LOW ENERGY SCATTERING ASYMPTOTICS FOR PLANAR OBSTACLES T. J. CHRISTIANSEN AND K. DATCHEV Abstract. We compute low energy asymptotics for the resolvent of a planar obstacle and deduce

2025-04-27 1 0 560.35KB 26 页 10玖币
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LOW ENERGY SCATTERING ASYMPTOTICS FOR PLANAR OBSTACLES
T. J. CHRISTIANSEN AND K. DATCHEV
Abstract. We compute low energy asymptotics for the resolvent of a planar obstacle, and deduce
asymptotics for the corresponding scattering matrix, scattering phase, and exterior Dirichlet-to-
Neumann operator. We use an identity of Vodev to relate the obstacle resolvent to the free resolvent
and an identity of Petkov and Zworski to relate the scattering matrix to the resolvent. The leading
singularities are given in terms of the obstacle’s logarithmic capacity or Robin constant. We expect
these results to hold for more general compactly supported perturbations of the Laplacian on R2,
with the definition of the Robin constant suitably modified, under a generic assumption that the
spectrum is regular at zero.
1. Introduction
1.1. Main results. Consider the Dirichlet Laplacian −∆ on a planar exterior domain Ω = R2\O,
where O⊂R2is a compact set. We study three fundamental objects in the scattering theory of
−∆ on Ω: the resolvent, the scattering matrix, and the scattering phase. Each is a function of
the frequency λ. Our main results are uniformly convergent series expansions for all three objects
near λ= 0. We also deduce asymptotics for the Dirichlet-to-Neumann operator and for its lowest
eigenvalue.
We begin with the resolvent R(λ), defined for Im λ > 0 to be the operator which takes f∈L2(Ω)
to the unique u∈ D := {u∈H1
0(Ω): ∆u∈L2(Ω)}solving (−∆−λ2)u=f. For every χ∈C∞
0(R2),
the product χR(λ)χextends meromorphically as an operator-valued function of λto Λ, the Riemann
surface of the logarithm: see Section 1.4 for more on the notation used here and below.
For our main results we assume that Ois not polar, i.e. that C∞
0(Ω) is not dense in H1(R2).
For example, by Lemma A.1, it is enough if Ocontains a line segment.
Theorem 1. Suppose Ois not polar. Then there are operators B2j,k :L2
c(Ω) → Dloc (i.e. mapping
compactly supported functions in L2(Ω) to functions which are locally in D) and a constant a, such
that, for every χ∈C∞
0(R2), we have
χR(λ)χ=
∞
X
j=0
j
X
k=−j−1
χB2j,kχλ2j(log λ−a)k
=χB0,0χ+χB0,−1χ(log λ−a)−1+χB2,1χλ2(log λ−a) + ··· ,
(1.1)
with the series converging absolutely in the space of bounded operators L2(Ω) → D, uniformly on
sectors near zero.
2020 Mathematics Subject Classification. 35P25, 47A40, 35J25.
Key words and phrases. resolvent, scattering matrix, scattering phase, Dirichlet boundary conditions, capacity.
1
arXiv:2210.05744v3 [math.AP] 9 Feb 2023
2 T. J. CHRISTIANSEN AND K. DATCHEV
Remarks. 1. Our proof also shows that if k6= 0, then B2j,k has finite rank. Moreover, there is a
unique harmonic function Gin Dloc such that log |x| − G(x) is bounded as |x|→∞, and
B0,−1=1
2πG⊗G, a = log 2 −γ−C(O) + πi
2, C(O) := lim
|x|→∞ log |x| − G(x).(1.2)
The quantity C(O) is important in potential theory. It is the negative of Robin’s constant and as
the logarithm of the logarithmic capacity: see Appendix A.
2. We used the following definition, which will recur below: Given functions fnmapping Λ to a
Banach space B, we say Pnfn(λ)converges absolutely in B, uniformly on sectors near zero if, for
any ϕ > 0, there is λ1>0 such that Pnkfn(λ)kBconverges uniformly on {λ∈Λ: 0 <|λ| ≤
λ1and |arg λ| ≤ ϕ}. Moreover, the series (1.1), as well as the series (1.6) below, may be freely
differentiated term by term, with each resulting series having a tail which converges absolutely in B,
uniformly on sectors near zero: see Appendix B.
