Dispersion Estimates for Spherical Schr\"odinger Equations: The Effect of Boundary Conditions

0.22 MB·

by  Markus Holzleitner

DISPERSION ESTIMATES FOR SPHERICAL SCHRO¨DINGER
EQUATIONS: THE EFFECT OF BOUNDARY CONDITIONS
6
1 MARKUSHOLZLEITNER,ALEKSEYKOSTENKO,ANDGERALDTESCHL
0
2 Dedicated with great pleasure toPetru A. Cojuhari on the occasion of his 65th birthday

t
c
O Abstract. We investigate the dependence of the L1 → L∞ dispersive esti-
mates for one-dimensional radial Schro¨dinger operators on boundary condi-
0 tionsat0.Incontrasttothecaseofadditiveperturbations,weshowthatthe
3 changeofaboundaryconditionatzeroresultsinthechangeofthedispersive
decay estimates if the angular momentum is positive, l ∈ (0,1/2). However,
] fornonpositiveangularmomenta,l∈(−1/2,0],thestandardO(|t|−1/2)decay
P
remainstrueforallself-adjointrealizations.
S
.
h
t
a
m 1. Introduction
[ We are concerned with the one-dimensional Schr¨odinger equation

2 d2 l(l+1)
v iψ˙(t,x)=Hαψ(t,x), Hα :=−dx2 + x2 , (t,x)∈R×R+, (1.1)
8
3 with the angular momentum l < 1 and self-adjoint boundary conditions at x=0
6 parameterized by a paramete|r|α 2[0,π) (the definition is given in Section 2, see
1 ∈
(2.1)–(2.2) — for recent discussion of this family of operators see [1, 4]). More
0
precisely,we areinterestedin the dependence ofthe L1 L∞ dispersiveestimates
.
1 associated to the evolution group e−itHα on the param→eters α [0,π) and l
0 ∈ ∈
( 1/2,1/2).
6 −
1 On the whole line such results have a long tradition and we refer to Weder [22],
: Goldberg and Schlag [9], Egorova, Kopylova, Marchenko and Teschl [5], as well as
v
the reviews [10, 18]. On the half line, the case l =0 with a Dirichlet boundary con-
i
X ditionwastreatedbyWeder[23].Thecaseofgenerall andtheFriedrichsboundary
r condition at 0 (α=0 in our notation)
a
1 1
limxl((l+1)f(x) xf′(x))=0, l , , (1.2)
x→0 − ∈ − 2 2
(cid:16) (cid:17)
wasrecentlyconsideredinKovaˇr´ıkandTruc[14]andtheyproved(seeTheorem2.4
in [14]) that
ke−itH0kL1(R+)→L∞(R+) =O(|t|−1/2), t→∞. (1.3)
It was proved in [13] that this estimate remains true under additive perturbations.
More precisely (see [13, Theorem 1.1]), let H = H +q, where the potential q is a
0
2010 Mathematics Subject Classification. Primary35Q41,34L25;Secondary81U30,81Q15.
Key words and phrases. Schro¨dingerequation, dispersiveestimates,scattering.
Research supported by the Austrian Science Fund (FWF) under Grants No. P26060 and
W1245.
OpusculaMath.36,no.6,769–786(2016).
1
2 M.HOLZLEITNER,A.KOSTENKO,ANDG.TESCHL
real integrable on R function. If in addition
+
1 ∞
q(x)dx< and xmax(2,l+1) q(x)dx< , (1.4)
| | ∞ | | ∞
Z0 Z1
and there is neither a resonance nor an eigenvalue at 0, then
e−itHP (H) = (t−1/2), t . (1.5)
c L1(R+)→L∞(R+) O | | →∞
Here P (H) is t(cid:13)he orthogona(cid:13)l projection in L2(R ) onto the continuous spectrum
c (cid:13) (cid:13) +
of H.
