ON Lp-ESTIMATES FOR THE TIME DEPENDENT
SCHRÖDINGER OPERATOR ON L2
Lp -Estimates for the
Schrödinger Operator
MOHAMMED HICHEM MORTAD
Département de Mathématiques
Université d’Oran (Es-senia)
B.P. 1524, El Menouar, Oran. Algeria.
EMail: mortad@univ-oran.dz
Received:
07 July, 2007
Accepted:
29 August, 2007
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
Primary 35B45; Secondary 35L10.
Key words:
Schrödinger Equation, Strichartz Estimates and Self-adjointness.
Abstract:
Let L denote the time-dependent Schrödinger operator in n space variables. We
consider a variety of Lebesgue norms for functions u on Rn+1 , and prove or
disprove estimates for such norms of u in terms of the L2 norms of u and Lu.
The results have implications for self-adjointness of operators of the form L + V
where V is a multiplication operator. The proofs are based mainly on Strichartztype inequalities.
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
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Acknowledgements:
I would like to express all my gratitude to my ex-Ph.D.-supervisor Professor
Alexander M. Davie for his helpful comments, especially in the counterexamples
section.
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Contents
1
Introduction
3
2
2
L∞
t (Lx ) Estimates.
6
3
Lqt (Lrx ) Estimates.
10
4
Counterexamples
14
5
Question
16
Lp -Estimates for the
Schrödinger Operator
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
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1.
Introduction
Let (x, t) ∈ Rn+1 where n ≥ 1. The Schrödinger equation ∂u
= i4x u has been
∂t
much studied using spectral properties of the self-adjoint operator 4x . When a
multiplication operator (potential) V is added, it becomes important to determine
whether 4x + V is a self-adjoint operator, and there is a vast literature on this question (see e.g. [9]).
∂
One can also, however, regard the operator L = −i ∂t
− 4x as a self-adjoint
operator on L2 (Rn+1 ), and that is the point of view taken in this paper. We ask what
can be said about the domain of L, more specifically, we ask which Lq spaces, and
more generally mixed Lqt (Lrx ) space, a function u must belong to, given that u is in
the domain of L (i.e. u and Lu both belong to L2 (Rn+1 )). We answer this question
and, using the Kato-Rellich theorem, deduce sufficient conditions on V for L + V to
be self-adjoint.
Our approach is based on the fact that any sufficiently well-behaved function u
on Rn+1 can be regarded as a solution of the initial value problem (IVP)
(
−iut − 4x u = g(x, t),
(1.1)
u(x, α) = f (x)
Lp -Estimates for the
Schrödinger Operator
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
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where α ∈ R, f (x) = u(x, α) and g = Lu.
To apply this, we will use estimates for u based on given bounds for f and g.
A number of such estimates are known and generally called Strichartz inequalities,
after [12] which obtained such an Lq bound for u. This has since been generalized to
give inequalities for mixed norms [13, 4]. The specific inequalities we use concern
the case g = 0 of (1.1) and give bounds for u in terms of kf kL2 (Rn ) - see (3.2) below.
The precise range of mixed Lqt (Lrx ) norms for which the bound (3.2) holds is known
as a result of [13, 4] and the counterexample in [6].
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In Section 2 we prove a special case of our main theorem, namely a bound for u
2
in L∞
t (Lx ), which does not require Strichartz estimates, only elementary arguments
using the Fourier transform. The main theorem, giving Lqt (Lrx ) bounds for the largest
possible set of (q, r) pairs, is proved in Section 3. In fact, we prove a somewhat
stronger bound, in a smaller space L2,q,r defined below. The fact that the set of pairs
(q, r) covered by Theorem 3.1 is the largest possible is shown in Section 4.
Some results on a similar question for the wave operator can be found in [7]. For
Strichartz-type inequalities for the wave operator, see e.g. [11, 12, 2, 3, 4].