3. If we used λ2j(log λ)kinstead of λ2j(log λ−a)kin our expansion, in place of (1.1) we would get
χR(λ)χ=
∞
X
j=0
j
X
k=−∞
χe
B2j,kχλ2j(log λ)k.(1.3)
Note that terms in (1.3) with j≥1 have infinitely many predecessors, and hence (1.3) is an
asymptotic expansion only as far as the terms with j= 0. The technique of using a shift of the
logarithm to reduce the number of terms comes from [Jen84].
Our second theorem concerns the scattering matrix S(λ), which is a meromorphic family of
operators L2(S1)→L2(S1). It can be defined for λ∈Λ by Petkov and Zworski’s formula
S(λ) = I+A(λ),where A(λ) = 1
4πiE(λ)[∆, χ1]R(λ)[∆, χ2]E(λ)∗.(1.4)
Here E(λ) is the operator from L2(R2) to L2(S1) which has integral kernel e−iλω·xχ3(x), and
χ1, χ2, χ3are radial functions in C∞
0(R2) obeying
χ1≡1 near Dρ, χi≡1 near supp χi−1,0≤χi≤1,(1.5)
where ρis large enough that O⊂Dρ, and Dρ={x∈R2:|x|< ρ}. The formula (1.4) comes from
Theorem 4.26 of [DyZw19], and is a variant of the original Proposition 2.1 of [PeZw01].
Theorem 2. Suppose Ois not polar. Then there are finite rank operators Sj,k :L2(S1)→L2(S1),
such that
S(λ) = I+i
2(1 ⊗1)(log λ−a)−1+S1,−1λ(log λ−a)−1+
∞
X
j=2
bj/2c
X
k=−bj/2c−1
Sj,kλj(log λ−a)k,(1.6)
with the series converging absolutely in the space of trace-class operators L2(S1)→L2(S1), uni-
formly on sectors near zero.
Our third theorem concerns the scattering phase σ(λ) for λnear zero, defined by
σ(λ) = 1
2πi log det S(λ) = 1
2πi tr log S(λ).(1.7)
Theorem 3. Suppose Ois not polar. Then there are complex numbers a2j,k and b2j,k such that
σ(λ) = 1
2πi log 1 + iπ
log λ−a+
∞
X
j=1
j
X
k=−∞
a2j,kλ2j(log λ−a)k,(1.8)
LOW ENERGY SCATTERING ASYMPTOTICS FOR PLANAR OBSTACLES 3
and
σ0(λ) = −1
2λ(log λ−a)(log λ−¯a)+
∞
X
j=1
j
X
k=−∞
b2j−1,kλ2j−1(log λ−a)k,(1.9)
with the series converging absolutely in C, uniformly on sectors near zero. In particular, as λ→0
through positive real values, we have
σ(λ) = 1
πarctan π
2 log(λ/2) + 2C(O)+2γ+O(λ2log λ),(1.10)
and
σ0(λ) = −2/λ
4(log(λ/2) + C(O) + γ)2+π2+O(λlog λ).(1.11)
Remark. In each of the theorems above, the assumption that Ois not polar is necessary as well as
sufficient. Indeed, if Ois polar, then the expansion near λ= 0 of R(λ) is not of the form (1.1) but
instead equals that of the free resolvent (2.4). Moreover, S(λ) = Iand σ(λ) = 0 for all λ. To see
this, note that if Ois polar, then H1
0(Ω), the form domain of the Dirichlet Laplacian on Ω, is dense
in H1(R2), the form domain of the free Laplacian on R2. Consequently the continuous extension
of R(λ) from L2(Ω) to L2(R2) equals the free resolvent R0(λ) of Section 2.1.
1.2. Background and context. Early low frequency resolvent expansions were obtained by Mac-
Camy [Mac65]. Vainberg [Vai75,Vai89] has very general results and many references. We focus on
dimension two because of its physical importance and because the problem is harder here than in
other dimensions; see for example Lemma 2.3 in [LaPh72] by Lax and Phillips for an expansion of
the scattering matrix in dimension three.
