Spectral theory for Schr\"odinger operators with $\delta$-interactions supported on curves in $\mathbb R^3$

0.44 MB·

by  Jussi Behrndt

SPECTRAL THEORY FOR SCHRO¨DINGER OPERATORS WITH
δ-INTERACTIONS SUPPORTED ON CURVES IN R3
6
1 JUSSIBEHRNDT,RUPERTL.FRANK,CHRISTIANKU¨HN,VLADIMIRLOTOREICHIK,
0
ANDJONATHANROHLEDER
2

p
Abstract. The main objective of this paper is to systematically develop a
e
spectral and scattering theory for selfadjoint Schro¨dinger operators with δ-
S
interactions supported on closed curves in R3. We provide bounds for the

9 numberofnegativeeigenvaluesdependingonthegeometryofthecurve,prove
2 an isoperimetric inequality for the principal eigenvalue, derive Schatten–von
Neumann properties for the resolvent difference with the free Laplacian, and

] establishanexplicitrepresentation forthescatteringmatrix.
P
S
.
h
t 1. Introduction
a
m
Schr¨odinger operators with singular interactions supported on sets of Lebesgue
[ measurezeroweresuggestedinthephysicsliteratureassolvablemodelsinquantum

mechanics in [12, 38, 46, 49, 61]. They appear, e.g., in the modeling of zero-
2
v range interactions of quantum particles [22, 23, 52, 53], in the theory of photonic
3 crystals[42], and inquantum few-body systems in strongmagnetic fields [20]. The
3 mathematicalinvestigationoftheirspectralandscatteringpropertiesattractedalot
4
of attention during the last decades. First studies were mostly devoted to singular
6
interactions supported on a discrete set of points, see the monograph [4] and [35,
0
. Chapter 5]. Later on, singular interactions supported on more general curves,
1
surfaces, and manifolds gained much attention; there is an extensive literature on
0
Schr¨odinger operators with δ-interactions supported on manifolds of codimension
6
1 one,see,e.g,[5,9,16,18,27,30,35,36,37]andthereferencestherein. Manifoldsof
: highercodimensionwerefirsttreatedin[17]intheveryspecialcaseofaninteraction
v
supported on a straightline in R3. More generalcurves were consideredin [13, 19,
i
X
28, 31, 32, 33, 34, 45, 47, 48, 54, 56, 60].
r In the present paper we systematically develop a spectral and scattering theory
a
for Schr¨odinger operators with singular interactions supported on curves in the
three-dimensionalspace. More specifically, for a compact, closed, regularC2-curve
Σ R3 we consider the selfadjoint Schr¨odinger operator ∆ in L2(R3), which
Σ,α
⊂ −
corresponds to the formal differential expression
1
(1.1) ∆ δ( Σ),
− − α ·−
whereα R 0 istheinversestrengthofinteraction. Themathematicallyrigorous
definition∈of\{∆} is more involved than in the case of, e.g., a curve in R2 or a
Σ,α
−
1991 Mathematics Subject Classification. Primary 81Q10; Secondary 35P15, 35P20, 35P25,
47F05,81Q15.
Key words and phrases. Schro¨dinger operator, δ-interaction, spectral theory, Birman–
Schwingeroperator,Schatten–von Neumannideal,spectralasymptotics,isoperimetricinequality,
scatteringmatrix.
1
2 J.BEHRNDT,R.L.FRANK,C.KU¨HN,V.LOTOREICHIK,ANDJ.ROHLEDER
hypersurface in R3. For our purposes an explicit characterization of the domain
and action of ∆ is essential; here the key difficulty is to define an appropriate
Σ,α
−
generalizedtrace mapforfunctions whicharenotsufficiently regular;see Section2
for the details. Our method is strongly inspired by [56] and the abstract concept
of boundary triples [7, 8, 21, 24, 25], and can also be viewed as a special case of
the more generalapproachin [54] (see Example 3.5 therein); cf. [19, 31, 34, 60] for
equivalent alternative definitions.
