Decay Rates and Probability Estimates for Massive Dirac Particles in the Kerr-Newman Black Hole Geometry

42 Pages·0.45 MB·English

by  Felix Finster

Decay Rates and Probability Estimates for Massive Dirac
Particles in the Kerr-Newman Black Hole Geometry
F. Finster, N. Kamran∗, J. Smoller†, and S.-T. Yau‡
July 2001 / January 2002
2
0
0
Abstract
2

n The Cauchy problem is considered for the massive Dirac equation in the non-
a extremeKerr-Newmangeometry,forsmoothinitialdatawithcompactsupportoutside
J
the event horizon and bounded angular momentum. We prove that the Dirac wave

1 function decays in L∞ at least at the rate t−5/6. For generic initial data, this rate
3 loc
of decay is sharp. We derive a formula for the probability p that the Dirac particle

escapes to infinity. For various conditions on the initial data, we show that p = 0,1
2
v or 0 < p < 1. The proofs are based on a refined analysis of the Dirac propagator
4 constructed in [4].
9
0
7 Contents
0
1
0 1 Introduction 2
/
c
q 2 The Long-Time Dynamics under a Spectral Condition 7
-
r
g 3 Decay Rates of Fourier Transforms – Basic Considerations 10
:
v
i 4 The Planar Equation 13
X
r 5 Uniform Bounds for the Potentials 15
a
6 The Region ω > m 26
7 The Region ω < m 30
8 Proof of the Decay Rates 36
9 Probability Estimates 37
References 41
∗Research supported by NSERCgrant # RGPIN 105490-1998.
†Research supported in part by theNSF,Grant No. DMS-0103998.
‡Research supported in part by theNSF,Grant No. 33-585-7510-2-30.
1
1 Introduction
The Cauchy problem for the massive Dirac equation in the non-extreme Kerr-Newman
black hole geometry outside the event horizon was recently studied [4], and it was proved
that for initial data in L∞ with L2 decay at infinity, the probability for the Dirac particle
loc
to be located in any compact region of space tends to zero as t . This result shows
→ ∞
that the Dirac particle must eventually either disappear into the event horizon or escape
to infinity. The questions of the likelihood of each of these possibilities and the rates
of decay of the Dirac wave function in a compact region of space were left open. In the
presentpaper,weshallstudythesequestionsbymeansofadetailedanalysisoftheintegral
representation of the Dirac propagator constructed in [4]. This analysis will also give us
some insight into the physical mechanism which leads to the decay.
Recall that in Boyer-Lindquist coordinates (t,r,ϑ,ϕ) with r > 0, 0 ϑ π, 0 ϕ <
≤ ≤ ≤
2π, the Kerr-Newman metric is given by [2]
ds2 = g dxjxk
jk
∆ dr2 sin2ϑ
= (dt a sin2ϑdϕ)2 U +dϑ2 (adt (r2+a2)dϕ)2 (1.1)
U − − ∆ − U −
(cid:18) (cid:19)
with
U(r,ϑ) = r2+a2 cos2ϑ , ∆(r) = r2 2Mr+a2+Q2 ,
−
and the electromagnetic potential is
Qr
A dxj = (dt a sin2ϑdϕ) ,
j
− U −
whereM, aM and Q denote the mass, the angular momentum and the charge of the black
hole, respectively. Here a and/or Q are allowed to be zero, so that our results apply also
to the Kerr, Reissner-Nordstr¨om, and Schwarzschild solutions. We shall restrict attention
to the non-extreme case M2 > a2 + Q2, in which case the function ∆ has two distinct
zeros,
r = M M2 a2 Q2 and r = M + M2 a2 Q2 ,
0 1
− − − − −
p p
corresponding to the Cauchy horizon and the event horizon, respectively. We will here
consider only the region r > r outside of the event horizon, and thus ∆> 0.
