α
t log2 (t)
where 0 < α < min( 41 , β − 32 ).
Our proof of Theorem 1.2 will be mostly self-contained. Since we can allow polynomial
growth in x and y for many terms that arise, the proof is somehow less technical than the
proof in [29].
To obtain Theorem 1.1, we use the inequality
min 1,
a log2 (a)
.
,
b
log2 (b)
a, b > 2,
and interpolate (2) with the result of Schlag in [29], which states that under the conditions
of Theorem 1.1, one has
Z
sup
L≥1
0
∞
1
2
eitλ λχ(λ/L)[RV+ (λ2 ) − RV− (λ2 )](x, y)dλ . .
t
4
M. B. ERDOĞAN, W. R. GREEN
2. The Free Resolvent
In this section we discuss the properties of the free resolvent, R0± (λ2 ) = [−∆−(λ2 ±i0)]−1 ,
in R2 . To simplify the formulas, we use the notation
e
f = O(g)
to denote
dj
dj f
=
O
g ,
dλj
dλj
j = 0, 1, 2, 3, ...
ek (g).
If the derivative bounds hold only for the first k derivatives we write f = O
Recall that
i
i
1
R0± (λ2 )(x, y) = ± H0± (λ|x − y|) = ± J0 (z) − Y0 (z).
4
4
4
(3)
Thus, we have
(4)
i
R0+ (λ2 )(x, y) − R0− (λ2 )(x, y) = J0 (λ|x − y|).
2
From the series expansions for the Bessel functions, see [1], we have, as z → 0,
(5)
(6)
1
1
e6 (z 6 ),
J0 (z) = 1 − z 2 + z 4 + O
4
64
2
2 1 2 e 4
Y0 (z) = (log(z/2) + γ)J0 (z) +
z + O4 (z )
π
π 4
2γ
2
e 2 log(z)).
+ O(z
= log(z/2) +
π
π
Further, for |z| > 1, we have the representation (see, e.g., [1])
(7)
H0± (z) = e±iz ω± (z),
e (1 + |z|)− 21 .
ω± (z) = O
This implies that for |z| > 1
(8)
C(z) = eiz ω+ (z) + e−iz ω− (z),
e (1 + |z|)− 21 ,
ω± (z) = O
for any C ∈ {J0 , Y0 } respectively with different ω± .
In particular, for λ|x − y| . 1, we have
(9)
i
γ
1
e λ2 |x − y|2 log(λ|x − y|) .
R0± (λ2 )(x, y) = ± −
−
log(λ|x − y|/2) + O
4 2π 2π
For λ|x − y| & 1, we have
(10)
R0± (λ2 )(x, y) = eiλ|x−y| ω+ (λ|x − y|) + e−iλ|x−y| ω− (λ|x − y|).
WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
5
3. Resolvent Expansion Around the Zero Energy
Let U (x) = 1 if V (x) ≥ 0 and U (x) = −1 if V (x) < 0, and let v = |V |1/2 . We have
V = U v 2 . We use the symmetric resolvent identity, valid for =λ > 0:
RV± (λ2 ) = R0± (λ2 ) − R0± (λ2 )vM ± (λ)−1 vR0± (λ2 ),
(11)
where M ± (λ) = U +vR0± (λ2 )v. The key issue in the resolvent expansions is the invertibility
of the operator M ± (λ) for small λ under various spectral assumptions at zero. Below,
using the properties of the free resolvent listed above, we provide an expansion for the free
resolvent around λ = 0, and then using it obtain analogous expansions of the operator
M ± (λ). Similar lemmas were proved in [18] and [29], however we need to obtain slightly
different error bounds. The following operator and the function arise naturally in the
expansion of M ± (λ) (see (6))
1
G0 f (x) = −
2π
(12)
Z
log |x − y|f (y) dy,
R2
i
1
γ .
g ± (λ) := kV k1 ± −
log(λ/2) −
4 2π
2π
(13)
Lemma 3.1. We have the following expansion for the kernel of the free resolvent
R0± (λ2 )(x, y) =
1 ±
g (λ) + G0 (x, y) + E0± (λ)(x, y).
kV k1
Here G0 (x, y) is the kernel of the operator G0 in (12), g ± (λ) is as in (13), and E0± satisfies
the bounds
1
1
|E0± | . λ 2 |x − y| 2 ,
1
1
|∂λ E0± | . λ− 2 |x − y| 2 ,
1
3
|∂λ2 E0± | . λ− 2 |x − y| 2 .
Proof. To obtain the expansions recall (9), which states that for λ|x − y| . 1, we have
i
γ
1
e λ2 |x − y|2 log(λ|x − y|)
R0± (λ2 )(x, y) = ± −
−
log(λ|x − y|/2) + O
4 2π 2π
g ± (λ)
e λ2 |x − y|2 log(λ|x − y|) .
=
+ G0 (x, y) + O
kV k1
For λ|x − y| & 1, using (10) we have
R0± (λ2 )(x, y) = eiλ|x−y| ω+ (λ|x − y|) + e−iλ|x−y| ω− (λ|x − y|)
=
g ± (λ)
eiλ|x−y|
e log(λ|x − y|) + O
e
.
+ G0 (x, y) + O
1/2
kV k1
(1 + λ|x − y|)
Let χ be a smooth cutoff for [−1, 1], and χ
e = 1 − χ. Using the formulas above we have
e λ|x − y|)2 log(λ|x − y|) ,
E0± (λ)(x, y)χ(λ|x − y|) = χ(λ|x − y|)O
6
M. B. ERDOĞAN, W. R. GREEN
eiλ|x−y|
.