The main results of this paper deal with spectral and scattering properties of
∆ and extend and complement results in [19, 26, 28, 29, 32, 45, 56]. First we
Σ,α
−
verifythattheoperator ∆ isinfactselfadjoint;alongwiththis,inTheorem3.1
Σ,α
−
weestablishaKreintypeformulafortheresolventdifferenceof ∆ andthefree
Σ,α
−
Laplacian ∆ . Using this formula we show that the resolvent difference
free
−
(1.2) ( ∆ λ) 1 ( ∆ λ) 1, λ ρ( ∆ ) ρ( ∆ ),
Σ,α − free − Σ,α free
− − − − − ∈ − ∩ −
iscompact;inparticular,theessentialspectrumof ∆ equals[0, ). Moreover,
Σ,α
− ∞
we provide a Birman–Schwinger principle for the negative eigenvalues of ∆
Σ,α
−
and employ this principle for a more detailed study of these eigenvalues. In fact,
in Theorem 3.3 we show that the negative spectrum is always finite and we prove
upper and lower estimates for the number of negative eigenvalues, depending on
the (inverse)strength of interactionα and the geometry of the curve; these results
complement the estimates in [19, 32, 44, 45]. In the case that Σ is a circle our
estimates lead to an explicit formula for the number of negative eigenvalues. As
a further main result, in Theorem 3.6 we prove that amongst all curves of a fixed
lengththeprincipleeigenvalueof ∆ ismaximizedbythecircle. Withthisresult
Σ,α
−
we give an affirmative answer to an open problem formulated in [27, Section 7.8].
Ourproofisinspiredbyrelatedconsiderationsforδ-interactionssupportedonloops
in the plane in [26, 29].
Another group of results focuses on a more detailed comparison of ∆ with
Σ,α
−
the free Laplacian. From a careful analysis of the operators involved in the Krein
typeresolventformulaweobtainanasymptoticupperboundforthesingularvalues
s (λ) s (λ) ... of the resolvent difference (1.2) in Theorem 3.2,
1 2
≥ ≥
1
(1.3) s (λ)=O as k + .
k k2lnk → ∞
(cid:18) (cid:19)
Inparticular,theresolventdifferencein(1.2)belongstotheSchatten–vonNeumann
class S for any p > 1/2; this improves the trace class estimate in [19] and is in
p
accordance with a previous observation for periodic curves in [28, Remark 4.1].
Notethat, asa consequenceof (1.3),the absolutelycontinuousspectrumof ∆
Σ,α
−
equals [0, ) and the waveoperators for the scattering pair ∆ , ∆ exist
free Σ,α
∞ {− − }
and are complete. In Theorem 3.8 a representation of the associated scattering
matrix is given in terms of an explicit operator function which acts in L2(Σ); this
complementsearlierinvestigationsin[19, Section3]. Its proofreliesonanabstract
approachdeveloped recently in [11].
The paper is organized as follows. In Section 2 we discuss in detail the math-
ematically rigorous definition of the operator ∆ . Section 3 contains all main
Σ,α
−
resultsofthispaper. Theirproofsarecarriedoutintheremainderofthispaper. In
fact, Section 4 is preparatory and contains the analysis of the Birman–Schwinger
operator. The actual proofs of Theorems 3.1–3.8 are contained in Section 5. In
a short appendix the notions of quasi boundary triples and their Weyl functions
3
fromextensiontheoryofsymmetricoperatorsarereviewedanditisshownhowthe
operators ∆ and ∆ fit into this abstract scheme.
free Σ,α
− −
Acknowledgements
Jussi Behrndt, Christian Ku¨hn, Vladimir Lotoreichik, and Jonathan Rohleder
gratefully acknowledge financial support by the Austrian Science Fund (FWF),
project P 25162-N26. Vladimir Lotoreichik also acknowledgesfinancial support by
theCzechScienceFoundation,project14-06818S.RupertFrankacknowledgessup-
port through NSF grant DMS-1363432. The authors also wish to thank Johannes
Brasche and Andrea Posilicano for helpful discussions and the anonymous referees
for their helpful comments which led to various improvements.
2. Definition of the operator −∆Σ,α
In this section we define the operator ∆ associated with the differential
Σ,α
expression (1.1) in L2(R3). On a formal lev−el we interpret the action of (1.1) as
1
(2.1) u:= ∆u u δ .