Our Theorem 1is a variant of Theorem 2 of [WeWi92] by Weck and Witsch, of Theorem 1 of
[KlVa94] by Kleinman and Vainberg, and of Theorem 1.7 of [StWa20] by Strohmaier and Waters.
More specifically, the results of [KlVa94] and [StWa20] cover problems which are more general
than ours in many respects, but specialized to our setting they require ∂Ω to be C∞, while our
assumption that Ois polar is optimal. The results of [WeWi92] expand only up to O(λ2). Our
methods are different from those in the papers mentioned above. Specifically, [KlVa94] and [StWa20]
rely on general theory developed in [Vai89] and [M¨uSt14] respectively, while our approach based on
algebra of series of operators as in Vodev [Vod99,Vod14] leads to a direct short proof of complete
resolvent asymptotics: see Section 2.
Our scattering phase asymptotic (1.10) improves previous results in [HaZe99] and [McG13], and
is implicit in the proof of Theorem 3.25 of [StWa20]. To compare the results, recall that in [McG13],
McGillivray computes asymptotics of the Krein spectral shift function ξdefined by
tr Jg(H)J∗−g(−∆)=Z∞
0
g0(µ)ξ(µ)dµ,
where Jis the operator taking functions on R2\Oto functions on R2by extending them by zero.
By the Birman–Krein formula (see [JeKa78] or Proposition 0.1 and Theorem 1.1 of [Chr98]), we
see that ξ(µ) = −σ(√µ). Putting b= log 4 −2C(O)−2γand applying (1.10) gives
ξ(µ) = 1
πarctan π
b−log µ+O(µlog µ) (1.12)
=1
−log µ+−b
(−log µ)2+b2−π2
3
(−log µ)3+O((log µ)−4).(1.13)
4 T. J. CHRISTIANSEN AND K. DATCHEV
Thus, (1.12) recovers and improves (1.13). The first term of (1.13) was first computed in
[HaZe99], and all three terms of (1.13) were computed in [McG13] (note that C(K) in [McG13]
corresponds to 2C(O) here). See Figure 1for a comparison of the approximations when the obstacle
Ois a disk of radius ρ, in which case the spectral shift function is given by
ξρ(µ) = 1
2π
∞
X
`=−∞
arg H(2)
`(ρ√µ)/H(1)
`(ρ√µ); (1.14)
see (C.1). Note that the scaling ξρ(µ) = ξ1(ρ2µ) exhibited by (1.14) is also respected by our
approximation in (1.12), but not by the individual terms of (1.13).
0.00 0.05 0.10 0.15 0.20
0.05
0.10
0.15
0.20
0.25
0.30
0.00 0.05 0.10 0.15 0.20
0.05
0.10
0.15
0.20
0.25
0.30
0.00 0.05 0.10 0.15 0.20
0.05
0.10
0.15
0.20
0.25
0.30
0.00 0.05 0.10 0.15 0.20
0.05
0.10
0.15
0.20
0.25
0.30
Figure 1. Graphs of the spectral shift function ξand its first three approximations,
when the obstacle Ois a disk of radius ρ= 0.15 (top left), 1.5 (top right), 15 (bottom
left), and 150 (bottom right). The horizontal axis is the dimensionless (scaling-
invariant) variable ρλ =ρ√µ. The blue is the true value (1.14). The green is the
first term from (1.13), first obtained in [HaZe99]. The red is the first three terms
from (1.13), first obtained in [McG13]. The yellow is the leading approximation
obtained in this paper, given by the first term of (1.12), which is implicit in the
proof of Theorem 3.25 of [StWa20]. Observe that the yellow and blue curves are
independent of the radius ρ, while the red and green depend on ρ.
More recently, in [GMWZ22], more accurate analytic and numerical investigations of the scatter-
ing phase have been conducted for both a disk and for other obstacles, including ones with different
kinds of stable and unstable trapping.
1.3. Discussion of methods and outline of proof. The method of proof is as follows. In
Section 2, we deduce the series for the resolvent R(λ) of Theorem 1from the series for the free
LOW ENERGY SCATTERING ASYMPTOTICS FOR PLANAR OBSTACLES 5
resolvent R0(λ), using a resolvent identity due to Vodev [Vod14], and techniques in part based on
Vodev’s work in [Vod99].