1/2
(1 + λ|x − y|)
e log(λ|x − y| + O
e
E0± (λ)(x, y)e
χ(λ|x − y|) = χ
e(λ|x − y|) O
Combining these bounds we have
1
|E0± (λ)(x, y)| . (λ|x − y|)2− χ(λ|x − y|) + (λ|x − y|)0+ χ
e(λ|x − y|) . (λ|x − y|) 2 .
For λ-derivatives, note that
|∂λ E0± (λ)(x, y)| .
1
1
(λ|x − y|)2−
|x − y|1/2
χ
e(λ|x − y|) . λ− 2 |x − y| 2 ,
χ(λ|x − y|) +
1/2
λ
λ
|∂λ2 E0± (λ)(x, y)| .
1
3
(λ|x − y|)2−
|x − y|3/2
χ(λ|x
−
y|)
+
χ
e(λ|x − y|) . λ− 2 |x − y| 2 .
λ2
λ1/2
and
The following corollary follows from the bounds for ∂λ E0± and ∂λ2 E0± .
Corollary 3.2. For 0 < α < 1 and b > a > 0 we have
1
1
|∂λ E0± (b) − ∂λ E0± (a)| . a− 2 |b − a|α |x − y| 2 +α .
Proof. The mean value theorem together with the bound on ∂λ2 E0± from Lemma 3.1 imply
that
3
|∂λ E0± (b) − ∂λ E0± (a)| . a−1/2 |b − a||x − y| 2 .
Interpolating this with the bound on ∂λ E0± from Lemma 3.1 yields the claim.
Lemma 3.3. Let 0 < α < 1. For λ > 0 define M ± (λ) := U + vR0± (λ2 )v. Let P =
vh·, vikV k−1
1 denote the orthogonal projection onto v. Then
M ± (λ) = g ± (λ)P + T + E1± (λ).
Here T = U + vG0 v where G0 is an integral operator defined in (12). Further, the error
term satisfies the bound
1
sup λ− 2 |E1± (λ)|
0<λ<λ1
HS
1
+
sup λ 2 |∂λ E1± (λ)|
0<λ<λ1
+
sup
0<λ<b<λ1
3
provided that v(x) . hxi− 2 −α− .
HS
1
λ 2 (b − λ)−α |∂λ E1± (b) − ∂λ E1± (λ)|
HS
. 1,
WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
7
Proof. Note that
E1± (λ) = M ± (λ) − [g ± (λ)P + T ] = vR0± (λ2 )v − g ± (λ)P − vG0 v = vE0± (λ)v.
Therefore the statement follows from Lemma 3.1 and Corollary 3.2, and the fact that for
k ≥ 0, v(x)|x − y|k v(y) is Hilbert-Schmidt on L2 (R2 ) provided that v(x) . hxi−k−1− .
Recall the following definition from [29] and [7].
Definition 3.4. We say an operator T : L2 (R2 ) → L2 (R2 ) with kernel T (·, ·) is absolutely
bounded if the operator with kernel |T (·, ·)| is bounded from L2 (R2 ) to L2 (R2 ).
It is worth noting that finite rank operators and Hilbert-Schmidt operators are absolutely
bounded. Also recall the following definition from [18], also see [29] and [7].
1 − P . We say zero is a regular point of the spectrum of
Definition 3.5. Let Q :=
H = −∆ + V provided QT Q = Q(U + vG0 v)Q is invertible on QL2 (R2 ).
In [29], it was proved that if zero is regular, then the operator D0 := (QT Q)−1 is absolutely bounded on QL2 .
Below, we discuss the invertibility of M ± (λ) = U + vR0± (λ2 )v, for small λ. This lemma
was proved in [18] and in [29]. We include the proof for completeness since we state slightly
different error bounds.
Lemma 3.6. Let 0 < α < 1. Suppose that zero is a regular point of the spectrum of
H = −∆ + V . Then for sufficiently small λ1 > 0, the operators M ± (λ) are invertible for
all 0 < λ < λ1 as bounded operators on L2 (R2 ). Further, one has
M ± (λ)−1 = h± (λ)−1 S + QD0 Q + E ± (λ),
(14)
Here h± (λ) = g ± (λ) + c (with c ∈ R), and
"
P
(15)
S=
−QD0 QT P
−P T QD0 Q
#
QD0 QT P T QD0 Q
is a finite-rank operator with real-valued kernel. Further, the error term satisfies the bounds
1
sup λ− 2 |E ± (λ)|
0<λ<λ1
HS
1
sup λ 2 |∂λ E ± (λ)|
+
0<λ<λ1
+
sup
0<λ<b.λ<λ1
3
provided that v(x) . hxi− 2 −α− .
HS
1
λ 2 +α (b − λ)−α |∂λ E ± (b) − ∂λ E ± (a)|
HS
. 1,
8
M. B. ERDOĞAN, W. R. GREEN
Proof. We will give the proof for M + and drop the superscript “+” from formulas. Using
Lemma 3.3, we write M (λ) with respect to the decomposition L2 (R2 ) = P L2 (R2 )⊕QL2 (R2 ).
"
M (λ) =
g(λ)P + P T P
PTQ
QT P
QT Q
#
+ E1 (λ).
Denote the matrix component of the above equation by A(λ) = {aij (λ)}2i,j=1 .