The main result of the present paper shows that the decay estimates (1.3) and
(1.5)arenolongertrueforα (0,π)ifl (0,1/2).Inotherwords,thismeansthat
∈ ∈
singular rank one perturbations destroy these decay estimates if l (0,1/2) (since
∈
thechangeofaboundaryconditioncanbeconsideredasarankoneperturbationin
the resolvent sense). Namely, consider first the operator H , which is associated
π/2
with the following boundary condition at x=0:
1 1
limx−l−1(lf(x)+xf′(x))=0, l , . (1.6)
x→0 ∈ − 2 2
(cid:16) (cid:17)
Theorem 1.1. Let l <1/2. Then
| |
ke−itHπ/2kL1(R+)→L∞(R+) =O(|t|−1/2), t→∞, (1.7)
for all l ( 1/2,0], and
∈ −
ke−itHπ/2kL1(R+,max(x−l,1))→L∞(R+,min(xl,1)) =O(|t|−1/2+l), t→∞, (1.8)
whenever l (0,1/2). The last estimate is sharp.
∈
In the remaining case α (0,π/2) (π/2,π), the decay estimate is given by the
∈ ∪
the next theorem.
Theorem 1.2. Let l <1/2 and α (0,π/2) (π/2,π). Then
| | ∈ ∪
ke−itHαPc(Hα)kL1(R+)→L∞(R+) =O(|t|−1/2), t→∞, (1.9)
for all l ( 1/2,0], and
∈ −
ke−itHαPc(Hα)kL1(R+,max(x−l,1))→L∞(R+,min(xl,1)) =O(|t|−1/2), t→∞, (1.10)
whenever l (0,1/2).
∈
Notice that in the case l (0,1/2) we need to consider weighted L1 and L∞
∈
spaces since functions contained in the domain of H might be unbounded near 0.
α
Finally, let us briefly outline the content of the paper. In the next section we
define the operator H and collect its basic spectral properties. Section 3 contains
α
the proof of Theorem 1.1. In particular, we compute explicitly the kernel of the
evolution group e−itHπ/2 and this enables us to prove (1.7) and (1.8) by using
the estimates for Bessel functions J (all necessary facts on Bessel functions are
ν
contained in Appendix A). Theorem 1.2 is proved in Section 4. Its proof is based
onthe useofaversionofthe vanderCorputlemma,whichis giveninAppendix B.
Also Appendix B contains necessary facts about the Wiener algebras (R) and
0
(R). In the final section we formulate some sufficient conditions forWa function
W
f(H) of a 1-D Schr¨odinger operator H to be an integral operator.
DISPERSION ESTIMATES: THE EFFECT OF BOUNDARY CONDITIONS 3
2. Self-adjoint realizations and their spectral properties
Let l ( 1/2,1/2)and denote by H the maximal operator associated with
max
∈ −
d2 l(l+1)
τ = +
−dx2 x2
in L2(R ). Note that τ is limit point at infinity and limit circle at x = 0 since
+
l <1/2.Therefore,self-adjointrestrictionsofH (orinotherwords,self-adjoint
max
|re|alizations of τ in L2(R )) form a 1-parameter family. More precisely (see, e.g.,
+
[7] and also [1]), the following limits
1
Γ f := limW (f,xl+1), Γ f := − limW (f,x−l) (2.1)
0 x 1 x
x→0 2l+1x→0
existandare finite for all f dom(H ). Self-adjointrestrictions H ofH are
max α max
∈
parameterized by the following boundary conditions at x=0:
dom(H )= f dom(H ): sin(α)Γ f =cos(α)Γ f , α [0,π). (2.2)
α max 1 0
{ ∈ } ∈
Note that the case α=0 corresponds to the Friedrichs extension of H =H∗ .
min max
Let φ(z,x) and θ(z,x) be the fundamental system of solutions of τu=zu given
by
πx
φ(z,x)=Cl−1 2 z−2l4+1Jl+12(√zx),
r (2.3)
πx z2l4+1
θ(z,x)=C J (√zx),
l 2 sin((l+1)π) −l−12
r 2
where J is the Bessel function of order ν (see Appendix A) and
ν
√π
C = . (2.4)
l Γ(l+ 3)2l+1
2
The Weyl solution normalized by Γ ψ =1 is given by
0
πx
ψ(z,x)=θ(z,x)+m(z)φ(z,x)=Cliz2l4+1 2 Hl(+1)1/2(√zx)∈L2(0,∞), (2.5)
r
where H(1) is the Hankel function of the first kind [17, Chapter X.2], and
ν
( z)l+1/2
m(z)= C2 − , z C R , (2.6)
− l sin((l+ 1)π) ∈ \ +
2
is the Weyl function associated with H . Here the branch cut of the root is taken
0
along the negative real axis. Notice that
C2
dρ(λ)= πl 1[0,∞)(λ)λl+12dλ (2.7)
is the corresponding spectral measure. It follows from (A.1) that
x−l
φ(z,x)=xl+1(1+o(1)), θ(z,x)= (1+o(1)),
2l+1
as x 0 and, moreover,
→
Γ θ =Γ φ=1, Γ θ =Γ φ=0.