α Σ Σ
A − − α | ·
It will be shown that gives rise to a selfadjoint operator in L2(R3). The key
α
A
difficultyinthedefinitionofthisoperatoristospecifyasuitabledomain. Notethat
theSobolevspaceH2(R3)isnotasuitabledomainasu δ L2(R3)forallthose
Σ Σ
u H2(R3) which do not vanish identically on Σ. On|th·e ot6∈her hand, any proper
su∈bspace of H2(R3) will turn out to be too small for ∆ to become selfadjoint
Σ,α
in L2(R3). Thus it is necessary to include suitable m−ore singular elements in the
domain of the operator. This requires the definition of a generalized trace u for
Σ
functions u L2(R3) which are not sufficiently regular. |
∈
Let us first fix some notation. We assume that Σ is a compact, closed, regular
C2-curve in R3 of length L without self-intersections and that σ :[0,L] R3 is a
→
C2-parametrizationof Σ with σ˙(s) =1 for all s [0,L]. Occasionally we identify
| | ∈
σ with its L-periodic extension. For h L2(Σ) we define the distribution hδ via
Σ
∈
(2.2) hδ ,ϕ = h(x)ϕ(x)dσ(x), ϕ H2(R3),
h Σ i 2,2 ∈
− ZΣ
whereϕ(x)istheevaluationofthecontinuousfunctionϕatx Σ, , denotes
2,2
thedualitybetweenH 2(R3)andH2(R3),anddσ denotesint∈egrathio·n·iw−ithrespect
−
to the arc length on Σ. Note that it follows from the continuity of the restriction
map H2(R3) ϕ ϕ L2(Σ) (see, e.g., [14, Theorem 24.3]) that hδ
Σ Σ
H 2(R3) and t∋hat h7→ h|δ ∈is a continuous mapping from L2(Σ) to H 2(R3). W∈e
− Σ −
will often use that hδ7→ L2(R3) if and only if h=0.
Σ
∈
For λ<0 we define the bounded operator
(2.3) γ :L2(Σ) L2(R3), h γ h=( ∆ λ) 1(hδ ),
λ λ − Σ
→ 7→ − −
where ∆ λ is viewed as an isomorphism between L2(R3) and H 2(R3). In the
−
followin−g le−mma useful representations of γ and its adjoint γ : L2(R3) L2(Σ)
are provided. We denote the selfadjoint Lapλlacian in L2(R3) wλ∗ith domain→H2(R3)
by ∆ .
free
−
Lemma 2.1. Let λ<0. Then
e √ λx y
− − | − |
(2.4) (γ h)(x)= h(y) dσ(y)
λ
4π x y
ZΣ | − |
4 J.BEHRNDT,R.L.FRANK,C.KU¨HN,V.LOTOREICHIK,ANDJ.ROHLEDER
holds for almost all x R3 and all h L2(Σ). Moreover,
∈ ∈
(2.5) γλ∗u= (−∆free−λ)−1u |Σ
holds for all u L2(R3). (cid:0) (cid:1)
∈
Proof. For h L2(Σ) and u L2(R3) we have
∈ ∈
hγλh,uiL2(R3) = γλh,(−∆free−λ)(−∆free−λ)−1u L2(R3)
=(cid:10)(−∆−λ)(γλh),(−∆−λ)−1u 2,2(cid:11)
−
=(cid:10)hδΣ,(−∆−λ)−1u 2,2 (cid:11)
−
(cid:10) (cid:11)
= h(y) ( ∆ λ) 1u (y)dσ(y)
− free− −
ZΣ
(cid:0) e √ λx y (cid:1)
− − | − |
= h(y) dσ(y)u(x)dx,
ZR3ZΣ 4π|x−y|
where we have used (2.2) and the integral representation of ( ∆ λ) 1, see,
free −
− −
e.g., [57, (IX.30)]. This proves both (2.4) and (2.5). (cid:3)
The identity (2.4) indicates that in general the trace of γ h on Σ does not
λ
exist due to the singularity of the integral kernel. This motivates the following
regularization. Here and in the following we denote by C0,1(Σ) the space of all
complex-valued Lipschitz continuous functions on Σ. Moreover, for x=σ(s ) Σ
0
∈
and δ >0 let
(2.6) IΣ(x)= σ(s):s (s δ,s +δ)
δ { ∈ 0− 0 }
be the open interval in Σ with center x and length 2δ. In order to define the trace
of γ h in a generalized sense, for λ 0, h C0,1(Σ) and x Σ we set
λ
≤ ∈ ∈
e √ λx y lnδ
− − | − |
(2.7) (B h)(x)= lim h(y) dσ(y)+h(x) ;
λ
δց0(cid:20)ZΣ\IδΣ(x) 4π|x−y| 2π (cid:21)
due to technical reasons the case λ= 0 is included here although γ is defined for
λ
λ<0only. Itwillbe showninProposition4.5thatB isawell-defined,essentially
λ
selfadjoint operator in L2(Σ) for each λ 0 and that the domain of its closure B
λ
≤
is independent of λ. Note that the basic idea in the definition of B is to remove
λ
the singularity of γ h on Σ. We remark that the limit in the definition of B can
λ λ
also be viewed as the finite part in the sense of Hadamard of the first summand
as δ 0; cf. [51, Chapter 5]. A procedure of this type is frequently employed to
ց
define hypersingular integral operators.