Our techniques can be applied to quite general compactly supported perturbations of −∆ on
R2, far beyond the specific, classical setting considered here, that of the Dirichlet problem on an
exterior domain. In a companion paper we will address the more general case. As demonstrated
already in [BGD88,JeNe01] for the case of Schr¨odinger operators on R2, different kinds of singular
behavior of the resolvent near zero can then appear.
We deduce the series for the scattering matrix and scattering phase of Theorems 2and 3in
Section 3, by inserting the resolvent expansion (1.1) into the formulas (1.4) and (1.7). Section 4
contains some related results for the Dirichlet-to-Neumann operator.
Appendix Acollects background on polar and nonpolar sets, Appendix Bcontains a lemma
about series with logarithmic terms, and Appendix Ccontains formulas for the scattering matrix
and scattering phase of a disk; see also Section 5.2 of [HaZe99] and Section 3 of [GMWZ22].
The presentation of Theorems 1,2, and 3is self-contained, based on standard complex and
functional analysis, except for a few ingredients. For the resolvent expansions we quote the series
(2.1) and (2.2) for the free resolvent. For the scattering matrix and scattering phase we quote
Petkov and Zworski’s formula (1.4). Our analysis of the Dirichlet-to-Neumann operator in Section 4
requires more, namely mapping properties from [GMZ07], results on solutions of boundary value
problems from [McL00], and results on analyticity of eigenvalues for which [ReSi78] is a reference.
1.4. Notation and conventions.
•C∞
0(U) is the set of functions in C∞(U) with compact support in U.
•O⊂R2is compact and Ω = R2\O. We say Ois polar if C∞
0(Ω) is dense in H1(R2). The Dirichlet
Laplacian is −∆: D → L2(Ω), with D={u∈H1
0(Ω): ∆u∈L2(Ω)}, and ∆ := ∂2
x1+∂2
x2.
•The Dirichlet resolvent R(λ) is defined for Im λ > 0 to be the operator which takes f∈L2(Ω) to
the unique u∈ D solving (−∆−λ2)u=f. For every χ∈C∞
0(R2), the product χR(λ)χcontinues
meromorphically from the upper half plane to Λ, the Riemann surface of the logarithm. The free
resolvent R0(λ) is defined in the same way but with L2(Ω) replaced by L2(R2) and Dreplaced
by H2(R2). See Sections 2.1 and 2.3 for a review of these facts.
•The mapping Λ →Cgiven by λ7→ log λis bijective, with the upper half plane Im λ > 0
identified with the subset of Λ where arg λ= Im log λtakes values in (0, π).
•The mapping λ7→ ¯
λon Λ is defined by |¯
λ|=|λ|and arg ¯
λ=−arg λ.
•The product of a function χon R2and a function fon Ω is the function χf on Ω obtained by
ignoring the values of χon R2\Ω.
•L2
c(Ω) is the set of functions f∈L2(Ω) such that χf =ffor some χ∈C∞
0(R2). Other function
spaces with a ‘c’ subscript are defined analogously.
• Dloc is the set of functions fon Ω such that χf ∈ D for all χ∈C∞
0(R2). Other function spaces
with a ‘loc’ subscript are defined analogously.
•Gis the unique harmonic function in Dloc such that log |x| − G(x) is bounded as |x| → ∞. We
construct Gand prove its uniqueness in Lemma 2.6.
•C(O) = lim|x|→∞ log |x| − G(x).
•γis Euler’s constant, given by γ=−Γ0(1) = 0.577 . . . .
•a= log 2 −γ−C(O) + πi
2.
摘要：

LOWENERGYSCATTERINGASYMPTOTICSFORPLANAROBSTACLEST.J.CHRISTIANSENANDK.DATCHEVAbstract.Wecomputelowenergyasymptoticsfortheresolventofaplanarobstacle,anddeduceasymptoticsforthecorrespondingscatteringmatrix,scatteringphase,andexteriorDirichlet-to-Neumannoperator.WeuseanidentityofVodevtorelatetheobstacle...

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