Since QT Q is invertible by the assumption that zero is regular, by the Fehsbach formula
−1
invertibility of A(λ) hinges upon the existence of d = (a11 − a12 a−1
22 a21 ) . Denoting D0 =
(QT Q)−1 : QL2 → QL2 , we have
a11 − a12 a−1
22 a21 = g(λ)P + P T P − P T QD0 QT P = h(λ)P
with h(λ) = g(λ) + T r(P T P − P T QD0 QT P ) = g(λ) + c, where c ∈ R as the kernels of T ,
QD0 Q and v are real-valued. The invertibility of this operator on P L2 for small λ follows
from (13). Thus, by the Fehsbach formula,
"
A(λ)−1 =
(16)
−da12 a−1
22
d
#
−1
−1
−1
−a−1
22 a21 d a22 a21 da12 a22 + a22
"
#
P
−P T QD0 Q
−1
= h (λ)
+ QD0 Q =: h−1 (λ)S + QD0 Q.
−QD0 QT P QD0 QT P T QD0 Q
Note that S has rank at most two. This and the absolute boundedness of QD0 Q imply that
A−1 is absolutely bounded.
Finally, we write
M (λ) = A(λ) + E1 (λ) = [1 + E1 (λ)A−1 (λ)]A(λ).
Therefore, by a Neumann series expansion, we have
(17)
−1
M −1 (λ) = A−1 (λ) 1 + E1 (λ)A−1 (λ)
= h(λ)−1 S + QD0 Q + E(λ),
The error bounds follow in light of the bounds for E1 (λ) in Lemma 3.3 and the fact that,
as an absolutely bound operator on L2 , |A−1 (λ)| . 1, |∂λ A−1 (λ)| . λ−1 , and (for 0 < λ <
b < λ1 )
|∂λ A−1 (λ) − ∂λ A−1 (b)| . (b − λ)α λ−1−α .
1
In the Lipschitz estimate, the factor λ− 2 −α arises from the case when the derivative hits
A−1 (λ).
WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
9
Remark. Under the conditions of Theorem 1.1, the resolvent identity
(18) RV± (λ2 ) = R0± (λ2 ) − R0± (λ2 )vM ± (λ)−1 vR0± (λ2 )
= R0± (λ2 ) − R0± (λ2 )
vSv ± 2
R (λ ) − R0± (λ2 )vQD0 QvR0± (λ2 ) − R0± (λ2 )vE ± (λ)vR0± (λ2 )
h± (λ) 0
1
1
holds as an operator identity between the spaces L2, 2 + (R2 ) and L2,− 2 − (R2 ), as in the
limiting absorption principle, [2].
We complete this section by noting that for fixed x, y the kernel RV± (λ2 )(x, y) of the
resolvent remains bounded as λ → 0. This is because of a cancellation between the first
and second summands of the second line in (18). A consequence of this cancellation will be
crucial in the next section, see Proposition 4.3 and Proposition 4.4.
4. Proof of Theorem 1.2 for Low Energies
In this section we prove Theorem 1.2 for low energies. Let χ be a smooth cut-off for
[0, λ1 ] as in the introduction, where λ1 is sufficiently small so that the expansions in the
previous section are valid. We have
3
Theorem 4.1. Fix 0 < α < 1/4. Let v(x) . hxi− 2 −α− . For any t > 2, we have
p
3
3
Z ∞
w(x)w(y) hxi 2 hyi 2
− 2
+ 2
itλ2
+
(19)
e λχ(λ)[RV (λ ) − RV (λ )](x, y)dλ .
.
t1+α
t log2 (t)
0
We start with a simple lemma:
Lemma 4.2. For t > 2, we have
Z
∞
2
eitλ λ E(λ)dλ −
0
iE(0)
1
.
2t
t
Z
0
t−1/2
|E 0 (λ)|dλ +
E 0 (t−1/2 )
1
+
t2
t3/2
∞
E 0 (λ) 0
t−1/2
λ
Z
dλ.
2
Proof. To prove this lemma we integrate by parts once using the identity eitλ λ =
2
∂λ eitλ /(2it), and then divide the integral into pieces on the sets [0, t−1/2 ] and [t−1/2 , ∞).
Finally integrate by parts once more in the latter piece:
Z
0
∞
Z t−1/2
Z ∞
0
iE(0)
i
i
2 E (λ)
itλ2 0
e λE(λ)dλ =
+
e E (λ)dλ +
eitλ λ
dλ
2t
2t 0
2t t−1/2
λ
Z t−1/2
Z ∞
0
0
iE(0)
i
1 E 0 (λ)
1
2
itλ2 E (λ)
=
+
eitλ E 0 (λ)dλ − 2
−
e
dλ.
2t
2t 0
4t λ λ=t−1/2 4t2 t−1/2
λ
itλ2
10
M. B. ERDOĞAN, W. R. GREEN
We start with the contribution of the free resolvent to (19). Note that it is easy to obtain
this statement for the free evolution using its convolution kernel. We choose to present the
proof below to introduce some of the methods we will employ throughout the paper.
Proposition 4.3. We have
Z ∞
hxi 32 hyi 23 1
+ 2
− 2
itλ2
e λχ(λ)[R0 (λ ) − R0 (λ )](x, y)dλ = − + O
.
5
4t
0
t4
Proof. Using Lemma 3.1, we have
R0+ − R0− =
i
+ E0+ (λ) − E0− (λ).
2
Therefore we can rewrite the λ integral above as
Z
Z ∞
i ∞ itλ2
2
e λχ(λ)dλ +
eitλ λχ(λ)(E0+ (λ) − E0− (λ))dλ =: A + B.
2 0
0
Note that by integrating by parts twice as in the proof of Lemma 4.2 we obtain
Z ∞
χ0 (λ) 1
i
1
2 d
(20)
A=− − 2
eitλ
dλ = − + O(t−2 ).