1
Our starting point is the representation of the Dirac propagator for a Dirac particle of
mass m and charge e established in [4, Thm. 3.6]
1 ∞ 2
Ψ(t,x) = dωe−iωt tkωnΨkωn(x)<Ψkωn Ψ > . (1.2)
π ab a b | 0
k,n∈ZZ−∞ a,b=1
X X
Here Ψ is the initial data, ω is the energy, and <..> is a positive scalar product (see [4]
0
|
for details). The quantum number k arises from the usual separation exp( i(k + 1))
∼ − 2
of the angular dependence around the axis of symmetry, whereas n labels the eigenval-
ues of generalized total angular momentum in Chandrasekhar’s separation of the Dirac
equation into ODEs [3]. The Ψkωn are solutions of the Dirac equation and tkωn are com-
a ab
plex coefficients; they can all be expressed in terms of the fundamental solutions to these
ODEs. We postpone the detailed formulas for Ψkωn and tkωn to later sections, and here
a ab
2
merely describe those qualitative properties of the wave functions Ψkωn which are needed
a
for understandingour results. Near the event horizon, the Ψkωn go over asymptotically to
a
spherical waves. In the region ω > m, the solutions for a= 1 are the incoming waves, i.e.
| |
asymptotically near theevent horizonthey arewaves movingtowards theblack hole. Con-
versely, the solutions for a= 2 are the outgoing waves, which near the event horizon move
outwards, away from the black hole. Asymptotically near infinity, the Ψkωn, ω > m, go
a | |
over to spherical waves. In the region ω < m, however, the fundamental solutions for
| |
a = 1,2 near the event horizon are both linear combinations of incoming and outgoing
waves, taken in such a way that Ψkωn and Ψkωn at infinity have exponential decay and
1 2
growth, respectively.
For technical simplicity, we make the assumption that Ψ is smooth and compactly
0
supported outside the event horizon. We point out that, while the assumption of compact
support is physically reasonable at infinity, it is indeed restrictive with respect to the be-
havior near the event horizon. Furthermore, we shall assume that the angular momentum
is bounded in the strict sense that there exist constants k and n such that
0 0
1 ∞ 2
Ψ(t,x) = dωe−iωt tkωnΨkωn(x)<Ψkωn Ψ > . (1.3)
π  ab a b | 0 
|kX|≤k0 |nX|≤n0Z−∞ aX,b=1
 
We expect that the rate of decay is the same if an infinite number of angular modes are
present. Namely, away from the event horizon, modes with large angular momentum feel
strong centrifugal forces and should therefore be quickly driven out to infinity, whereas
the behavior near the event horizon is independent of the angular momentum. However,
it seems a very delicate problem to rigorously establish decay rates without the assump-
tion(1.3), becausethiswouldmakeitnecessarytocontrolthedependenceofourestimates
on k and n. Finally, we assume that the charge of the black hole is so small that the grav-
itational attraction is the dominant force at a large distance from the black hole. More
precisely, we shall assume throughout this paper that
mM > eQ . (1.4)
| |
We now state our main results and discuss them afterwards.
Theorem 1.1 (Decay Rates) Consider the Cauchy problem
(iγjD m)Ψ(t,x) = 0, Ψ(0,x) = Ψ (x)
j 0
−
for the Dirac equation in the non-extreme Kerr-Newman black hole geometry with small
charge (1.4). Assume that the Cauchy data Ψ is smooth with compact support outside
0
the event horizon and has bounded angular momentum (1.3).
(i) If for any k and n,
limsup <Ψkωn Ψ > = 0 or limsup <Ψkωn Ψ > = 0, (1.5)
| 2 | 0 | 6 | 2 | 0 | 6
ωցm ωր−m
then for large t,
Ψ(t,x) = ct−56 + (t−65−ε) , (1.6)
| | O
with c= c(x) = 0 and any ε < 1 .
6 30
3
(ii) If for all k,n and a= 1,2,
<Ψkωn Ψ > = 0
a | 0
for all ω in a neighborhood of m, then Ψ(t,x) has rapid decay in t (i.e. for any
± | |
fixed x, Ψ(t,x) decays in t faster than polynomially).
