Resonances for Schrodinger operators with compactly supported potentials

0.25 MB·English

by  T. J. Christiansen

RESONANCES FOR SCHRO¨DINGER OPERATORS WITH
COMPACTLY SUPPORTED POTENTIALS
9 T. J. CHRISTIANSENANDP. D. HISLOP
0
0
Abstract. We describe the generic behavior of the resonance count-
2
ing function for a Schr¨odinger operator with a bounded, compactly-

n supported real or complex valued potential in d ≥ 1 dimensions. This
a
notecontainsasketchoftheproofofourmainresults[5,6]thatgeneri-
J
callytheorderofgrowthoftheresonancecountingfunctionisthemax-
8 imal value d in the odd dimensional case, and that it is the maximal
value d on each nonphysical sheet of the logarithmic Riemann surface
]
h in the even dimensional case. We include a review of previous results
p concerning the resonance counting functions for Schr¨odinger operators
- with compactly-supported potentials.
h
t
a
m
[

1. Introduction and Survey of Results
1
v
We are interested in the resonance counting function for Schro¨dinger op-
3
0 erators HV = ∆+V, acting on L2(Rd), with compactly supported real-
1 or complex-valu−ed potentials V L∞(Rd). We give a survey of results and
1 ∈ 0
discuss on our recent results on the generic behavior of the resonance count-
.
1 ing function [5] and [6]. We refer the reader to [3, 5, 6] for complete proofs
0
of Theorems 1 and 2.
9
0 Resonances are the poles of the meromorphic continuation of the cut-
: off resolvent of H as we now explain. For any V L∞(Rd), we denote
v V ∈ 0
i by χ C∞(Rd) any smooth, compactly supported function such that
X V ∈ 0
χ V = V. The cut-off resolvent for the Laplacian H = ∆ is defined to
V 0
r −
a be 0(λ) χV(H0 λ2)−1χV. This is a holomorphic, bounded operator-
valuRed fun≡ction on L−2(Rd) for λ C for d 1 odd, and for λ Λ, for d 4
∈ ≥ ∈ ≥
even. For d = 2 it is holomorphic on Λ 0 , with a logarithmic singularity
\{ }
at λ = 0. The unperturbed cut-off resolvent (λ) has the representation
0
R
(λ) = E (λ)+(λd−2logλ) E (λ), (1)
0 1 2
R
where E (λ) and E (λ) are entire operator-valued functions with E (λ) = 0
1 2 2
ford 1odd,andford = 2,theoperatorE (λ = 0)isafinite-rankoperator
2
≥
This follows, for example, from the formula for the Green’s function for the
Laplacian
(d−2)/2
i λ
(1)
R (λ) = H (λ x y ), (2)
0 4 2π x y (d−2)/2 | − |
(cid:18) | − |(cid:19)
2 T.J.CHRISTIANSENANDP.D.HISLOP
(1)
where the Hankel function of the first kind is defined by H (z) = J (z)+
ν ν
iN (z), and the expansion of the Neumann Bessel function N (z), when
ν ν
ν N.
∈
An application of the second resolvent formula allows us to write the
perturbed, cut-off resolvent (λ) for H as
V V
R
(λ)(1+V (λ)) = (λ). (3)
V 0 0
R R R
The resolvent R (λ) is analytic for Imλ >> 0, and it is meromorphic for
V
Imλ > 0 with at most finitely-many poles with finite multiplicities corre-
sponding to the eigenvalues of H . Thanks to (3), the perturbed, cut-off
V
resolvent (λ) extends as a meromorphic, bounded operator-valued func-
V
tion on CRif d 1 is odd, and onto the Riemannn surface Λ if d 4 is even,
≥ ≥
withanadditionallogarithmicsingularityatλ = 0whend= 2. Thepolesof
these continuations are the resonances of H . They are independent of the
V
choice of χ satisfying the above conditions. It is seen that they correspond
V
to the values of λ for which 1+V (λ) is not boundedlyinvertible. We will
0
R
use this fact in section 2.1.