4t 8t 0
dλ
λ
4t
Using the bounds in Lemma 3.1 for E(λ) = χ(λ)(E0+ (λ) − E0− (λ)), we see that E(0) = 0,
and
1
1
1p
|∂λ E(λ)| . λ− 2 |x − y| 2 . λ− 2 hxihyi,
∂ E(λ) 1
3
3
5
3
3
5
λ
∂λ
. χ(λ)[λ− 2 |x − y| 2 + λ− 2 |x − y| 2 ] . λ− 2 hxi 2 hyi 2 .
λ
Applying Lemma 4.2 with these bounds we obtain
1
|B| .
t
Z
0
t−1/2
Z
E 0 (t−1/2 )
1 ∞ E 0 (λ) 0
|E (λ)|dλ +
+ 2
dλ
t t−1/2
λ
t3/2
p
p
3
3 Z
3
3
Z −1/2
hxihyi t
hxihyi hxi 2 hyi 2 ∞ − 5
hxi 2 hyi 2
− 12
2 dλ .
.
λ dλ +
+
λ
.
5
5
t
t2
t−1/2
0
t4
t4
0
We now consider the contribution of the second term in (18) to (19):
Z Z ∞
2
(21)
eitλ λχ(λ)[R− − R+ ]v(x1 )S(x1 , y1 )v(y1 )dλdx1 dy1 ,
R4
0
where
(22)
R± =
R0± (λ2 )(x, x1 )R0± (λ2 )(y1 , y)
.
h± (λ)
WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
11
3
Proposition 4.4. Let 0 < α < 1/4. If v(x) . hxi− 2 −α− , then we have
pw(x)w(y) hxi 21 +α+ hyi 12 +α+ 1
(21) =
.
+O
+O
4t
t1+α
t log2 (t)
Proof. Recall from Lemma 3.1 that
R0± (λ2 )(x, x1 ) =
1 ±
g (λ) + G0 (x, x1 ) + E0± (λ)(x, x1 ).
kV k1
Also recall that h± (λ) = g ± (λ) + c with c ∈ R. Therefore
R± =
e
e
1 ±
e 0 (x, x1 ) + G
e 0 (y, y1 ) + G0 (x, x1 )G0 (y, y1 ) + E ± (λ),
g
(λ)
+
c
+
G
2
2
±
g (λ) + c
kV k1
where
(23) E2± (λ) :=
e 0 (x, x1 ) e 0 (y, y1 ) 1 1 G
G
1+ ±
E0± (λ)(y, y1 ) +
1+ ±
E ± (λ)(x, x1 )
kV k1
g (λ) + c
kV k1
g (λ) + c 0
+
E0± (λ)(x, x1 )E0± (λ)(y, y1 )
,
g ± (λ) + c
e 0 = kV k1 G0 − c. Using this and (13), we have
and G
R− − R + = −
e 0 (x, x1 )G
e 0 (y, y1 )
i
G
+ c3
+ E2− (λ) − E2+ (λ),
2kV k1
(log(λ) + c1 )2 + c22
where c1 , c2 , c3 ∈ R.
Accordingly we rewrite the λ–integral in (21) as a sum of the following
Z ∞
i
2
−
(24)
eitλ λχ(λ)dλ,
2kV k1 0
Z ∞
e 0 (x, x1 )G
e 0 (y, y1 )
G
2
eitλ λχ(λ)
(25)
dλ,
(log(λ) + c1 )2 + c22
0
Z ∞
2
(26)
eitλ λχ(λ)[E2− (λ) − E2+ (λ)]dλ.
0
Note that by (20) we have
(27)
(24) =
1
+ O(t−2 ).
4tkV k1
The leading term above will cancel the boundary term that arose in Proposition 4.3.
The decay rate
1
t log2 (t)
appears because of the following lemma, which seems to be optimal.
Define
k(x, x1 ) := 1 + log− (|x − x1 |) + log+ (|x1 |),
where log− (x) = | log(x)|χ(0,1) (x) and log+ (x) = log(x)χ(1,∞) (x).
12
M. B. ERDOĞAN, W. R. GREEN
Lemma 4.5. For t > 2, we have the bound
|(25)| .
p
1
k(x,
x
)k(y,
y
)
w(x)w(y).
1
1
t log2 (t)
Lemma 4.6. Let 0 < α < 1/4. For t > 2, we have the bound
|(26)| . t−1−α k(x, x1 )k(y, y1 ) hxihyihx1 ihy1 i
1 +α+
2
.
We will prove Lemma 4.5 and Lemma 4.6 after we finish the proof of the proposition.
Using the bounds we obtained in (27), Lemma 4.5, Lemma 4.6 in (21), we obtain
Z
1
(21) =
v(x1 )S(x1 , y1 )v(y1 )dx1 dy1
4tkV k1 R4
pw(x)w(y) Z
+O
k(x,
x
)v(x
)|S(x
,
y
)|v(y
)k(y,
y
)dx
dy
1
1
1 1
1
1
1 1
t log2 (t)
R4
hxihyi 21 +α+ Z
1
1
+α+
+α+
2
2
k(x,
x
)hx
i
+O
v(x
)|S(x
,
y
)|v(y
)k(y,
y
)
dx
dy
.
1
1
1
1
1
1
1
1
1
t1+α
R4
Note that the integrals in the error terms are bounded in x, y, since
1
kv(y1 )hy1 i 2 +α+ k(y, y1 )kL2y . 1.
1
Also note that we can replace S with P in the first integral since the other parts of the
operator S contains Q on at least one side and that Qv = 0. Therefore,
1
(21) =
4tkV k1
Z
v(x1 )P (x1 , y1 )v(y1 )dx1 dy1 + O
hxihyi 23 + pw(x)w(y) +O
5
t log2 (t)
t4
pw(x)w(y) hxihyi 23 + 1
=
+O
+O
.