We assume notions and definitions about the Fourier Transform and unbounded
operators and for a reference one may consult [8], [5] or [10]. We also use on several
occasions the well-known Duhamel principle for the Schrödinger equation (see e.g.
[1]).
Notation. The symbol û stands for the Fourier transform of u in the space (x)
variable while the inverse Fourier transform will be denoted either by F −1 u or ǔ.
We denote by C0∞ (Rn+1 ) the space of infinitely differentiable functions with compact support.
We denote by R+ the set of all positive real numbers together with +∞.
For 1 ≤ p ≤ ∞, k · kp is the usual Lp -norm whereas k · kLpt (Lqx ) stands for the
mixed spacetime Lebesgue norm defined as follows
1q
q
ku(t)kLrx dt .
Z
kukLqt (Lrx ) =
kukLqt,k (Lrx ) =
k
vol. 8, iss. 3, art. 80, 2007
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We also define some modified mixed norms. First we define, for any integer k,
k+1
Mohammed Hichem Mortad
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R
Z
Lp -Estimates for the
Schrödinger Operator
1q
ku(t)kqLrx dt
,
and then
! p1
kukLp,q,r =
X
kukpLq
r
t,k (Lx )
.
k∈Z
We note that kukLp,q1 ,r ≥ kukLp,q2 ,r if q1 ≥ q2 , and that kukLqt (Lrx ) ≤ kukLp,q,r if
q ≥ p.
Finally we define
MLn = {f ∈ L2 (Rn+1 ) : Lf ∈ L2 (Rn+1 )},
Lp -Estimates for the
Schrödinger Operator
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
where L is defined as in the abstract and where the derivative is taken in the distributional sense. We note that MLn = D(L), the domain of L, and also that C0∞ (Rn+1 )
is dense in MLn in the graph norm kukL2 (Rn+1 ) + kLukL2 (Rn+1 ) .
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2.
2
L∞
t (Lx ) Estimates.
Before stating the first result, we are going to prepare the ground for it. Take the
Fourier transform of the IVP (1.1) in the space variable to get
(
−iût + η 2 û = ĝ(η, t),
Lp -Estimates for the
Schrödinger Operator
û(η, α) = fˆ(η)
which has the following solution (valid for all t ∈ R):
Z t
2
−iη 2 t
ˆ
(2.1)
û(η, t) = f (η)e
+i
e−iη (t−s) ĝ(η, s)ds,
vol. 8, iss. 3, art. 80, 2007
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α
where η ∈ Rn .
Duhamel’s principle gives an alternative way of writing the part of the solution
depending on g. Taking the case f = 0, the solution of (1.1) can be written as
Z t
(2.2)
u(x, t) = i
us (x, t)ds,
α
where us is the solution of
(
Mohammed Hichem Mortad
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Lus = 0,
t > s,
us (x, s) = g(x, s).
Now we state a result which we can prove using (2.1). In the next section we
prove a more general result using Strichartz inequalities and Duhamel’s principle
(2.2).
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Proposition 2.1. For all a > 0, there exists b > 0 such that
kukL2,∞,2 ≤ akLuk2L2 (Rn+1 ) + bkuk2L2 (Rn+1 )
for all u ∈ MLn .
Proof. We prove the result for u ∈ C0∞ (Rn+1 ) and a density argument allows us to
deduce it for u ∈ MLn .
We use the fact that any such u is, for any α ∈ R, the unique solution of (1.1),
where f (x) = u(x, α) and g = Lu, and therefore satisfies (2.1).
Let k ∈ Z and let t and α be such that k ≤ t ≤ k + 1 and k ≤ α ≤ k + 1.
Squaring (2.1), integrating with respect to η in Rn , and using Cauchy-Schwarz (and
the fact that |t − α| ≤ 1), we obtain
Z
Z Z t
2
2
(2.3)
kû(·, t)kL2 (Rn ) ≤ 2
|û(η, α)| dη + 2
|ĝ(η, s)|2 dsdη.