With the help of B we can make the following definition.
λ
Definition2.2. Letλ<0. Forh domB wedefinethegeneralizedtrace(γ h)
λ λ Σ
∈ |
of γ h on Σ by
λ
(γ h) =B h L2(Σ), h domB .
λ Σ λ λ
| ∈ ∈
Accordingly, for a function u = u +γ h with u H2(R3) and h domB we
c λ c λ
∈ ∈
define its generalized trace u on Σ by
Σ
|
(2.8) u =u +(γ h) =u +B h.
Σ c Σ λ Σ c Σ λ
| | | |
5
Note that u is well-defined. Indeed, the representationofu asa sumis unique
Σ
since γ h H2|(R3) implies h = 0. Moreover, the definition of u is independent
λ Σ
∈ |
of the choice of λ<0; cf. Section 4.3.
Furthermore,notethattheexpression in(2.1)isnolongerformal,butmakes
α
A
sense as we have defined the generalized trace u . Now we are able to define the
Σ
|
Schr¨odingeroperator ∆ correspondingtothedifferentialexpressionin(1.1)in
Σ,α
−
a rigorous way.
Definition 2.3. For α R 0 the Schr¨odinger operator ∆ in L2(R3) with
Σ,α
∈ \{ } −
δ-interaction of strength 1 supported on Σ is defined by
α
1
∆ u= u= ∆u u δ ,
Σ,α α Σ Σ
− A − − α | ·
dom( ∆ )= u=u +γ h:u H2(R3), h domB , u L2(R3) ,
Σ,α c λ c λ α
− ∈ ∈ A ∈
where λ<0 is arbit(cid:8)rary and the generalized trace uΣ is defined in (2.8). (cid:9)
|
Observethattheoperator ∆ iswell-definedsincedomB andthetraceu
Σ,α λ Σ
− |
do not depend on the choice of λ. Note also that for α=+ we formally have
∞
∆ u= ∆u, dom( ∆ )=H2(R3),
Σ,+ Σ,+
− ∞ − − ∞
so that the Schr¨odinger operator with δ-interaction of strength 0 on Σ coincides
withthefreeLaplacian ∆ ;thiswillbemadepreciseinTheorem3.1(ii)below.
free
−
Remark 2.4. The definition of ∆ relies on the generalized trace in Defini-
Σ,α
−
tion 2.2 and, thus, on the operator B . As mentioned above, the operator B is
λ λ
designed in such a way that the singularity of γ h on Σ is removed; this is done
λ
here by the term lnδ. However, an alternative choice lnδ +c with an arbitrary δ-
2π 2π
independentconstantc Rcanbemade. Thisleadstoadifferentoperator ∆ ,
Σ,α
∈ −
which can be transformed into the operator in Definition 2.3 by adding the same
constant c to α. For instance, for c = ln2 one obtains the family of operators
−2π
considered in [60].
Remark 2.5. For a function u = u +γ h dom( ∆ ) with h C0,1(Σ) we
c λ Σ,α
∈ − ∈
denote by u(s,δ), s [0,L),the meanvalue ofu overacircle ofa sufficiently small
∈
radius δ >0 centered at σ(s) and being orthogonalto Σ in σ(s). According to [60,
Remark 3] (see also [28, 31]) the functions
b
u(s,δ) h (s) 1
0
h (s):=2πlim and h (s):= lim u(s,δ) ln
0 1
δ 0ln(1/δ) δ 0 − 2π δ
ց ց (cid:20) (cid:18) (cid:19)(cid:21)
are well-defined and cobntinuous on Σ and the function u satisfies the following
b
boundary condition
ln2
h (s)= α+ h (s).