5
4t
t log2 (t)
t4
R4
Proof of Lemma 4.5. First note that
p
e 0 (x, x1 )| . 1 + | log |x − x1 || . k(x, x1 ) w(x).
|G
(28)
Second, we bound the λ-integral by using Lemma 4.2 with E(λ) =
χ(λ)
,
λ| log(λ)|3
|∂λ E(λ)| .
∂λ
χ(λ)
.
(log(λ)+c1 )2 +c22
Note that
∂ E(λ) χ(λ)
λ
. 3
.
λ
λ | log(λ)|3
Applying Lemma 4.2 with these bounds we obtain
Z
∞
e
0
itλ2
1
λ E(λ) dλ .
t
Z
0
t−1/2
E 0 (t−1/2 )
1
|E (λ)|dλ +
+ 2
3/2
t
t
0
∞
E 0 (λ) 0
t−1/2
λ
Z
dλ
WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
.
1
t
t−1/2
Z
0
1
χ(λ)
1
dλ +
+ 2
3
3
λ| log(λ)|
t log (t) t
13
∞
χ(λ)
t−1/2
λ3 | log(λ)|3
Z
dλ.
It is easy to calculate that
1
t
t−1/2
Z
0
1
1
dλ ∼
.
λ| log(λ)|3
t log2 (t)
It remains to bound the integral on [t−1/2 , ∞):
1
t2
Z
∞
t−1/2
Z
1
1 1/10
χ(λ)
1
dλ . 2 + 2
dλ
λ3 | log(λ)|3
t
t t−1/2 λ3 | log(λ)|3
Z
Z −1/4
1
1
1 1/10 1
1 t
1
1
. 2+ 2
dλ + 2
dλ . 3/2 +
.
3
3
3
t
t t−1/4 λ
t t−1/2 λ | log(t)|
t| log(t)|3
t
1
, ∞) converges.
The first inequality follows since the integral on [ 10
Combining the bounds we obtained above finishes the proof of the lemma.
Before we prove Lemma 4.6, we discuss the following variant of Lemma 4.2:
Lemma 4.7. Assume that E(0) = 0. For t > 2, we have
Z
∞
e
(29)
0
itλ2
p
√
√
Z
Z
1 ∞ |E 0 ( s)|
1 ∞ E 0 ( s + πt ) − E 0 ( s)
√
√
λ E(λ)dλ .
ds +
ds
t 0
t π
s(1 + st)
s
t
Z
Z
p
1 ∞ |E 0 (λ)|
1 ∞
.
dλ
+
E 0 (λ 1 + πt−1 λ−2 ) − E 0 (λ) dλ.
2
t 0 (1 + λ t)
t t−1/2
2
2
Proof. As before we integrate by parts once using the identity eitλ λ = ∂λ eitλ /(2it), and
then let s = λ2 to obtain
Z ∞
Z ∞
Z ∞
Z 2π Z ∞
0 √
t
i
i
itλ2
itλ2 0
its E ( s)
√ ds =
e λE(λ)dλ =
e E (λ)dλ =
e
+
.
2π
2t 0
4t 0
s
0
0
t
The contribution of the first integral is bounded by the first integral on the right hand side
of (29). We rewrite the second integral as
p
Z ∞
Z ∞
Z ∞
π
0
0 √
0 √
its E ( s)
it(s− πt ) E ( s)
its E ( s + t )
√ ds = −
√ ds = −
p
e
e
e
ds.
2π
2π
π
s
s
s + πt
t
t
t
Therefore it suffices to consider (the integral on [π/t, 2π/t] is bounded by the first integral
on the right hand side of (29))
Z ∞
π
t
The claim follows from
E 0 (√s) E 0 (ps + π ) t
√
e
− p
ds.
s
s + πt
its
14
M. B. ERDOĞAN, W. R. GREEN
p
p
√
√
√
|E 0 ( s) − E 0 ( s + πt )|
E 0 ( s) E 0 ( s + πt )
1
1
√
p
− p
+ |E 0 ( s)| √ − p
.
π
π
s
s
s+ t
s+ t
s + πt
p
√
√
|E 0 ( s) − E 0 ( s + πt )| |E 0 ( s)|
√
.
+
.
3
s
ts 2
Proof of Lemma 4.6. We will only consider the following part of (26):
Z
(30)
0
∞
Z ∞
e 0 (x, x1 ) G
2
2
eitλ λχ(λ) 1 +
eitλ λ E(λ) dλ.
E0 (λ)(y, y1 )dλ =:
g(λ) + c
0
The other parts are either of this form or much smaller. We also omit the ± signs since we
can not rely on a cancellation between ’+’ and ’-’ terms.
Using Lemma 3.1, Corollary 3.2, and (28), we estimate (for 0 < λ < b . λ < λ1 )
|∂λ E(λ)| . k(x, x1 )
p
p
1
1
1
w(x)χ(λ)λ− 2 hy − y1 i 2 . k(x, x1 ) w(x)hyihy1 iλ− 2 ,
∂λ E(b) − ∂λ E(λ) . χ(λ)k(x, x1 )
p
1
1
w(x)λ− 2 −α (b − λ)α hy − y1 i 2 +α
p
1
1
. χ(λ)k(x, x1 ) w(x)(hyihy1 i) 2 +α λ− 2 −α (b − λ)α .