We will always assume V L∞(Rd). The resonance counting function
∈ 0
n (r) for d 1 and odd is defined to be the number of poles λ of (λ),
V j V
≥ R
including multiplicities, with λ r and Imλ < 0. For d 2 even,
j j
| | ≤ ≥
the resonance counting function n (r) for the mth-sheet Λ , m = 0, is
V,m m
6
definedas thenumberofpolesλ of (λ)with λ r andmπ < argλ <
j V j j
R | | ≤
(m+1)π.
The best results on the asymptotic behavior of n (r) is due to Zworski
V
[42] for d = 1. He proved that
2
n (r)= suppV r+o(r). (4)
V
π| |
He obtained a similar asymptotic expansion in Rd, d 3 odd, for a family
≥
of spherically symmetric potentials V(r) with compact support [0,a] and a
discontinuity at r = a [43]. The exact form of the constant of the leading
term in this expansion was identified by Stefanov [33]. An asymptotic ex-
pansionof theform(4)foraclass ofsuper-exponentiallydecaying potentials
in one-dimension was proved by Froese [7] and Simon [28]. In what follows,
we consider dimensions d 2.
≥
Wemention thatmanyresultsareknownfortheresonancecountingfunc-
tion for Schro¨dinger operators H (h) = h2∆ + V in the semiclassical
V
−
regime of small h > 0, but we will not discuss these here. We refer to
the extensive papers of Sj¨ostrand [31] and of Sj¨ostrand and Zworski [32] for
the details and references. We also refer the reader to two review articles
[45, 38]. Many aspects of resonances for Schro¨dinger operators were ex-
plored by physicists. Regge [21] studied the Jost function and proved that
there are infinitely many resonances for compactly supported, real-valued
potentials. This and other aspects of resonances are discussed by Newton
[15]. Nussenzveig [16] studied analytically and numerically the poles of the
RESONANCES FOR SCHRO¨DINGER OPERATORS 3
S-matrix for square well and square barrier potentials of the type described
in section 6.
1.1. Upper Bounds. Inanalogy withthecelebratedWeylupperboundfor
theeigenvaluecountingfunctionforellipticoperatorsoncompactmanifolds,
upperboundsfortheresonancecountingfunctionwereprovenintheeighties
and nineties. Because the meromorphic continuation of the cut-off resolvent
dependson the parity of the dimension, the results are different for odd and
even dimensions. The upper bounds hold for both complex and real-valued
potentials.
1.1.1. Upper bounds - odd dimensions d 3. Polynomial upper bounds on
≥
the resonance counting function n (r) were first proved by Melrose [13]. He
V
showed that n (r) C r d+1. Thisupperboundwas improved by Zworski
V d
≤ h i
[41] who obtained the optimal upper bound n (r) C r d. Another proof
V d
≤ h i
was given by Vodev [35] and by Sj¨ostrand and Zworski [32]. The optimal
constant C was computed by Stefanov [33].
d
1.1.2. Upper bounds - even dimensions d 2. The first upper bounds were
≥
proven by Intissar [9]. Intissar defined, for any ǫ > 0 and r > 1, a resonance
counting function N (ǫ,r) λ r−ǫ < λ < rǫ, argλ < ǫlogr . For
In j j j
≡ { | | | | | }
even dimensions d 4, and for any ǫ (0,√2/2), he proved the polynomial
≥ ∈
upperboundN (ǫ,r) C r d+1. Vodevconsideredtheresonancecounting
In ǫ
≤ h i
function N (r,a) defined as the number of resonances λ satisfying 0 <
Vo j
λ < r and argλ < a, for r,a > 1. This function counts the number
j j
| | | |
of resonances in a fixed sector of opening angle a. Vodev proved the upper
bound for even dimensions d 2 [36, 37]:
≥
N (r,a) C a(rd+(loga)d). (5)
Vo d
≤
Note the explicit dependance of the coefficient on a. Since the Green’s
function has a logarithmic singularity at zero energy in the two-dimensional
case,thisrequiredaseparateargument. However, thisupperbounddoesnot
distinguish between the resonances that might occur on each sheet Λ ,m
m
Z∗ Z 0 . ∈
≡ \{ }
1.2. Lower bounds. There are few lower bounds in dimensions d 2. As
≥
above, there are different bounds depending upon whether the dimension
is even or odd. Because of the existence of complex potentials with no
resonances [4], these results hold only for nontrivial real-valued potentials.