1 0
2π
(cid:18) (cid:19)
In many-body physics with zero-range interactions a boundary condition of this
type is known as Skorniakov–Ter-Martirosiancondition; see [59] and also [22, 50].
3. Main results
In this section we present all main results of this paper. It will be shown that
∆ is selfadjoint and its spectral and scattering properties will be analyzed.
Σ,α
−
This section is focused on the main statements and does not contain their proofs;
these are postponed to Section 5 below. In the following we denote by σ ( ∆ ),
p Σ,α
−
6 J.BEHRNDT,R.L.FRANK,C.KU¨HN,V.LOTOREICHIK,ANDJ.ROHLEDER
σ ( ∆ ), and ρ( ∆ ) the point spectrum, essential spectrum, and resolvent
ess Σ,α Σ,α
− −
set of ∆ , respectively.
Σ,α
In t−he first theorem we check that ∆ is a selfadjoint operator in L2(R3),
Σ,α
−
prove a Birman–Schwinger principle for its negative eigenvalues and compare its
resolvent to the resolvent of the free Laplacian ∆ in a Krein type formula,
free
−
which also implies that the difference of the resolvents is compact.
Theorem 3.1. The Schro¨dinger operator ∆ in L2(R3) in Definition 2.3 is
Σ,α
−
selfadjoint. Moreover, the following assertions hold.
(i) For each λ < 0 the operator γ is an isomorphism between ker(α B )
λ λ
−
and ker( ∆ λ). In particular, for each λ<0
Σ,α
− −
λ σ ( ∆ ) if and only if α σ (B ).
p Σ,α p λ
∈ − ∈
(ii) The set ρ( ∆ ) ( ,0) is nonempty and for each λ ρ( ∆ )
Σ,α Σ,α
− ∩ −∞ ∈ − ∩
( ,0) the resolvent formula
−∞
(3.1) (−∆Σ,α−λ)−1 =(−∆free−λ)−1+γλ α−Bλ −1γλ∗
is valid. Furthermore, ∆Σ,α converges to (cid:0)∆free in(cid:1)the norm resolvent
− −
sense as α + .
→ ∞
(iii) For each λ ρ( ∆ ) ρ( ∆ ) the resolvent difference
Σ,α free
∈ − ∩ −
(3.2) ( ∆ λ) 1 ( ∆ λ) 1
Σ,α − free −
− − − − −
is compact and, in particular, σ ( ∆ )=[0, ).
ess Σ,α
− ∞
Next we investigate the resolvent difference of ∆ and the free Laplacian in
Σ,α
−
more detail.
Theorem 3.2. Let s (λ) s (λ) ... be the singular values of the resolvent
1 2
≥ ≥
difference of ∆ and ∆ in (3.2), counted with multiplicities. Then
Σ,α free
− −
1
s (λ)=O as k + .
k k2lnk → ∞
Inparticular,(3.2)belongstotheS(cid:16)chatten–(cid:17)vonNeumannidealS (L2(R3))foreach
p
p>1/2.
The logarithmic factor in the estimate for the singular values in the above theo-
remisrelatedtothefactthattheeigenvaluesofB behaveasymptoticallyas lnk,
λ − 2π
see Proposition 4.5 (iii).
Inthe followingtheoremweshowthatthe discretespectrumof ∆ isalways
Σ,α
−
finite and give estimates for the number N of negative eigenvalues, counted with
α
multiplicities. Let R= L and define the intervals
2π
ln(4R) ln(4R) 1 ln(4R)
I = ,+ , I = , ,
1 0
− 2π ∞ 2π − π 2π
(cid:20) (cid:19) (cid:20) (cid:19)
and
r+1 r
ln(4R) 1 1 ln(4R) 1 1
I = , , r =1,2,...,
r
2π − π 2j 1 2π − π 2j 1
(cid:20) j=1 − j=1 − (cid:19)
X X
which are disjoint and satisfy R= ∞r= 1Ir. Moreover,set
−
L L 1S 1 2
(3.3) d = dsdt 0,
Σ
4π σ(t) σ(s) − 4π τ(t) τ(s) ≥
Z0 Z0 (cid:12) | − | | − |(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
7
where σ is the parametrization of Σ fixed in the beginning of Section 2 and τ
denotes an arc length parametrization of a circle of radius R.