Noting that E(0) = 0 we can use Lemma 4.7 to obtain
Z
Z
p
1 ∞ |E 0 (λ)|
1 ∞
0
E
(λ
|(30)| .
dλ
+
1 + πt−1 λ−2 ) − E 0 (λ) dλ.
t 0 (1 + λ2 t)
t t−1/2
Using the bounds above, we estimate the first integral by
p
p
Z
k(x, x1 ) w(x)hyihy1 i ∞
k(x, x1 ) w(x)hyihy1 i
1
√
dλ .
.
t
t5/4
λ(1 + λ2 t)
0
To estimate the second integral, we apply the Lipschitz bound with
b−λ=λ
p
1
1 + πt−1 λ−2 − 1 ∼ ,
tλ
and get
p
p
1
1
Z
k(x, x1 ) w(x)(hyihy1 i) 2 +α λ1 − 1 −α
k(x, x1 ) w(x)(hyihy1 i) 2 +α
−α
λ 2 (tλ) dλ .
,
t
t1+α
t−1/2
since α ∈ (0, 1/4).
Taking into account the contribution of the term with the roles of x and y switched, we
obtain the assertion of the lemma.
WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
15
Next we consider the contribution of the third term in (18) to (19):
Z Z ∞
2
+
eitλ λχ(λ)[R−
(31)
2 − R2 ]v(x1 )[QD0 Q](x1 , y1 )v(y1 )dλdx1 dy1 ,
R4
0
where
± 2
± 2
R±
2 = R0 (λ )(x, x1 )R0 (λ )(y1 , y).
(32)
Recall from Lemma 3.1 that
R0± (λ2 )(x, x1 ) = c[a log(λ|x − x1 |) + b ± i] + E0± (λ)(x, x1 ),
where a, b, c ∈ R. Therefore
2
R±
2 = c (a log(λ|x − x1 |) + b)(a log(λ|y − y1 |) + b) − 1
± ic2 a log(λ|x − x1 |) + a log(λ|y − y1 |) + 2b + E3± (λ),
where
(33) E3± (λ) := c[a log(λ|x − x1 |) + b ± i]E0± (λ)(y, y1 )
+ c[a log(λ|y − y1 |) + b ± i]E0± (λ)(x, x1 ) + E0± (λ)(x, x1 )E0± (λ)(y, y1 ).
Using this, we have
+
−
+
2
R−
2 − R2 = −2c (a log(λ|x − x1 |) + a log(λ|y − y1 |) + 2b) + E3 (λ) − E3 (λ).
Using this in (31), and noting that the contribution of the first summand vanishes since
Qv = 0, we obtain
Z Z
(34)
(31) =
R4
0
∞
2
eitλ λχ(λ)[E3− (λ) − E3+ (λ)]v(x1 )[QD0 Q](x1 , y1 )v(y1 )dλdx1 dy1 .
3
Proposition 4.8. Let 0 < α < 1/4. If v(x) . hxi− 2 −α− , then we have
(31) = O
hxi 21 +α+ hyi 21 +α+ t1+α
.
Proof. Let E(λ) = χ(λ)E3 (λ) (we dropped the ’±’ signs). Using
p
| log |x − x1 || . k(x, x1 ) w(x),
and the bounds in Lemma 3.1 and Corollary 3.2 we estimate (for 0 < λ < b . λ < λ1 )
1
1
|∂λ E(λ)| . χ(λ)λ− 2 − (hyihxihy1 ihx1 i) 2 + k(x, x1 )k(y, y1 ),
1
1
∂λ E(b) − ∂λ E(λ) . χ(λ)k(x, x1 )k(y, y1 )(hxihx1 ihyihy1 i) 2 +α+ λ− 2 −α− (b − λ)α .
16
M. B. ERDOĞAN, W. R. GREEN
Applying Lemma 4.7 together with these bounds as in the proof of the previous lemma, we
bound the λ-integral by
1
1
k(x, x1 )k(y, y1 )(hxihx1 ihyihy1 i) 2 +α+
t1+α
.
Therefore,
(31) . t
−1−α
Z
1
k(x, x1 )k(y, y1 )(hyihy1 ihxihx1 i) 2 +α+ v(x1 )|QD0 Q(x1 , y1 )|v(y1 )dx1 dy1
R4
1
.
1
hxi 2 +α+ hyi 2 +α+
,
t1+α
1
since kv(x1 )k(x, x1 )hx1 i 2 +α+ kL2x . 1.
1
We now turn to the contribution of the error term E ± (λ) from Lemma 3.6 in (18).
Dropping the ’±’ signs, we need to consider
Z Z ∞
2
(35)
eitλ λ E(λ)v(x1 )v(y1 ) dλ dx1 dy1 ,
R4
0
where
E(λ) := χ(λ)R0 (λ2 )(x, x1 )E(λ)(x1 , y1 )R0 (λ2 )(y, y1 ).
3
Proposition 4.9. Let 0 < α < 1/4. If v(x) . hxi− 2 −α− , then we have
(35) = O
hxi 21 +α+ hyi 21 +α+ t1+α
.
Proof. Let
1
1
T0 := sup λ− 2 |E ± (λ)| + sup λ 2 |∂λ E ± (λ)|
0<λ<λ1
0<λ<λ1
1
+
sup
0<λ<b.λ<λ1
λ 2 +α
|∂λ E ± (b) − ∂λ E ± (λ)|.
(b − λ)α
By Lemma 3.6, we see that T0 is Hilbert-Schmidt on L2 (R2 ), and hence we have the
following bounds for the kernels
1
1
|E ± (λ)| . λ 2 T0 , |∂λ E ± (λ)| . λ− 2 T0 ,
1
|∂λ E ± (b) − ∂λ E ± (λ)| . λ− 2 −α (b − λ)α T0 ,
if 0 < λ < b . λ < λ1 .