1.2.1. Lower bounds - odd dimensions d 3. The best known lower bounds
≥
for odd dimensions, other than the asymptotic results for certain radial
potentials [43], hold for potentials V L∞(Rd) of fixed sign. In the odd
∈ 0
dimensional case, Lax and Phillips [10] proved that if V 0 is bounded
≥
below by a characteristic function on a ball of radius R > 0, then that
the number of purely imaginary poles in the lower half complex plane of
modulus less or equal to r > 0 is bounded from below by c rd−1, with
d
4 T.J.CHRISTIANSENANDP.D.HISLOP
c > 0. Menzala and Schonbek [12] allow potentials with a small negative
d
component so that ∆ + V > 0 in the quadratic form sense. Vasy [34]
−
proved the existence of an infinite number of purely imaginary poles when
V 0, extending [10] to the case of a strictly negative potential. These
± ≥
results imply that n (r) is similarly bounded from below. The idea of these
V
proofs is to note that the scattering amplitude is a monotone function of
V at purely imaginary energies. This reduces the problem to an explicit
calculation of the number of purely imaginary poles for the symmetric step
potential V(x) = V χ (x), for any V > 0, where B (x) is the ball of
0 BR(0) | | 0 R
radius R > 0 centered at x.
In the general case of nontrivial, smooth, real-valued V C∞(Rd), Mel-
∈ 0
rose [14] observed from the Poisson formula that there must be infinitely-
many resonances if any of thecoefficients in theexpansion of theheat kernel
for k > d 2 are nonvanishing. In particular, for d= 3, there are an infinite
−
number of resonances since a = V2 = 2. This was extended to a class of
1
6
super-exponentially decaying, smooth, real-valued potentials in [1, 25, 26].
R
The first quantitative lower bounds for the resonance counting function for
nontrivial, smooth, real-valued V C∞(Rd), not of fixed sign, were proved
∈ 0
in [2]. In particular, it was proved there that
n (r)
V
limsup = , (6)
r(logr)−p ∞
r→∞
for all p > 1. For the same family of potentials, Sa´ Barreto [22] improved
this to
n (r)
V
limsup > 0. (7)
r
r→∞
We mention that, in particular, all these lower bounds require the potential
to be smooth.
1.2.2. Lower bounds - even dimensions d 2. There are only two results
≥
on lower bounds in the even dimensional case for d 4. Sa´ Barreto and
≥
Tang [24] proved the existence of at least one resonance for a real-valued,
compactly-supported, smooth nontrivial potential. Sa´ Barreto [23] studied
the resonance counting function N (r) defined to be the number of reso-
SaB
nances λ with 1/r < λ < r and argλ < logr. As r , this region
j j j
| | | | → ∞
in the Riemann surface Λ opens like logr. Sa´ Barreto proved that for even
d 4,
≥
N (r)
SaB
limsup = , (8)
(logr)(loglogr)−p ∞
r→∞
for all p > 1.
1.3. Generic behavior. The order of growth of a positive, real-valued,
monotone increasing function n(r) is defined by
logn(r)
ρ limsup , (9)
≡ logr
r→∞
RESONANCES FOR SCHRO¨DINGER OPERATORS 5
when it is finite. Roughly speaking, we prove that the order of growth of
theresonancecountingfunctionismaximalformostreal-valued orcomplex-
valued potentials in L∞(Rd) in the even and odd dimensional cases. Here,
0
the term “most potentials” is meant in the Borel sense. Let X be a metric
space. A subset is said to be Baire typical or generic if it is a dense G
δ
subset of X.
We refer to the beginning of section 1 for the definition of the resonance
counting function n (r) in odd dimensions, and of n (r) for even dimen-
V V,m
sions. The order of growth of the resonance counting function n (r) or
V
n (r) is defined by
V,m
logn (r)
V
ρ limsup , d odd, (10)
V
≡ logr
r→∞
or, by
logn (r)
V,m
ρ limsup , d even. (11)
V,m
≡ logr
r→∞
As discussed above, it is known that both ρ and ρ are bounded above
V V,m
by d. We are interested in the values of these order of growth exponents for
generic potentials. The notion of generic used here was employed by Simon
[28] in his study of singular continuous spectrum for Schro¨dinger operators.