Theorem 1.2 (Probability Estimates) Consider the Cauchy problem as in Theorem
1.1, with initial data Ψ normalized by <Ψ Ψ > = 1. Let p be the probability for the
0 0 0
|
Dirac particle to escape to infinity, defined for any R > r by
1
p = lim (ΨγjΨ)(t,x)ν dµ , (1.7)
j
t→∞
Z{R<r<∞}
where ν denotes the future directed normal to the hypersurface t = const and dµ denotes
the induced invariant measure on that hypersurface. Then p is given by
1 1 2
p = dω 2 tkωn 2 <Ψkωn Ψ > . (1.8)
π 2 − | 12 | 2 | 0
|kX|≤k0 |nX|≤n0ZIR\[−m,m] (cid:18) (cid:19)(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Accordingly, 1 p gives the probability that the Dirac particles disappears into the event
−
horizon. Furthermore,
(i) Suppose that the outgoing energy distribution for ω > m is
| |
non-zero, i.e.
<Ψkωn Ψ > = 0
2 | 0 6
for some ω with ω > m. Then p > 0.
| |
(ii) If the energy distribution of the Cauchy data has a non-zero contribution in the in-
terval [ m,m], then p < 1.
−
(iii) If the energy distribution of the Cauchy data is supported in [ m,m], then p = 0.
−
(iv) If (1.5) holds, then 0 < p < 1.
The decay rate of t−65 obtained in Theorem 1.1 quantifies the effect of the black hole’s
gravitational attraction on the long-time behavior of massive Dirac particles. Before dis-
cussing this effect in detail, it is instructive to recall the derivation of the decay rates
in Minkowski space. We denote the plane-wave solutions of the Dirac equation by Ψ ,
~ksǫ
where ~k is momentum, ǫ = 1 is the sign of energy, and s = refers to the two spin
± ±
orientations. The plane-wave solutions are normalized according to
(Ψ Ψ ) = δ(~k ~k′)δ δ ,
~ksǫ| ~k′s′ǫ′ − ss′ ǫǫ′
where (..) is the usual spatial scalar product
|
(Ψ Φ)(t) = Ψ(t,~x)γ0Φ(t,~x)d~x .
|
Z
The Dirac propagator is obtained by decomposing the initial data into the plane-wave
solutions,
Ψ(t,~x) = d~k Ψ (t,~x)(Ψ (t = 0) Ψ ),
~ksǫ ~ksǫ | 0
s,ǫ Z
X
4
andastraightforward calculation usingtheexplicit formof theplane-wave solutions yields
that
d4k
Ψ(t,~x) = 2π (k/+m)δ(k2 m2)Γ(ω)e−ikxγ0Ψˆ (~k) , (1.9)
(2π)4 − 0
Z
where Ψˆ (~k) is the Fourier transform of Ψ (~x) (and as usual x = (t,~x), k = (ω,~k),
0 0
k/ = k γj, and Γ is the step function Γ(x) = sgn(x)). Let us assume for simplicity that
j
the initial data is a Schwartz function. We write (1.9) as a Fourier integral in ω,
∞
Ψ(t,~x) = Ψ˜(ω,~x)e−iωtdω (1.10)
Z−∞
where
Ψ˜(ω,~x) = d~k (k/+m)δ(ω2 ~k 2 m2)Γ(ω)ei~k~xγ0Ψˆ (~k).
(2π)3 −| | − 0
Z
We consider the ω-dependence of Ψ˜ for fixed ~x. The δ-distribution gives a contribution to
themomentum integral only for~k on the sphere ~k 2 = ω2 m2. Thus Ψ˜(ω,~x)vanishes for
| | −
ω < m and has rapid decay at infinity. Furthermore, Ψ˜ is clearly smooth in the region
| |
ω > m. For ω near m, Ψ˜ has the expansion
| | | |
∞ k2dk
Ψ˜(ω,~x) = (ωγ0+m)δ(ω2 k2 m2)Γ(ω)γ0Ψˆ (0) (1+ (k))
4π2 − − 0 O
Z0
Γ(ω)
= (ω+mγ0)Ψˆ (0) ω2 m2 + (ω2 m2).