Rn
Rn
Rn
k=−∞
vol. 8, iss. 3, art. 80, 2007
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k
|ĝ(η, s)| dηds.
Rn
ess sup ku(·, t)k2L2 (Rn ) ≤ 2kLuk2L2 (Rn+1 ) + 2kuk2L2 (Rn+1 ) .
k≤t≤k+1
Finally to get an arbitrarily small constant in the Lu term we use a scaling argument: let m be a positive integer and let v(x, t) = u(mx, m2 t). Then we find
kvkL2 (Rn+1 ) = m−1−n/2 kukL2 (Rn+1 )
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Now take the essential supremum of both sides in t over [k, k + 1], then sum in k
over Z to get (recalling that g = Lu)
∞
X
Mohammed Hichem Mortad
α
Now integrating against α in [k, k + 1] allows us to say that
Z k+1 Z
Z k+1 Z
2
2
ku(·, t)kL2 (Rn ) ≤ 2
|û(η, α)| dηdα + 2
k
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and
kLvkL2 (Rn+1 ) = m1−n/2 kLukL2 (Rn+1 ) .
Also,
kv(·, t)kL2 (Rn ) = m−n/2 ku(·, m2 t)kL2 (Rn )
and so
sup kv(·, t)k2L2 (Rn ) = m−n
k≤t≤k+1
sup
m2 k≤t≤m2 (k+1)
ku(·, t)k2L2 (Rn )
Mohammed Hichem Mortad
m2 (k+1)−1
≤ m−n
Lp -Estimates for the
Schrödinger Operator
vol. 8, iss. 3, art. 80, 2007
X
sup ku(·, t)k2L2 (Rn ) .
j=m2 k
j≤t≤j+1
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Summing over k gives
kvk2L2,∞,2
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≤m
−n
kuk2L2,∞,2
≤m
−n
≤ 2m
−2
2kLuk2L2 (Rn+1 )
kLvk2L2 (Rn+1 )
+
2kuk2L2 (Rn+1 )
+ 2m
2
kvk2L2 (Rn+1 )
and choosing m so that 2m−2 < a completes the proof.
Now we recall the Kato-Rellich theorem which states that if L is a self-adjoint
operator on a Hilbert space and V is a symmetric operator defined on D(L), and if
there are positive constants a < 1 and b such that kV uk ≤ akLuk + bkuk for all
u ∈ D(L), then L + V is self-adjoint on D(L) (see [9]).
Corollary 2.2. Let V be a real-valued function in L∞,2,∞ . Then L+V is self-adjoint
on D(L) = MLn .
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Proof. One can easily check that
kV ukL2 (Rn+1 ) ≤ kV kL∞,2,∞ kukL2,∞,2 .
Choose a < kV k−1
L∞,2,∞ and then Proposition 2.1 shows that L + V satisfies the
hypothesis of the Kato-Rellich theorem.
In particular, it follows that L + V is self-adjoint whenever V ∈ L2t (L∞
x ).
Lp -Estimates for the
Schrödinger Operator
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
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3.
Lqt (Lrx ) Estimates.
Now we come to the main theorem in this paper, which depends on the following
Strichartz-type inequality. Suppose n ≥ 1 and q and r are positive real numbers
(possibly infinite) such that q ≥ 2 and
2 n
n
(3.1)
+ = .
q
r
2
When n = 2 we exclude the case q = 2, r = ∞. Then there is a constant C such
that if f ∈ L2 (Rn ) and g = 0, the solution u of (1.1) satisfies
(3.2)
kukLqt (Lrx ) ≤ Ckf kL2 (Rn ) .
This result can be found in [13] for q > 2; the more difficult ‘end-point’ case
where q = 2, n ≥ 3 is treated in [4]. That (3.2) fails in the exceptional case n =
2, q = 2, r = ∞ is shown in [6].