0 1 1 0
4 M.HOLZLEITNER,A.KOSTENKO,ANDG.TESCHL
Set
φ (z,x):=cos(α)φ(z,x)+sin(α)θ(z,x),
α
(2.8)
θ (z,x):=cos(α)θ(z,x) sin(α)φ(z,x),
α
−
for all z C. Therefore, W(θ ,φ )=1 and
α α
∈
m(z)cos(α)+sin(α)
ψ (z,x):=θ (z,x)+m (z)φ (z,x), m (z)= , (2.9)
α α α α α
cos(α) m(z)sin(α)
−
is a Weyl solution normalized by W(ψ ,φ )=1. Hence
α α
φ (z,x)ψ (z,y), x y,
α α
G (z;x,y)= ≤ (2.10)
α
(φα(z,x)ψα(z,y), x y,
≥
is the Green’s function of H . The absolutely continuous spectrum remains un-
α
changed, σ (H )=[0, ), but there is one additional eigenvalue
ac α
∞
2
cot(α)cos(lπ) 2l+1
E = (2.11)
α − C2
(cid:18) l (cid:19)
if π <α<π. Finally, since
2
Imm(z)
Imm (z)= , (2.12)
α cos(α) m(z)sin(α)2
| − |
we get the absolutely continuous partof the correspondingspectralmeasure of the
operator H :
α
1
ρ′ (λ)dλ= Imm (λ+i0)dλ
α π α
(2.13)
1 C2λl+1/2 (λ)
= l 1[0,∞) dλ.
π(cos(α) C2sin(α)tan(πl)λl+1/2)2+C4sin2(α)λ2l+1
− l l
3. Proof of Theorem 1.1
Similar to the case α = 0 (see [14]), the kernel of the evolution group e−itHπ/2
can be computed explicitly.
Lemma 3.1. Let l < 1/2. Then the evolution group e−itHπ/2 is an integral oper-
| |
ator for all t=0 and its kernel is given by
6
[e−itHπ/2](x,y)= il−21t/2eix24+ty2√xyJ−l−1/2 x2yt , (3.1)
(cid:16) (cid:17)
for all x, y >0 and t=0.
6
Proof. First, notice that
φ (z,x)=θ(z,x), m (z)= 1/m(z),
π/2 π/2
−
and then define the spectral transformation U: L2(R ) L2(R ;ρ ) by
+ + π/2
→
U: f fˆ, fˆ(λ):= θ(λ,x)f(x)dx,
7→ R
Z +
for every f L2(R ). Notice that U extends to an isometry on L2(R ) and its
inverse U−1:∈L2(cR +;ρ ) L2(R ) is given by +
+ π/2 +
→
U−1: g gˇ, gˇ(x):= θ(λ,x)g(λ)dρ (λ),
π/2
7→ R
Z +
DISPERSION ESTIMATES: THE EFFECT OF BOUNDARY CONDITIONS 5
for all g L2(R ;ρ ). Therefore, we get by using (2.3) and (2.13)
∈ c + π/2
(e−(it+ε)Hπ/2f)(x)=(U−1e−(it+ε)λUf)(x)=(U−1e−(it+ε)λfˇ)(x)
= θ(λ,x)e−(it+ε)λ θ(λ,y)f(y)dydρ (λ)
π/2
R R
Z + Z +
√xy
= e−(it+ε)λ J (√λx)J (√λy)f(y)dydλ.