We proved the following theorem in [5, 6].
Theorem 1. Let K Rd be a fixed, compact set with nonempty interior.
There is a dense G ⊂set (K) L∞(K;F), for F = R or F = C, such
that if V (K),δand ifVFd 2⊂is ev0en, then ρ = d for all m Z 0 ,
F V,m
∈ V ≥ ∈ \{ }
or, if d 3 is odd, then ρ = d.
V
≥
For odd dimensions, a similar result holds with C∞(K;F) in place of
0
L∞(K;F) [3, 5]. For the proof of Theorem 1, it is essential to have an ex-
0
plicit example of a potential for which the order of growth of the resonance
counting function is d. For odd dimensions, Zworski [43] proved the asymp-
totic form of n (r) for a class of positive, nontrivial, spherically symmetric
V
potentials. This result is, of course, stronger than the lower bound stated
in Theorem 2. Since, for our purposes, only a lower bound is required, this
gives an alternate, and perhaps simpler, proof. For the even dimensional
case, we proved alower boundon theresonance counting function for spher-
ically symmetric, constant, positive potentials on each nonphysical sheet.
Let B (x) be the ball of radius R > 0 centered at x Rd.
R
∈
Theorem 2. Let V(x) = V χ (x) for V > 0. Then, for d even, there
0 BR(0) 0
is a constant c > 0 so that n (r) c rm, for m Z∗. For d odd, there
m V,m m
≥ ∈
is a constant c > 0 so that n (r) > c rd.
0 V 0
In the remainder of this article, we sketch the main ideas behind the
proofsofTheorems1and2. Section2presentsthereductiontoaproblemof
countingzerosofafunctionholomorphiconahalf-plane. WeproveTheorem
1insection3moduloresultsconcerningplurisubharmonicfunctionsthatare
6 T.J.CHRISTIANSENANDP.D.HISLOP
presented in sections 4 and 5. The proof of Theorem 2 is sketched in section
6.
2. Reduction to a zero counting problem
As is typical for these problems, we reduce the estimate on the resonance
counting function to one on the number of zeros of a function holomorphic
in the half plane Imλ > 0. In our papers, we took different approaches
depending upon whether the dimension is even or odd. Here, we unify the
approaches and show how to use the method of [6] for either parity of the
dimension. For odd dimensions, in [5] and [3], resonances were treated by
considering the S-matrix and its scattering phase. Crucial for the argument
is the fact that the S-matrix admits a Weierstrass product representation
in the complex plane. For the even dimensional case, we work directly with
the second resolvent formula and we do that here for any dimension.
2.1. Resolvent approach: any dimension. Let V L∞(Rd) and let
∈ 0
χ be any compactly-supported, smooth function χ C∞(Rd) so that
V V ∈ 0
χ V = V. The cut-off free resolvent (λ) χ R (λ)χ has a meromor-
V 0 V 0 V
R ≡
phic continuation to Λ, the logarithmic cover of the plane, for d 4 even,
or to C, for d 1, odd (see, for example, [14, 27]). For d = 2 ≥there is a
≥
logarithmic singularity at λ = 0. For d odd, there are two distinct sheets,
the physical sheet Λ , and Λ . For d even, there is the physical sheet and
0 −1
infinitely-many distinct nonphysical sheets Λ , m Z∗. In order to unify
m
notation, we define m(d) to be equal to m Z whe∈n d is even, and, when
∈
d is odd, we take m = 1 to be the lower-half complex plane, so m(d) is
−
mmod2. Inthis way, thesheets Λ areallidentified withΛ andthesheets
2k 0
Λ are all identified with Λ .
2k+1 −1
We use the following key identity, that follows from (2) and the formulas
for the meromorphic continuation of Hankel functions (see [6, section 6] or
[17, chapter 7]), relating the free resolvent on Λ to that on Λ , for any
m 0
m Z,
∈
m mod 2 d odd
R (eimπλ) = R (λ) m(d)T(λ), where m(d) =
0 0 − m d even.