For n ≥ 1 we define a region Ωn ∈ R+ × R+ as follows: for n 6= 2,
2 n
n
+
+
(3.3)
Ωn = (q, r) ∈ R × R : + ≥ , q ≥ 2, r ≥ 2
q
r
2
and for n = 2, Ω2 is defined by the same expression, with the omission of the point
(2, ∞).
The sets Ωn are probably most easily visualized in the ( 1q , 1r )-plane. Then Ω1
is a quadrilateral with vertices ( 41 , 0), ( 12 , 0), (0, 12 ), ( 12 , 12 ) and for n ≥ 2, Ωn is a
triangle with vertices ( 12 , n−2
), (0, 12 ), ( 12 , 12 ), the point ( 12 , 0) being excluded in the
2n
case n = 2.
Theorem 3.1. Let n ≥ 1, and let (q, r) ∈ Ωn . Then for all a > 0, there exists b > 0
such that
(3.4)
kukL2,q,r ≤ akLukL2 (Rn+1 ) + bkukL2 (Rn+1 )
n
for all u ∈ ML .
Lp -Estimates for the
Schrödinger Operator
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
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Proof. By the inclusion L2,q1 ,r ⊆ L2,q2 ,r , when q1 ≥ q2 it suffices to treat the case
where 2q + nr = n2 , for which (3.2) holds.
Let k ∈ Z and let α ∈ [k, k + 1]. As in the proof of Proposition 2.1 we use the
fact that u is the solution of (1.1) with f = u(·, α) and g = Lu. Now we split u into
two parts u = u1 + u2 , where u1 , u2 are the solutions of
(
(
Lu1 = g,
Lu2 = 0,
u1 (x, α) = 0,
u2 (x, α) = f.
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
The estimate for u2 is deduced from (3.2):
(3.5)
Lp -Estimates for the
Schrödinger Operator
ku2 kLqt (Lrx ) ≤ Ckf kL2 (Rn ) ≤ Cku(·, α)kL2 (Rn ) .
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For u1 we apply (2.2) to obtain
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Z
t
u1 (x, t) = i
(3.6)
us (x, t)ds,
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from which we deduce
Z
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k+1
kus (·, t)kLr (Rn ) ds
ku1 (·, t)kLr (Rn ) ≤
k
for t ∈ [k, k + 1], and hence
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Z
ku1 k
Lqt,k (Lrx )
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k+1
≤
kus kLqt (Lrx ) ds
k
Z
≤C
k+1
kg(·, s)kL2 (Rn ) ds
k
≤ CkgkL2 (Rn ×[k,k+1]) .
Close
Combining this with (3.5) we have
kuk2Lq
r
t,k (Lx )
≤ 2C 2 ku(·, α)k2L2 (Rn ) + 2C 2 kLuk2L2 (Rn ×[k,k+1]) .
Integrating w.r.t. α from k to k + 1 gives
kuk2Lq
r
t,k (Lx )
≤ 2C 2 kuk2L2 (Rn ×[k,k+1]) + 2C 2 kLuk2L2 (Rn ×[k,k+1]) .
Lp -Estimates for the
Schrödinger Operator
Summing over k, we obtain
kuk2L2,q,r ≤ 2C 2 kukL2 (Rn+1 ) + 2C 2 kLukL2 (Rn+1 ) ,
and the proof is completed by a similar scaling argument to that used in Proposition
2.1.
Using the inclusion L2,q,r ⊆
Lqt (Lrx )
for all u ∈
vol. 8, iss. 3, art. 80, 2007
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for q ≥ 2 we deduce
Corollary 3.2. Let n ≥ 1, and let (q, r) ∈ Ωn . Then for all a > 0, there exists b > 0
such that
kukLqt (Lrx ) ≤ akLukL2 (Rn+1 ) + bkukL2 (Rn+1 )
(3.7)
Mohammed Hichem Mortad
MLn .