R R 2 −l−21 −l−12
Z +Z +
Since l <1/2, (A.1) implies that
| |
2l+1/2
J (k) (1+ (k)) (3.2)
| −l−1/2 |≤ Γ(1/2 l)kl+1/2 O
−
as k 0. Noting that f L2(R ) and using (3.2), Fubini’s theorem implies
→ ∈ c +
√xy
(e−(it+ε)Hπ/2f)(x)= R f(y) R e−(it+ε)λ 2 J−l−12(√λx)J−l−21(√λy)dλdy.
Z + Z +
(3.3)
The integral
√xy ∞
[e−(it+ε)Hπ/2](x,y):= 2 Z0 e−itλJ−l−21(√λx)J−l−21(√λy)dλ (3.4)
is knownas Weber’s second exponential integral[21, 13.31](cf. also [6, (4.14.39)])
§
and hence
(e−(it+ε)Hπ/2f)(x)= ε+1itZ0∞e−4x(2ε++yit2)√2xyI−l−21(cid:16)2(εx+yit)(cid:17)f(y)dy,
where I is the modified Bessel function (see [17, Chapter X] and in particular
ν
formula (10.27.6) there)
∞ (z/2)ν+2n
I (z)= =e∓iνπ/2J ( iz), π arg(z) π/2. (3.5)
ν ν
n!Γ(ν+m+1) ± − ≤ ≤
n=0
X
The estimate (A.2) implies
J (k) k−1/2(1+ (k−1)) (3.6)
−l−1/2
| |≤ O
as k . Therefore, there is C >0 which depends only on l and such that
→∞
1+k l
√kJ (k) C , k >0. (3.7)
| −l−1/2 |≤ k
(cid:18) (cid:19)
By (3.7) we deduce
2|√ε+xyit|(cid:12)(cid:12)e−4x(2ε++yit2)I−l−12(cid:16)2(εx+yit)(cid:17)(cid:12)(cid:12)≤Cs|ε+1it|(cid:12)(cid:12)1+ 2(εx+yit)(cid:12)(cid:12)l,
whichisuniforml(cid:12)(cid:12)y(wrt.ε)boundedoncom(cid:12)(cid:12)pactsetsK (cid:12)(cid:12) R+ R+.Th(cid:12)(cid:12)uswecan
apply dominated convergence and hence the claim follow⊂s⊂. × (cid:3)
In particular, we immediately arrive at the following estimate.
6 M.HOLZLEITNER,A.KOSTENKO,ANDG.TESCHL
Corollary 3.2. Let l <1/2. Then there is a constant C >0 which depends only
| |
on l and such that the inequality
C 2t+xy l
[e−itHπ/2](x,y) (3.8)
≤ √2t xy
(cid:18) (cid:19)
(cid:12) (cid:12)
holds for all x, y >0 an(cid:12)d t>0. (cid:12)
Proof. Applying (3.7) to (3.1), we arrive at (3.8). (cid:3)
Remark 3.3. For any fixed x and y R , we get from (A.1)
+
∈
√xy xy −l−1/2 1 xy −l
e−itHπ/2(x,y) = (3.9)
∼ 2t 4t t1/2−l 2
(cid:12) (cid:12) (cid:16) (cid:17) (cid:16) (cid:17)
Moreover, in vi(cid:12)ew of (A.1) o(cid:12)ne can see that
(cid:12) (cid:12)
xy −l
e−itHπ/2(x,y) cltl−1/2 , (3.10)
≥ 2
(cid:12) (cid:12) (cid:16) (cid:17)
whenever xy <t with so(cid:12)me constant c(cid:12)l >0, which depends only on l.
(cid:12) (cid:12)
Now we are ready to prove our first main result.
Proof of Theorem 1.1. If l ( 1/2,0], then
∈ −
2t+xy l
1
xy ≤
(cid:18) (cid:19)
for all x,y >0 and t 0. This immediately implies (1.7).