8π2 0 − O −
p
A typical plot of Ψ˜(ω,~x) is shown in Figure 1(a). If Ψˆ vanishes in a neighborhood of
0
~k = 0, then Ψ˜(.,~x)| is a Sch|wartz function, and thus its Fourier transform (1.10) has rapid
decay. This is the analogue of Case (ii) of Theorem 1.1. However, if Ψ˜ (0) = 0, the decay
0
6
rate is determined by the square root behavior of Ψ˜ for ω near m. A change of variables
| |
gives that for any test function η which is supported in a neighborhood of the origin,
∞ ∞ u
√ω mη(ω m)e−iωtdω = e−imt t−23 √uη e−iu du.
− − t
Zm Z0 (cid:16) (cid:17)
An integration-by-parts argument shows that the last integral is bounded uniformly in t,
and is non-zero for large t if η(0) = 0. From this we conclude that in Minkowski space,
6
Ψ(t,~x) decays polynomially at the rate t−32.
| |
Wenowproceedwithamoredetaileddiscussionofourresults,beginningwiththerates
ofdecayobtainedinTheorem1.1. Naively speaking,amassiveDiracparticlebehavesnear
the event horizon similar to a massless particle, i.e. like a solution of the wave equation.
In Minkowski space, solutions of the wave equation decay rapidly in time according to the
Huygens principle. On the other hand, at large distance from the black hole the solutions
shouldbehavelikethoseof themassiveDiracequation inMinkowski space, whichdecay at
theratet−23. ItisthustemptingtoexpectthatthesolutionsofthemassiveDiracequation
in the Kerr-Newman black hole geometry should decay at a rate which “interpolates”
between the behavior of a massive particle in Minkowski space and that of a massless
particle, and should thus decay at a rate no slower than t−32. However, Theorem 1.1
shows that this naive picture is incorrect, since the rate of decay we have established
for a massive Dirac particle in the Kerr-Newman black hole geometry is actually slower
than that of a massive particle in Minkowski space. Thus the gravitational field of the
5
Ž Ž
Y Y
È È HaL È È HbL
~ sin€€€€€€€€€1€€€€€€€€€€€€
(cid:143)!m!!!!-!!!!!Ω!!!!
~ (cid:143)!Ω!!!!-!!!!!m!!!!
Ω Ω
m m
Figure 1: Typical plot for Ψ˜ in Minkowski space (a) and in the Kerr-Newman black hole
geometry (b).
black hole affects the behavior of massive Dirac particles in a more subtle way. One can
understand this fact by comparing the plots in Figure 1, which give typical examples for
the energy distribution of the Dirac wave function in Minkowski space and in the Kerr-
Newman geometry. One sees that in the Kerr-Newman geometry, there is a contribution
to the energy distribution for ω < m, which oscillates infinitely fast as ω approaches m.
| |
When taking the Fourier transform, these oscillations lead to the decay rate t−65 given
Theorem 1.1 (see the rigorous saddle point argument in Lemma 3.3).
The oscillations in the energy distribution in Figure 1(b) are a consequence of the field
behavior near spatial infinity. On a qualitative level, they can already be understood in
Newtonian gravity and the semi-classical approximation. Namely, in the Newtonian limit
of General Relativity, the momentum ~k of a relativistic particle is related to its energy ω
by
2
mM
~k 2 = ω+ m2 .
| | r −
(cid:18) (cid:19)
Thus the particle has positive momentum even if ω < m, provided that the Newtonian
potential is large enough, mM > m ω. This means in the semi-classical approximation
r −
that the wave function Ψ(r) has an oscillatory behavior near the black hole,
r mM
Ψ(r) exp i ~kd~s for r < R ,
∼ ± ≡ m ω
(cid:18) Z (cid:19) −
and will fall off exponentially for r > R. As a consequence, the fundamental solutions
Ψkωn for ω < m involve phase factors exp( i R~kd~s). In the limit ω m, R ,
a | | ∼ ± ր → ∞
leading to infinitely fast oscillations in our integral representation. This simple argument
even gives the correct quantitative behavior of theRphases (m ω)−21.