In particular, we get such a bound for kukLq (Rn+1 ) whenever 2 ≤ q ≤ (2n + 4)/n.
By applying the Kato-Rellich theorem we can deduce a generalization of Corollary 2.2 from Theorem 3.1. We first define
2 n
∗
+
+
(3.8)
Ωn = (p, s) ∈ R × R : + ≤ 1, p ≥ 2, s ≥ 2
p s
for n 6= 2, and for n = 2, Ω2 is defined by the same expression, with the omission of
the point (2, ∞).
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Corollary 3.3. Let n ≥ 1 and let (p, s) ∈ Ω∗n . Let V be a real-valued function
belonging to L∞,p,s . Then L + V is self-adjoint on MLn .
2p
2s
Proof. Let q = p−2
and r = s−2
. Then (q, r) ∈ Ωn and the conclusion (3.4) of
Theorem 3.1 applies. Now we have
Z k+1
Z k+1
2
kV u(·, t)kL2 (Rn ) ≤
ku(·, t)k2Lr (Rn ) kV (·, t)k2Ls (Rn )
k
k
≤ kuk2Lq
r
t,k (Lx )
kV k2Lp
s
t,k (Lx )
Lp -Estimates for the
Schrödinger Operator
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
and summation over k gives
kV ukL2 (Rn+1 ) ≤ kukL2,q,r kV kL∞,p,s .
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Then, using (3.4), the result follows in the same way as Corollary 2.2.
It follows from Corollary 3.3 that L + V is self-adjoint whenever V ∈ Lpt (Lsx )
for (p, s) ∈ Ω∗n . Taking the case s = p, we find that L + V is self-adjoint if V ∈
Lp (Rn+1 ) for some p ≥ n + 2.
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4.
Counterexamples
Now we show that Theorem 3.1 is sharp, as far as the allowed set of q, r is concerned.
Proposition 4.1. Let n ≥ 1 and let q and r be positive real numbers, possibly infinite,
such that (q, r) ∈
/ Ωn . Then there are no constants a and b such that (3.7) holds for
n
all u ∈ ML .
Proof. For (q, r) to fail to be in Ωn one of the following three possibilities must
occur: (i) q < 2 or r < 2; (ii) 2q + nr < n2 ; (iii) n = 2, q = 2 and r = ∞. We consider
these cases in turn.
(i) If q < 2, choose a sequence (βk )k∈Z which is in l2 but not in lq . Let φ(x, t) be a
n+1
smooth function
which vanishes for t outside [0, 1], and
P of compact support on R
let u(x, t) = k∈Z βk φ(x, t − k). Then u ∈ MLn , but u ∈
/ Lqt (Lrx ) for any r.
The case r < 2 can be treated similarly. We chose a sequence βk which is in l2
r
but
set u(x, t) =
P not l , and a smooth φ which vanishes for x1 outside [0, 1], then
n
β
φ(x
−
ke
,
t),
where
e
is
the
unit
vector
(1,
0,
.
.
.
,
0)
in
R
.
Then
u ∈ MLn ,
1
1
k∈Z k
but u ∈
/ Lqt (Lrx ) for any q.
(ii) In this case we use the scaling argument which shows that the Strichartz estimates
fail, together with a cutoff to ensure u and Lu are in L2 .
We start with a non-zero f ∈ L2 (Rn ), and let u be the solution of (1.1) with
2
α = 0 and g = 0. (An explicit example would be f (x) = e−|x| and then u(x, t) =
2
(1 + 4it)−n/2 e−|x| /(1+4it) ). Choose a smooth function φ on R such that φ(0) 6= 0
and such that φ and φ0 are in L2 . Then for λ > 0 define
vλ (x, t) = λn/2 u(λx, λ2 t)φ(t).