≥
Assume now that l (0,1/2). Clearly,
∈
2t+xy t
=1+2 3tmax(x−1,1)max(y−1,1)
xy xy ≤
for all t 1 and x, y >0. Indeed, the latter follows from the weaker estimate
≥
t
tmax(x−1,1)max(y−1,1), t 1, x,y >0,
xy ≤ ≥
which is equivalent to 1 max(x,1)max(y,1) for all x, y >0. Therefore,
≤
l
2t+xy
3tlmax(x−l,1)max(y−l,1), t 1, x,y >0,
xy ≤ ≥
(cid:18) (cid:19)
which proves (1.8). Remark 3.3 shows that (1.8) is sharp. (cid:3)
4. Proof of Theorem 1.2
Let us consider the following improper integrals:
I (t;x,y):=√xy e−itk2J (kx)J (ky)Imm (k2)k−2ldk, (4.1)
1 l+1 l+1 α
R 2 2
Z +
I (t;x,y):=√xy e−itk2J (kx)J (ky)Imm (k2)kdk, (4.2)
2 l+1 −l−1 α
R 2 2
Z +
I (t;x,y):=√xy e−itk2J (kx)J (ky)Imm (k2)k2l+2dk, (4.3)
3 −l−1 −l−1 α
R 2 2
Z +
where x, y > 0 and t = 0. Moreover, here and below we shall use the convention
Imm (k2) := Imm (k6 2 +i0) = lim Imm (k2 +iε) for all k R. Denote the
α α ε↓0 α
∈
corresponding integrand by A , that is, I (t)= e−itk2A (k;x,y)dk. Our aim is
j j R j
+
R
DISPERSION ESTIMATES: THE EFFECT OF BOUNDARY CONDITIONS 7
to use Lemma B.2 (plus the remarks after this lemma) and hence we need to show
thateachA belongstothe Wieneralgebra (R),thatis,coincidewithafunction
j
W
which is the Fourier transform of a finite measure.
We also need the following estimates, which follow from (2.13)
C2 k 2l+1, α=0,
Imm (k2)= l| | k , (4.4)
α  cos2(πl) k −2l−1+ (k −4l−2), α=0, →∞
Cl2sin2(α)| | O | | 6
and

Cl2 k 2l+1+ (k 4l+2), α=π/2,
Imm (k2)= cos(α)2| | O | | 6 k 0. (4.5)
α C−2cos2(πl)k −2l−1, α=π/2, →
 l | |
4.1. The integral I1. Consider the function
rl+1 ∞ ( r2/4)n
J(r):=√rJ (r)= − , r 0.
l+12 2l+1/2 n!Γ(ν+n+1) ≥
n=0
X
Note that J(r) rl+1 as r 0 and J(r) = 2 sin(r lπ)+O(r−1) as r +
∼ → π − 2 → ∞
(see (A.2)). Moreover, J′(r) rl as r 0 aqnd J′(r) = 2 cos(r lπ)+O(r−1)
∼ → π − 2
as r + (see (A.4)). In particular, J˜(r):=J(r) 2qsin(r lπ) is in H1(R ).
→ ∞ − π − 2 +
Moreover, we can define J(r) for r < 0 such that itqis locally in H1 and J(r) =
2 sin(r lπ) for r < 1. By construction we then have J˜ H1(R) and thus
π − 2 − ∈
qJ˜ is the Fourier transform of an integrable function (see Lemma B.3). Moreover,
sin(r lπ) is the Fourier transform of the sum of two Dirac delta measures and so
− 2
J is the Fourier transformofafinite measure.By scaling,the totalvariationofthe
measures corresponding to J(kx) is independent of x.
Next consider the function
Imm (k2) C2
F(k):= α = l .
k 2l+1 (cos(α) C2sin(α)tan(πl)k 2l+1)2+C4sin2(α)k 4l+2
| | − l | | l | |
By Corollary B.6, F is in the Wiener algebra (R).
0
W
Now it remains to note that
I (t)= e−itk2A (k2;x,y)dk = e−itk2J(kx)J(ky)F(k)dk, (4.6)
1 1
R R
Z + Z +
and applying Lemma B.2 we end up with the estimate
I (t;x,y) Ct−1/2, t>0, (4.7)
1
| |≤
with a positive constant C >0 independent of x, y >0.