(cid:26)
(12)
The operator T(λ) on L2(Rd) has a Schwartz kernel
T(λ,x,y) = iπ(2π)−dλd−2 eiλ(x−y)·ωdω, (13)
Sd−1
Z
see [14, Section 1.6]. We note that for any χ C∞(Rd), χT(λ)χ is a
holomorphic trace-class operator for λ C. The∈ope0rator T has a kernel
∈
proportional to x y (−d+2)/2J (λ x y ) when d is odd, and to x
(d−2)/2
| − | | − | | −
y (−d+2)/2N (λ x y ) when d is even. The different behavior of the
(d−2)/2
| | − |
free resolvent for d odd or even is encoded in (12).
By thesecond resolvent formula(3), thepoles of (λ) with multiplicity,
V
R
correspond to the zeros of I+VR (λ)χ . We can reduce the analysis of the
0 V
RESONANCES FOR SCHRO¨DINGER OPERATORS 7
zeros of the continuation of I +VR (λ)χ to Λ to the analysis of zeros
0 V m
of a related operator on Λ using (12). If 0 < argλ < π and m Z, then
0
∈
eimπλ Λ , and
m
∈
I +VR (eimπλ)χ = I +V(R (λ) mT(λ))χ
0 0 V
−
= (I +VR (λ)χ )(I m(I +VR (λ)χ )−1VT(λ)χ ).
0 V 0 V V
−
For any fixed V L∞(Rd), there are only finitely many poles of (I +
∈ 0
VR (λ)χ )−1 with 0 < argλ < π. Thus
0 V
f (λ) = det(I m(I +VR (λ)χ )−1VT(λ)χ ) (14)
V,m 0 V V
−
is a holomorphic function of λ when 0 < argλ < π and λ > c0 V L∞ .
| | hk k i
Moreover, with at most a finite number of exceptions, the zeros of f (λ),
V,m
with0< argλ< π correspond,withmultiplicity, to thepolesof (λ)with
V
R
mπ < argλ< (m+1)π. Henceforth, we will consider the function f (λ),
V,m
for m Z∗ Z 0 , on Λ . For d odd, we are only interested in m = 1.
0
∈ ≡ \{ } −
In this case, the zeros of f (λ), for λ Λ , correspond to the resonances.
V,−1 0
∈
A similar approach in the d odd case was employed by Froese [7].
2.2. The S-matrix approach in odd dimensions. We comment on the
case of odd dimensions used in [3, 5]. We denote the scattering matrix for
the pair H = ∆ and H = H + V by S (λ), acting on L2(Sd−1). In
0 V 0 V
the case that V−is real-valued, this is a unitary operator for λ R. The
∈
S-matrix is given explicitly by
S (λ) = I +c λd−2π (V VR (λ)V)πt I + , (15)
V d λ − V −λ ≡ Tλ
where π : L2(Rd) L2(Sd−1) is defined by (π f)(ω) = e−iλx·ωf(x) dx
λ λ
→
[40]. Under the assumption that supp V is compact, the operator :
λ
L2(Sd−1) L2(Sd−1) is trace class. The S-matrix has a mReromorphic Tcon-
→
tinuation to the entire complex plane with finitely many poles for Imλ > 0
corresponding to eigenvalues of H , and resonance poles in Imλ < 0. We
V
recall that if Imλ0 c0 V L∞ , the multiplicities of λ0, as a zero of
≥ hk k i
detS (λ), and of λ , as a pole of the cut-off resolvent (λ), coincide.
V 0 V
− R
Consequently, the function
f (λ) detS (λ), (16)
V V
≡
is holomorphic for Imλ > c0 V L∞ , and well-defined for Imλ 0 with
hk k i ≥
finitely many poles. Hence, the problem of estimating the number of zeros
of f (λ) in the upper half plane is the same as estimating the number of
V
resonances in the lower half plane.
This is facilitated in the odd dimensional case by the well-known repre-
sentation of f (λ) in terms of canonical products. Let G(λ;p) be defined
V
for integer p 1, by
≥
G(λ;p) = (1 λ)eλ+λ2/2+···+λp/p, (17)
−
and define
P(λ) = Π G(λ/λ ;d 1). (18)
λj∈RV,λj6=0 j −
8 T.J.CHRISTIANSENANDP.D.HISLOP
Then the function f (λ) may be written as
V
P( λ)
f (λ) = αeig(λ) − , (19)
V
P(λ)
whereg(λ)isapolynomialoforderatmostd. Carefulstudyofthescattering
matrixandtheupperboundofsection1.1.1 maybeusedtoshowthatf (λ)
V
is of order at most d in the half-plane Imλ > c V , see [44]. It is the
0 ∞
hk k i
representation (19) that is not available in the even dimensional case. This
facilitates theconstruction ofaplurisubharmonicfunction(seesections 4–5)
in the odd dimensional case as done in [3, 5].