Then (using Lu = 0) we find Lv(x, t) = −iλn/2 u(λx, λ2 t)φ0 (t). We calculate
Lp -Estimates for the
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Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
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kvλ kL2 (Rn+1 ) = kf kL2 (Rn ) kφkL2 and kLvλ kL2 (Rn+1 ) = kf kL2 (Rn ) kφ0 kL2 . Also
kvλ kLqt (Lrx ) = λ
β
1q
Z
ku(·, t)kqLr (Rn ) |φ(λ−2 t)|q dt
,
R
where β = n2 − nr − 2q > 0. So λ−β kvλ kLqt (Lrx ) → |φ(0)|kukLqt (Lrx ) (note that the norm
on the right may be infinite) and hence kvλ kLqt (Lrx ) tends to ∞ as λ → ∞, completing
the proof.
Lp -Estimates for the
Schrödinger Operator
Mohammed Hichem Mortad
(iii) This exceptional case we treat in a similar fashion to (ii), but we need the result
from [6], that the Strichartz inequality fails in this case. We start by fixing a smooth
function φ on R such that φ = 1 on [−1, 1] and φ and φ0 are in L2 .
Now let M > 0 be given and we use [6] to find f ∈ L2 (R2 ) with kf kL2 (R2 ) = 1
such that the solution u of (1.1) with α = 0 and g = 0 satisfies kukL2t (L∞
> M.
x )
RR
2
2
1/2
Then we can find R > 0 so that −R ku(·, t)kL∞ (R2 ) dt > M . Let λ = R and
define v(x, t) = λn/2 u(λx, λ2 t)φ(t). Then kvkL2 (R3 ) = kφkL2 , kLvkL2 (R3 ) = kφ0 kL2
and
Z 1
2
kvkL2t (L∞
≥
kv(·, t)k2L∞ (R2 ) dt > M 2 ,
x )
vol. 8, iss. 3, art. 80, 2007
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−1
which completes the proof, since M is arbitrary.
We remark that [6] also gives an example of f ∈ L2 (R2 ) such that u ∈
/ L2t (BM Ox )
and the argument of part (iii) can then be applied to show that no inequality
kukL2t (BM Ox ) ≤ akLukL2 (R3 ) + bkukL2 (R3 )
can hold.
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5.
Question
We saw as a result of Corollary 3.3 that if (p, s) ∈ Ω∗ , then L + V is self-adjoint
on MLn whenever V ∈ Lpt (Lsx ). One can ask whether this can be extended to a
larger range of (p, s) with p, s ≥ 2. If one asks whether L + V is defined on MLn ,
then we would require a bound kV ukL2 (Rn+1 ) ≤ akLukL2 (Rn+1 ) +bkuk to hold for all
u ∈ MLn . If such a bound is to hold for all V ∈ Lpt (Lsx ), then, in fact, we require (3.7)
2p
2s
to hold for q = p−2
and r = s−2
, which we know cannot hold unless (p, s) ∈ Ω∗ .
One can instead ask for L + V , defined on say C0∞ (Rn+1 ), to be essentially selfadjoint. This is equivalent to saying that the only (distribution) solution in L2 (Rn+1 )
of the PDE
−iut − 4x u + V u = ±iu
is u = 0 (see e.g. [8]).
We do not know if there are any values of (p, s) not in Ω∗n such that this holds for
all V ∈ Lpt (Lsx ). The analogous question for the Laplacian is extensively discussed
in [9].
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Schrödinger Operator
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
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References
[1] L.C. EVANS, Partial Differential Equations, American Mathematical Society
(G.S.M. Vol. 19), 1998.
[2] J. GINIBRE AND G. VELO, Generalized Strichartz inequalities for the wave
equation, J. of Functional Analysis, 133 (1995), 50–68.
[3] J. HARMSE, On Lebesgue space estimates for the wave equation, Indiana
Univ. Math. J., 39(1) (1990) 229–248.
Lp -Estimates for the
Schrödinger Operator
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
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Lp -Estimates for the
Schrödinger Operator
Mohammed Hichem Mortad
vol. 8, iss. 3, art. 80, 2007
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