4.2. The integral I . Assume first that l (0,1/2) and write
2
∈
χ (k) Imm (k2)
A (k2;x,y)=J(kx)Y(ky) l α ,
2
χ (ky) χ (k)
l l
where
rl
J(r)=√rJ (r), Y(r)=χ (r)√rJ (r), χ (r)= | | .
l+12 l −l−12 l 1+ rl
| |
8 M.HOLZLEITNER,A.KOSTENKO,ANDG.TESCHL
The asymptotic behavior (4.4) and (4.5) of Imm shows that
α
Imm (k2) k 1+l, k 0,
α
M(k)= = | | →
χl(k) (k −2l−1, k ,
| | | |→∞
and hence M H1(R), which implies that M is in the Wiener algebra (R).
0
∈ W
We continue J(r), Y(r) to the region r < 0 such that they are continuously
differentiable and satisfy
2 πl 2 πl
J(r)= sin r , Y(r)= cos r+ ,
π − 2 π 2
r (cid:18) (cid:19) r (cid:18) (cid:19)
forr < 1.ThenJ˜(r):=J(r) 2 sin(r πl)andY˜(r):=Y(r) 2 cos r+ πl
− − π − 2 − π 2
are in H1(R). In fact, they are qcontinuously differentiable and henqce it suffices to
(cid:0) (cid:1)
look at their asymptotic behavior.For r < 1 they are zero and for r >1 they are
−
O(r−1) andtheir derivativeis O(r−1)as canbe seenfromthe asymptoticbehavior
of Bessel functions (see Appendix A). Hence both J and Y are Fourier transforms
of finite measures. By scaling the total variation of the measures corresponding to
J(kx) and Y(ky) are independent of x and y, respectively.
It remains to consider the function χ (k)/χ (ky). Observe that
l l
χ (k) 1+ ky l 1 y−l
h (k):=1 l =1 | | = − =(1 y−l)(1 χ (k)).
y,l − χ (ky) − yl+ ky l 1+ k l − − l
l
| | | |
By CorollaryB.6, 1 χ (R). Therefore, applying Lemma B.2, we obtain the
l 0
− ∈W
following estimate
I (t;x,y) Ct−1/2max(1,y−l), t>0, (4.8)
2
| |≤
whenever l (0,1/2).
∈
Consider now the remaining case l ( 1/2,0]. Write
∈ −
A (k2;x,y)=J(kx)Y(ky)Imm (k2),
2 α
where
J(r)=√rJ (r), Y(r)=√rJ (r).
l+1 −l−1
2 2
Noting that Y(r) r−l as r 0 and using Lemma B.3, we can continue J and
∼ →
Y to the region r < 0 such that both J and Y are Fourier transforms of finite
measures.
It remains to consider Imm (k2) given by (2.13). However, by Corollary B.6,
α
this function is in the Wiener algebra (R) and hence applying Lemma B.2, we
0
W
end up with the estimate
I (t;x,y) Ct−1/2, t>0, (4.9)
2
| |≤
whenever l ( 1/2,0].
∈ −
4.3. The integral I . Again let us consider two cases. Assume first that l
3
∈
( 1/2,0]and then write
−
A (k2;x,y)=Y(kx)Y(ky)Imm (k2)k2l+1,
3 α
where
Y(r)=√rJ (r), r>0.
−l−1
2
DISPERSION ESTIMATES: THE EFFECT OF BOUNDARY CONDITIONS 9
Notice that
C2k4l+2
k 2l+1Imm (k2)= l ,
| | α (cos(α) C2sin(α)tan(πl)k2l+1)2+C4sin2(α)k4l+2
− l l
which is the sum of a constant and a function of the form (B.5), and hence it
belongs to the Wiener algebra (R) by Corollary B.6. Arguing as in the previous
W
subsection and applying Lemma B.2, we arrive at the following estimate
I (t;x,y) Ct−1/2, t>0, (4.10)
3
| |≤
whenever l ( 1/2,0].