∼ −
Thefact thatthedecay rate inthepresenceof ablack holeis slower than inMinkowski
space has the following direct physical interpretation. One can view the gravitational
attraction of the black hole and the tendency of quantum mechanical wave functions to
spread out in space as competing with each other over time. The component of the wave
function for ω near m and ω < m, which is responsible for the decay rate t−65, has not
enough energy to propagate out to infinity. But since it is an outgoing wave near the
event horizon (note that in (1.5) the fundamental solutions Ψkωn enter only for a = 2),
a
it is driven outwards and resists the gravitational attraction for a long time before it will
eventually be drawn into the black hole. As a result, the Dirac particle stays in any
6
compact region of space longer than it would in Minkowski space, and thus the rate of
decay of the wave function is slower.
According to this interpretation, our decay rates are a consequence of the far-field
behavior of the black hole. Similar to the “power law tails” in the massless case (see [8]),
our effect can be understood as a “backscattering” of the outgoing wave from the long-
range potential, but clearly the rest mass drastically changes the behavior of the wave
near infinity. We expect that result for the decay rates should be valid even in a more
general setting, independent of the details of the local geometry near the event horizon.
Furthermore, the decay rates should be independent of the spin. This view is supported
by [6, 7], who obtained the rate t−65 for massive scalar fields in a spherically symmetric
geometry using asymptotic expansions of the Green’s functions.
Theorem 1.2 gives a precise formula for the probability that the Dirac particle either
disappears into the black hole or escapes to infinity. In cases (i)–(iv) we give sufficient
conditions for these probabilities to occur. These results are consistent with the general
behavior of quantum mechanical particles in thepresence ofa potential barrier andcan be
thought of as a tunnelling effect. In case (iii), the particle does not have enough energy to
escape to infinity. Thinkingagain in terms of a tunnellingeffect, the Dirac particle cannot
tunnel to infinity because the potential barrier (which has finite height m) has infinite
width. Finally, one might ask whether p = 1 can occur; i.e., that the particles escapes to
infinity with probability one. This is indeed the case for very special initial data, whose
energy distribution is supported outside the interval [ m,m] (see Corollary 9.3 below).
−
We conclude by remarking that a number of significant results are known for the long-
term behavior of massless fieldsin black hole geometries. Theseresults do notcapture our
effect, which is intimately related to the presence of a mass gap in the energy spectrum.
Price [8] discussed the rates of decay of massless fields in the Schwarzschild background
for special choices of initial data. His decay rates depend on the angular momentum and
are faster than the ones we have derived. A rigorous proof of the boundedness of the
solutions of the wave equation in the Schwarzschild geometry has been given by Kay and
Wald [5]. Beyer pursues an approach using C0-semigroup theory, which also applies to
the Kerr metric and the massive case [1]. An important contribution to the long-time
behavior of gravitational perturbations of the Kerr metric has been given by Whiting [9].
2 The Long-Time Dynamics under a Spectral Condition
We begin the analysis with the case when the energy distribution of the Cauchy data is
zero in a neighborhood of ω = m. The following theorem is an equivalent formulation
±
of Theorem 1.1(ii).
Theorem 2.1 Consider the Cauchy problem
(iγjD m)Ψ(t,x) = 0 , Ψ(0,x) = Ψ (x)
j 0
−
for smooth initial data with compact support outside the event horizon. Assume that
angular momentum is bounded and that the energy is supported away from ω = m, i.e.
±
1 −m−ε m−ε ∞ 2
Ψ = + + dω tkωnΨkωn<Ψkωn Ψ > (2.1)
0 π ab a b | 0
|kX|≤k0 |nX|≤n0(cid:18)Z−∞ Z−m+ε Zm+ε(cid:19) aX,b=1
for suitable ε > 0. Then for all x, Ψ(t,x) has rapid decay in t.