Moreover, using Lemma 3.1 and Corollary 3.2, we have (for 0 < λ < b . λ < λ1 )
p
|R0 (λ2 )(x, x1 )| . (1 + | log λ|)k(x, x1 ) w(x) . λ0− k(x, x1 )hxi0+ ,
1p
1
|∂λ R0 (λ2 )(x, x1 )| . + λ− 2 hxihx1 i,
λ
WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
17
1
h 1
|x − x1 | 2 +α i
|∂λ R0 (λ )(x, x1 ) − ∂λ R0 (b )(x, x1 )| . (b − λ)
+
.
1
λ1+α
λ2
2
2
α
Therefore we have the bounds (for 0 < λ < b . λ < λ1 )
1
1
|∂λ E(λ)| . λ− 2 − (hyihxihy1 ihx1 i) 2 k(x, x1 )k(y, y1 )T0 (x1 , y1 ),
1
1
|∂λ E(b) − ∂λ E(λ)| . λ− 2 −α− (b − λ)α (hyihxihy1 ihx1 i) 2 +α+ k(x, x1 )k(y, y1 )T0 (x1 , y1 ).
Applying Lemma 4.7 as above yields the claim of the proposition.
Note that Proposition 4.3, Proposition 4.4, Proposition 4.8, and Proposition 4.9 yield
Theorem 1.1.
5. Proof of Theorem 1.2 For Energies Away From Zero
In this section we prove Theorem 1.2 for energies separated from zero:
Theorem 5.1. Under the assumptions of Theorem 1.1, we have for t > 2
Z
(36)
e
sup
L≥1
3
∞
itλ2
0
λe
χ(λ)χ(λ/L)[RV+ (λ2 )
−
RV− (λ2 )](x, y)dλ
.
3
hxi 2 hyi 2
3
t2
where χ
e = 1 − χ.
Proof. We start with the resolvent expansion
(37)
RV± (λ2 )
=
2M
+2
X
R0± (λ2 )(−V R0± (λ2 ))m
m=0
+ R0± (λ2 )(V R0± (λ2 ))M V RV± (λ2 )V (R0± (λ2 )V )M R0± (λ2 ).
(38)
We first note that the contribution of the term m = 0 can be handled as in Proposition 4.3
3
and it can be bounded by
3
hxi 2 hyi 2
t2
. For the case m > 0 we won’t make use of any cancellation
between ‘±’ terms. Thus, we will only consider R0− , and drop the ‘±’ signs. Using (3), (5),
(6), and (7) we write
R0 (λ2 )(x, y) = e−iλ|x−y| ρ+ (λ|x − y|) + ρ− (λ|x − y|),
where ρ+ and ρ− are supported on the sets [1/4, ∞) and [0, 1/2], respectively. Moreover,
we have the bounds
(39)
e + | log y|),
ρ− (y) = O(1
e (1 + |y|)−1/2
ρ+ (y) = O
18
M. B. ERDOĞAN, W. R. GREEN
We first control the contribution of the finite born series, (37), for m > 0. Note that
the contribution of the mth term of (37) to the integral in (36) can be written as a sum of
integrals of the form
(40)
Z
R2m
Z
∞
2
eitλ λe
χ(λ)χ(λ/L)e−iλ
P
j∈J
0
dj
Y
ρ+ (λdj )
Y
m
Y
ρ− (λd` )
`∈J ∗
j∈J
V (xn ) dλ dx1 . . . dxm ,
n=1
where dj = |xj−1 − xj | and J ∪ J ∗ is a partition of {1, ..., m, m + 1}. Let
E(λ) := χ
e(λ)χ(λ/L)e−iλ
P
j∈J
dj
Y
ρ+ (λdj )
Y
ρ− (λd` ).
`∈J ∗
j∈J
To estimate the derivatives of E, we note that
dkj
∂λk ρ+ (λdj ) .
,
(1 + λdj )k+1/2
1
∂λk ρ− (λdj ) . k , k = 1, 2, ...
λ
k = 0, 1, 2, ...,
Using the monotonicity of log− function, we also obtain
χ
e(λ) ρ− (λdj ) . χ
e(λ)(1 + | log(λdj )|)χ{0<λdj ≤1/2} . χ
e(λ)(1 + log− (λdj )) . 1 + log− (dj ).
It is also easy to see that
dk
χ(λ/L) . λ−k .
dλk
Finally, noting that (e
χ)0 is supported on the set {λ ≈ 1}, we can estimate
1 X
Y
1
dk Y
e(λ)
∂λ E . χ
+
dk +
(1 + log− (d` ))
1/2
λ
1 + λdk
(1
+
λd
)
j
j∈J
k∈J
`∈J ∗
1 X
Y
Y
X
1
1
dk
−
−1
2 −2
.χ
e(λ) +
(1+log
(d
))
.
χ
e
(λ)
λ
+
d
λ
(1+log− (d` ))
`
k
1/2
λ
(1
+
λd
)
k
k∈J
`∈J ∗
k∈J
`∈J ∗
(41)
1
.χ
e(λ)λ− 2
m+1
Y
1
hxk i 2
k=0
m+1
Y
(1 + log− (d` )).
`=1
We also have
(42)
1
X
Y
Y
d2k
1
∂λ2 E . χ
e(λ) 2 +
d2k +
(1 + log− (d` ))
1/2
λ
(1 + λdk )2
(1
+
λd
)
j
j∈J
k∈J
`∈J ∗
−2
.χ
e(λ) λ
+
X
k∈J
3
2
dk λ
− 12
Y
`∈J ∗
−
(1 + log (d` )) . χ
e(λ)λ
− 21
m+1
Y
hxk i
k=0
3
2
m+1
Y
(1 + log− (d` )).