∈ −
If l (0,1/2), write
∈
χ (k) χ (k) Imm (k2)
A (k2;x,y)=Y(kx)Y(ky) l l α ,
3 χ (kx)χ (ky) χ2(k)
l l l
where
rl
Y(r)=χ (r)√rJ (r), χ (r)= | | .
l −l−21 l 1+ rl
| |
Notice that
Imm (k2)k 2l+1
α
M(k):= | |
χ2(k)
l
C2 k 2l+2(1+kl)2
= l| |
(cos(α) C2sin(α)tan(πl)k 2l+1)2+C4sin2(α)k 4l+2
− l | | l | |
Clearly,byCorollaryB.6,M (R).Therefore,similartotheprevioussubsection,
∈W
we end up with the estimate
I (t;x,y) Ct−1/2max(1,x−l)max(1,y−l), t>0, (4.11)
3
| |≤
whenever l (0,1/2).
∈
4.4. Proof of Theorem 1.2. We begin with the representation of the integral
kernel of the evolution group.
Lemma 4.1. Let l <1/2 and α [0,π). Then the evolution group e−itHαPc(Hα)
| | ∈
is an integral operator and its kernel is given by
2
[e−itHαP (H )](x,y)= e−itk2φ (k2,x)φ (k2,y)Imm (k2)kdk, (4.12)
c α α α α
π R
Z +
where the integral is to be understood as an improper integral.
Proof. By (2.3) and (2.8),
φ (k2,x)=cos(α)φ(k2,x)+sin(α)θ(k2,x)
α
πx sin(α)
= C−1cos(α)k−l−1/2J (kx)+C kl+1/2 J (kx) ,
2 l l+21 l cos(πl) −l−21
r (cid:18) (cid:19)
10 M.HOLZLEITNER,A.KOSTENKO,ANDG.TESCHL
and hence
π cos2(α)
φ (k2,x)φ (k2,y)= √xy k−2l−1J (kx)J (ky) (4.13)
α α 2 C2 l+21 l+21
(cid:18) l
sin(2α)
+ (J (kx)J (ky)+J (kx)J (ky)) (4.14)
2cos(πl) l+12 −l−21 −l−12 l+21
sin2(α)
+C2k2l+1 J (kx)J (ky) . (4.15)
l cos2(πl) −l−12 −l−21
(cid:19)
By our considerations in the previous subsections, we have
φ (k2,x)φ (k2,y)Imm (k2)k (R)
α α α
∈W
with norm uniformly bounded for x,y restricted to any compact subset of (0, ).
∞
Moreover, we have e−i(t−iε)HαPc(Hα) e−itHαPc(Hα) as ε 0 in the strong op-
→ ↓
erator topology. By Lemma C.1, e−i(t−iε)HαPc(Hα) is an integral operator for all
ε > 0 and, moreover, the kernel converges uniformly on compact sets by Lemma
C.2.Hencee−itHαPc(Hα)isanintegraloperatorwhosekernelisgivenbythelimits
of the kernels of the approximating operators,that is, by (4.12). (cid:3)
Proof of Theorem 1.2. Combining(4.7),(4.8),(4.9),(4.10)and(4.11),wearriveat
the following decay estimate for the kernel of the evolution group
1, l ( 1/2,0],
[e−itHαP (H )](x,y) Ct−1/2 ∈ −
c α ≤ ×(max(1,x−l)max(1,y−l), l (0,1/2).
(cid:12) (cid:12) ∈
(cid:12) (cid:12) (4.16)
Thi(cid:12)s completes the proo(cid:12)f of Theorem 1.2. (cid:3)
Appendix A. Bessel functions
Here we collect basic formulas and information on Bessel functions (see, e.g.,
[17, 21]). We start with the definition:
z ν ∞ ( z2/4)n
J (z)= − . (A.1)
ν
2 n!Γ(ν+n+1)
(cid:16) (cid:17) nX=0
The asymptotic behavior as z is given by
| |→∞
2
J (z)= cos(z νπ/2 π/4)+e|Imz| (z −1) , argz <π. (A.2)
ν
πz − − O | | | |
r
(cid:16) (cid:17)
Noting that
ν ν
J′(z)= J (z)+ J (z)=J (z) J (z), (A.3)
ν − ν+1 z ν ν−1 − z ν
one can show that the derivative of the reminder satisfies
′
πz 1 1
J (z) cos(z νπ π) =e|Imz| (z −1), z . (A.4)
ν
2 − − 2 − 4 O | | | |→∞
(cid:18)r (cid:19)