7
Before giving the proof, we recall a few basic formulas from [4]. The separation ansatz
for the fundamental solutions Ψkωn is
a
Xkωn(r)Ykωn(ϑ)
− −
Ψkaωn(t,r,ϑ,ϕ) = e−iωte−i(k+12)ϕ XX+kkωωnn((rr))YY+kkωωnn((ϑϑ)) , (2.2)
+ −
 Xkωn(r)Ykωn(ϑ) 
 − + 
 
where X = (X ,X ) and Y = (Y ,Y ) are the radial and angular components, respec-
+ − + −
tively. The radial part X(u) is a solution of the radial Dirac equation [4, eqn (3.7)]
d 1 0 √∆ 0 imr λ
+iΩ(u) X = − X , (2.3)
du 0 1 r2+a2 imr λ 0
(cid:20) (cid:18) − (cid:19)(cid:21) (cid:18) − − (cid:19)
where
(k+ 1)a+eQr
Ω(u) = ω+ 2 , ∆ = r2 2Mr+a2+Q2 ,
r2+a2 −
λ is the angular eigenvalue (which depends smoothly on ω), and u ( , ) is related
∈ −∞ ∞
to the radius by
du r2+a2
= . (2.4)
dr ∆
To analyze X in the asymptotic region u , one employs for X the ansatz
→ −∞
e−iΩ0uf+(u)
X(u) = (2.5)
eiΩ0uf−(u)
(cid:18) (cid:19)
and obtains for f the equation
d 1 0
f = i(Ω Ω(u))
du 0− 0 1
(cid:20) (cid:18) − (cid:19)
√∆ 0 e−2iΩu(imr λ)
+ − f . (2.6)
r2+a2 e2iΩu( imr λ) 0
#
(cid:18) − − (cid:19)
StandardGronwallestimatesyieldthatthefundamentalsolutionsof(2.3)havetheasymp-
totic form [4, Lemma 3.1]
e−iΩ0uf+
X (u) = 0a + R (u) , (2.7)
a eiΩ0uf− 0
(cid:18) 0a (cid:19)
where R (u) c exp(du) for suitable constants c,d > 0 and
0
| | ≤
1 0
f = , f = . (2.8)
01 0 02 1
(cid:18) (cid:19) (cid:18) (cid:19)
In the asymptotic region u , one transforms the spinor basis with a matrix B(u) such
→ ∞
that the matrix potential in (2.3) becomes equal to the diagonal matrix iΩ(u)σ3 (σj are
−
the Pauli matrices). One must distinguish between the two cases ω < m and ω > m. In
| | | |
the first case, Ω(u) is imaginary for large u, and thus there are two fundamental solutions
X and X with exponential decay and growth, respectively, and we normalize them such
1 2
that
lim X(u) = 1 . (2.9)
u→−∞| |
8
In the case ω > m, Ω(u) is real for all u. The ansatz
| |
e−iΦ f+(u)
X = B with Φ′(u) = Ω(u) (2.10)
eiΦf−(u)
(cid:18) (cid:19)
gives the differential equation
d c
f = M(u)f with M(u) , (2.11)
du | | ≤ u2
which can again be controlled by Gronwall estimates. Thus one obtains the asymptotic
formula [4, Lemma 3.5]
coshΘ sinhΘ e−iΦ(u)f+
X (u) = ∞a + R (u) , (2.12)
a sinhΘ coshΘ eiΦ(u) f− ∞
(cid:18) (cid:19)(cid:18) ∞a (cid:19)
where R c/u for suitable c > 0 and
∞
| |≤
1 ω+m ωeQ+Mm2
Θ = log , Φ = Γ(ω) ω2 m2u + logu . (2.13)
4 ω m − √ω2 m2
(cid:18) − (cid:19) (cid:18)p − (cid:19)
Thecomplexfactorsf± in(2.12)aretheso-calledtransmission coefficients. Furthermore,
∞a
we introduced the functions t (α), 0 α 2π, in terms of the transmission coefficients
a
≤ ≤
by [4, eqn (3.47)]
t (α) = f+ e−iα f− eiα , t (α) = f+ e−iα + f− eiα . (2.14)
1 ∞2 − ∞2 2 − ∞1 ∞1
Finally, the coefficients (t ) are given by
ab a,b=1,2
δ δ if ω m
a1 b1
| |≤
tab =  1 2π tatb (2.15)
dα if ω > m.