3. Proof of Generic Behavior
The proof of Theorem 1 consists of two components. The first is the
G -property of the set of all potentials in L∞(K;F), for F = R or F = C
δ 0
separately, and K Rd a compact set with nonempty interior, with the
⊂
correct order of growth. Thisis fairly easy to prove. Thesecond is the proof
that this G -set is dense. The density argument is the difficult part.
δ
Many of the statements in the proof of the G -property do not depend
δ
on whether d is even or odd. We write f (λ) with the understanding that
V,m
m Z∗ for d even, and m = 1 for d odd. We use the same convention for
∈ −
the counting function n (r), so that n (r) = n (r).
V,m V,−1 V
We first need a Jensen-type theorem that relates the order of growth of
the zero counting function n (r) to the asymptotic behavior of f (λ).
V,m V,m
In general, we consider a function h holomorphic in the set λ C : λ
{ ∈ | | ≥
R 0, Imλ 0 . For r > R, we define n (h,r) to be the number of
+,R
≥ ≥ }
zeros of h, counted with multiplicities, in the closed upper half plane with
norm between R and r, inclusive.
Lemma 1. Let R > 0, let h be holomorphic in λ C λ R > 0,Imλ
{ ∈ | | | ≥ ≥
0 . Suppose, in addition, that h has only finitely many zeros on the real
}
axis. Suppose that for some p >0, and for some ǫ > 0, we have
r h′(s) −R h′(s)
ds = (rp−ǫ)and ds = (rp−ǫ).
h(s) O h(s) O
ZR (cid:12) (cid:12) Z−r (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Then n+,R(h,(cid:12)r) has(cid:12) order p if and only if (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
log πlog h(reiθ)dθ
lim sup 0 | | = p.
logr
r→∞ R
We will apply this lemma with h taken to be f . For positive constants
V,m
N,M,q > 0 and j > 2N, we define subsets of L∞(K;F) by
0
Am(N,M,q,j) ≡ {V ∈ L∞0 (K;F) : hkVkL∞i ≤ N,
π
log f (reiθ) dθ Mrq,
V,m
| | ≤
Z0
for 2N r j . (20)
≤ ≤ }
RESONANCES FOR SCHRO¨DINGER OPERATORS 9
Lemma 2. The set A (N,M,q,j) L∞(K;F) is closed.
m ⊂ 0
The proof is a continuity argument in the potential V. Suppose V
k
∈
A (N,M,q,j) converges to V in the L∞-norm. We show that the corre-
m
sponding functions f (z) converge using the basic bound [29]
m
det(1+A) det(1+B) A B ekAk1+kBk1+1, (21)
1
| − | ≤ k − k
with A= (I +V R (λ)χ)−1V T(λ)χ and B = (I +VR (λ)χ)−1VT(λ)χ.
j 0 j 0
In the next step, we characterize those V L∞(K;F) for which the
∈ 0
resonance counting function exponent is strictly less than the dimension d.
For N, M, q > 0, let
B (N,M,q) = A (N,M,q,j). (22)
m m
j≥2N
\
Note that B (N,M,q) is closed by Lemma 2.
m
Lemma 3. Let V L∞(K;F), with
∈ 0
logn (r)
V,m
limsup < d. (23)
logr
r→∞
Then there exist N, M N, l N, such that V B (N,M,d 1/l).
m
∈ ∈ ∈ −
The proof of this lemma uses Lemma 1 with h = f . Condition (23)
V,m
implies that
log πlog f (reiθ) dθ
limsup 0 | V,m | = p < d. (24)
logr
r→∞ R
From this it follows that there are constants p′,M > 0, with p p′ < d, so
≤
that
π
log f (reiθ) dθ Mrp′
V,m
| | ≤
Z0
when r c0 V L∞ . The lemma then follows by choosing ℓ N so that
≥ hk k i ∈
p′ d 1/ℓ.