`=1
WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
19
Using Lemma 4.7 (and taking the support condition of χ
e into account), we can bound the
λ integral in (40) by
Z
Z
1 ∞ |E 0 (λ)|
1 ∞ 0 p
(43)
dλ +
E (λ 1 + πt−1 λ−2 ) − E 0 (λ) dλ,
t2 0
λ2
t 0
Using (41), we can bound the first integral in (43) by
(44)
m+1
Y
1
hxk i 2
m+1
Y
(1 + log− (d` ))
k=0
∞
Z
χ
e(λ)λ−5/2 dλ .
0
`=1
m+1
Y
1
hxk i 2
m+1
Y
(1 + log− (d` )).
`=1
k=0
To estimate the second integral in (43) first note that
(45)
λ
p
1
1 + πt−1 λ−2 − λ ≈ .
tλ
Next using (45), (41) and (42), we have (for any 0 ≤ α ≤ 1)
(46)
E 0 (λ
p
1 + πt−1 λ−2 ) − E 0 (λ)
− 12
.χ
e(2λ)λ
m+1
Y
hxk i
1
2
m+1
1 Y
hxk i
(1 + log (d` )) min 1,
tλ
m+1
Y
k=0
`=1
−
k=0
1
. t−α χ
e(2λ)λ− 2 −α
m+1
Y
1
hxk i 2 +α
k=0
m+1
Y
(1 + log− (d` )).
`=1
Using this bound for α ∈ (1/2, 1], we bound the second integral in (43) by
(47) t−α
m+1
Y
1
hxk i 2 +α
k=0
m+1
Y
(1 + log− (d` ))
Z
0
`=1
∞
1
χ
e(2λ)λ− 2 −α .
. t−α
m+1
Y
1
hxk i 2 +α
m+1
Y
(1 + log− (d` )).
`=1
k=0
Combining (44) and (47), we obtain
|(43)| . t
−1−α
m+1
Y
hxk i
k=0
Using this (with
|(40)| . t−1−α
1
2
Z
1
+α
2
m+1
Y
(1 + log− (d` ))
`=1
< α < 2β − 52 ) in (40), we obtain
m+1
Y
R2m k=0
1
hxk i 2 +α
m+1
Y
m
Y
`=1
n=1
(1 + log− (d` ))
|V (xn )| dx1 . . . dxm
1
1
.
hx0 i 2 +α hxm+1 i 2 +α
3
t2
.
20
M. B. ERDOĞAN, W. R. GREEN
To control the remainder of the born series, (38), we employ the limiting absorption
principle, see [2],
k∂λk RV± (λ2 )kL2,σ (R2 )→L2,−σ (R2 ) < ∞,
(48)
for k = 0, 1, 2 with σ > k + 21 . Similar bounds hold for the derivatives of the free resolvent.
In addition, for the free resolvent one has
kR0± (λ2 )kL2,σ (R2 )→L2,−σ (R2 ) . λ−1+ ,
(49)
which is valid for σ > 21 . Using the representation (39), we note the following bounds on
the free resolvent which are valid on λ > λ1 > 0,
(
| log(λ|x − y|)| 0 < λ|x − y| <
±
|∂λk R0 (λ2 )(x, y)| . |x − y|k
1
(λ|x − y|)− 2
λ|x − y| & 1
Thus, for σ >
(50)
1
2
1
2
1
1
. λ− 2 |x − y|k− 2 .
+ k,
1
k∂λk R0± (λ2 )(x, y)hyi−σ kL2y . λ− 2
hZ
R2
1
|x − y|2k−1 i 12
dy . λ− 2 hximax(0,k−1/2) .
hyi2σ
Once again, we estimate the RV+ and RV− terms separately and omit the ‘±’ signs.
We write the contribution of (38) to (36) as
Z ∞
2
(51)
eitλ λ E(λ)(x, y) dλ,
0
where
E(λ)(x, y) = χ
e(λ)χ(λ/L) V RV± (λ2 )V (R0± (λ2 )V )M R0± (λ2 )(·, x), (R0± (λ2 )V )M R0± (λ2 )(·, y) .
Using (48), (49), and (50) (provided that M ≥ 2) we see that
(52)
3
3
∂λk E(λ)(x, y) . χ
e(λ)χ(λ/L)hλi−2− hxi 2 hyi 2 ,
k = 0, 1, 2.
This requires that |V (x)| . hxi−3− . One can see that the requirement on the decay rate of
the potential arises when, for instance, both λ derivatives act on one resolvent, this twice
5
5
5
differentiated resolvent operator maps L2, 2 + → L2,− 2 − by (48), or is in L2,− 2 − by (50).
5
1
The potential then needs to map L2,− 2 − → L2, 2 + for the next application of the limiting
absorption principle. This is satisfied if |V (x)| . hxi−3− .
The required bound now follows by integrating by parts twice:
Z ∞
3
3
∂λ E(λ)(x, y)
−2
|(51)| . |t|
∂λ
dλ . |t|−2 hxi 2 hyi 2 .
λ
0
WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
21
Acknowledgment.
The authors would like to thank Wilhelm Schlag for suggesting this problem. The first
author was partially supported by National Science Foundation grant DMS-0900865.
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WEIGHTED DISPERSIVE ESTIMATE FOR THE SCHRÖDINGER EQUATION
23
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Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.
E-mail address: berdogan@math.uiuc.edu
Department
of
Mathematics
Charleston, IL 61920, U.S.A.
E-mail address: wrgreen2@eiu.edu
and
Computer
Science,
Eastern
Illinois
University,