 2π t 2+ t 2 | |
Z0 | 1| | 2|

Proof of Theorem 2.1. Since (2.1) contains only finite sums, we can fix k,n and consider
onesummand. Thecoefficients inthedifferentialequation (2.6) aresmoothinω, andtheir
ω-derivatives are integrable on the half-lines u ( ,u ] for u sufficiently small. Apart
0 0
∈ −∞
from the singularities at ω = m, the same is true for the differential equation (2.11) for
±
u on the half line [u , ) and u sufficiently large. Since furthermore the ansatz (2.5)
1 1
∞
is smooth in ω R (( m ε, m+ε) (m ε,m+ε)) and (2.10) is smooth in ω
∈ \ − − − ∪ − ∈
( , m ε] [m+ε, ),andalsothecoefficientsofthedifferentialequation(2.3)depend
−∞ − − ∪ ∞
smoothly on ω in the bounded interval u [u ,u ], we conclude that the fundamental
0 1
∈
solutions Ψk0ωn0(x) and the transmission coefficients fk0ωn0 depend smoothly on ω
a a ∈
R (( m ε, m+ε) (m ε,m+ε)). Hence the integrand in (2.1) is a smooth function
\ − − − ∪ −
in ω (which vanishes for ω m< ε). SinceΨ has compact supportandthe fundamental
0
| |−
solutions for ω go over to plane waves, it is clear that the ω-derivatives of the
→ ∞
integrand in (2.1) are all integrable. It follows that the Fourier integral (1.2) has rapid
decay.
Accordingtothistheoremandusinglinearity, itremainstoanalyzetheenergydistribution
inaneighborhoodofω = m. Sinceallconstructionsandestimatesaresimilarforpositive
±
and negative ω, we can in what follows restrict attention to a neighborhood of ω = +m.
9
3 Decay Rates of Fourier Transforms – Basic Considerations
In this section, we derive estimates of some elementary Fourier integrals. Our decay rate
t−56 ultimately comes from Lemma 3.3. We always denote by ε a parameter in the range
0 < ε< 1 .
30
Lemma 3.1 Let g L∞(R) C1((0, )) with compact support and assume that for a
∈ ∩ ∞
suitable constant C,
C
g′(α) for all α > 0. (3.1)
| | ≤ α
Then there is a constant c = c(g) such that for all t > 0,
∞
eiαtg(α)dα ct−56−ε .
≤
(cid:12)Z0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. Assume that suppg (cid:12) [ L,L]. For giv(cid:12)en δ > 0, we split up the integral as
⊂ −
∞ δ ∞
eiαtg(α)dα = eiαtg(α)dα + eiαt g(α)dα .
Z0 Z0 Zδ
The first term can be estimated by
δ
eiαtg(α)dα c δ
1
≤
(cid:12)Z0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
with c1 = sup g . In the second (cid:12)term, we integra(cid:12)te by parts,
| |
∞ 1 ∞ d
eiαtg(α)dα = eiαt g(α)dα
it dα
Zδ Zδ (cid:18) (cid:19)
1 1 ∞
= eiδtg(δ) eiαtg′(α)dα ,
−it − it
Zδ
and estimate using (3.1),
∞ c C
eiαtg(α)dα 1 + (logL logδ) .
≤ t t −
(cid:12)Zδ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
We choose δ = t−65−ε(cid:12)to conclude that(cid:12)
∞ c C
eiαtg(α)dα c1t−56−ε + 1 + (logL logt−65−ε) ,
≤ t t −
(cid:12)Z0 (cid:12)
(cid:12) (cid:12)
and this has t(cid:12)he required deca(cid:12)y properties in t.
(cid:12) (cid:12)
In the next lemma we insert into the Fourier integral a phase factor which oscillates
infinitely fast as α 0.
ց
Lemma 3.2 Let g be as in Lemma 3.1. Then there is a constant c = c(g) such that for
all t > 0,
∞ i
exp iαt g(α)dα ct−65−ε . (3.2)
− √α ≤
(cid:12)Z0 (cid:18) (cid:19) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
10