≤ −
Lemma 4. For d 2 even and m Z∗ or d 1 odd and m = 1, the sets
≥ ∈ ≥ −
logn (r)
= V L∞(K;F) : lim sup V,m = d ,
Mm ∈ 0 logr
r→∞
(cid:26) (cid:27)
are G -sets. Furthermore, for d 2 even, the set
δ
≥
=
m
M M
m∈Z∗
\
is also a G -set.
δ
Proof. Lemma 3 shows that the complement of is contained in
m
M
B (N,M,d 1/l),
m
−
(N,M[,l)∈N3
which is an F set since it is a countable union of closed sets. By Lemma 1,
σ
if V , then V B (N,M,d 1/l) for any N, M, l N. Thus
m m m
is the∈cMomplement of6∈an F set. − ∈ M(cid:3)
σ
10 T.J.CHRISTIANSENANDP.D.HISLOP
Theorem 1 follows from Lemmas 1 - 4 and the density argument presented
here that relies on sections 4–6.
Proof of Theorem 1. Since Lemma 4 shows that is a G set, we need
m δ
M
only showthat each is densein L∞(K;F). To dothis, we follow theproofof
0
[3, Corollary 1.3] with appropriate modifications. Let V L∞(K;F) and
0 ∈ 0
let ǫ > 0. By Theorem 2, proved in section 6, we may choose a nonzero,
real-valued, spherically symmetric V L∞(K;R) so that V , for
m Z∗. We consider the holomorphi1c∈func0tion V(z) = V(z,x1) ∈= zMVm(x)+
1
(1 ∈ z)V (x), for z C. This function satisfies Assumptions (V) of section
0
− ∈
5, with V(1) = V and V(0) = V . Thus, by [3, Theorem 1.1], for d odd,
1 0
and by [6, Theorem 3.8], for d even, there exists a pluripolar set E C
m
(see Definition 4.2), so that for z C E , we have ⊂
m
∈ \
logn (r)
V(z),m
limsup = d.
logr
r→∞
If we set E = m∈Z∗Em, the set E C is also pluripolar [11, Proposition
1.37]. Since E∪ R R has Lebesgu⊂e measure 0 (e.g. [20, Section 12.2]),
we may find z0∩ R⊂, z0 E, with z0 < ǫ/(1+ V0 L∞ + V1 L∞). Then
V(z0) m for∈all m 6∈Z∗, and |V(|z0) V0 Lk∞ <k ǫ. Mokreokver, if V0 is
real-va∈lueMd, so is V(z ).∈ k − k (cid:3)
0
4. Plurisubharmonic functions and pluripolar sets
A key role is played in the density part of the proof of Theorem 1 by the
theoryofplurisubharmonicfunctions. Plurisubharmonicfunctionswerefirst
usedinthestudyof theresonancecountingfunctionin[3]. Abasicreference
is the book by Lelong and Gruman [11]. We refer to an open connected set
Ω Ck asadomain. For twodomainsΩ′ andΩ,weusethenotation Ω′ ⋐ Ω
⊂
if Ω′ is compact and Ω′ Ω.
⊂
Definition 4.1. A real-valued function φ(z) taking values in [ , ) is
plurisubharmonic in a domain Ω Ck, and we write φ PSH(−Ω)∞, if∞:
⊂ ∈
φ is upper semicontinuous and φ ;
• for every z Ω, for every w 6≡C−k,∞and every r > 0 such that
• z+uw : u ∈ r,u C Ω we∈have
{ | |≤ ∈ } ⊂
1 2π
φ(z) φ(z+reiθw) dθ. (25)
≤ 2π
Z0
If the dimension k = 1, then this is just the definition of a subharmonic
function. A basic example of a PSH function is φ (z) = log f(z) where
f
| |
f is holomorphic in a domain Ω. If f(z ) = 0, for some z Ω, then
0 0
∈
φ(z ) = . This shows that there is a connection between the behavior
0
−∞
of this PSH function φ (z) on Ω and the zeros of f on Ω. The points where
f
a nontrivial PSH function takes the value are special and rare.
−∞