Theorem 3.3. Let α=0 and denote by N the number of negative eigenvalues of
α
6
∆ , counted with multiplicities. If α d ln(4R) then N =0. Otherwise,
− Σ,α − Σ ≥ 2π α
2r+1 N 2l+1,
α
≤ ≤
where r 1 and l 0 are such that α+d I and α d I . In particular,
Σ r Σ l
≥− ≥ ∈ − ∈
N is finite and the operator ∆ is bounded from below.
α Σ,α
−
In the next corollarythe upper and lowerbounds onthe number N of negative
α
eigenvalues in Theorem 3.3 are made more explicit. This also leads to an asymp-
totic bound N = e 2πα+O(1) as α . We mention that a slightly better
α −
→ −∞
asymptotic bound was obtained in [32]. For convenience we make a very small
technical restriction and consider the case α+d < ln(4R) 1 only.
Σ 2π − π
Corollary 3.4. Let α = 0 be such that α+d < ln(4R) 1 and denote by N
6 Σ 2π − π α
the number of negative eigenvalues of ∆ , counted with multiplicities. Then the
Σ,α
−
estimate
(3.4) 2Rc−1e−2πα−γ 1 4(e912 1)<Nα <2Rce−2πα−γ +1
− − −
holds, where γ 0.577216 is the Euler–Mascheroni constant and c := e2πdΣ. In
≈
particular, N =e 2πα+O(1) as α .
α −
→−∞
In the case where Σ is a circle we have d = 0 and hence from Theorem 3.3
Σ
and Corollary 3.4 we immediately obtain the following explicit expressions for the
number of negative eigenvalues. For a similar formula in a related context see [45]
(cf. also [19]).
Corollary 3.5. Let Σ be a circle of radius R in R3, let α = 0, and denote by
6
N the number of negative eigenvalues of ∆ , counted with multiplicities. If
α Σ,α
−
α ln(4R) then N =0. Otherwise,
≥ 2π α
N =2r+1, where r 0 is such that α I .
α r
≥ ∈
If α< ln(4R) 1 then the estimate
2π − π
Nα 2Re−2πα−γ <1+4(e912 1)
| − | −
holds.
Next, we investigate the behavior of the smallest eigenvalue of ∆ when
Σ,α
−
varying Σ among all curves of a given length L. It turns out that circles are the
unique maximizers of the minimum of the spectrum σ( ∆ ) in the case that
Σ,α
−
negative eigenvalues exist. The analog of the following theorem for curves in the
two-dimensional space was shown in [26, 29].
Theorem 3.6. Let be a circle in R3 of radius R= L and assume that Σ is not
T 2π
a circle. Let α< ln(4R). Then
2π
minσ( ∆ )<minσ( ∆ ),
Σ,α ,α
− − T
where ∆ denotes the Schro¨dinger operator with δ-interaction of strength 1
support−ed oTn,αthe circle . α
T
8 J.BEHRNDT,R.L.FRANK,C.KU¨HN,V.LOTOREICHIK,ANDJ.ROHLEDER
Finally, we regard the pair ∆ , ∆ as a scattering system consist-
free Σ,α
{− − }
ing of the unperturbed Laplacian ∆ and the singularly perturbed operator
free
−
∆ . The following corollary is an immediate consequence of Theorem 3.2 and
Σ,α
−
the Birman–Kreintheorem [15].
Corollary 3.7. The absolutely continuous spectrum of ∆ is given by
Σ,α
−
σ ( ∆ )=[0,+ ).
ac Σ,α
− ∞
Moreover, the wave operators for the scattering pair ∆ , ∆ exist and are
free Σ,α
{− − }
complete.
In the next theorem we express the scattering matrix of the scattering system
∆ , ∆ intermsofthelimitsofacertainexplicitoperatorfunction,using
free Σ,α
{− − }
a result in [11]; we refer to [6, 43, 58, 62] and Appendix A for more details on
scattering theory. For our purposes it is convenient to consider the symmetric
operator S in L2(R3) defined as
Su= ∆u, domS = u H2(R3):u =0 ,
Σ
− ∈ |
which turns out to be the intersection o(cid:8)f the selfadjoint oper(cid:9)ators ∆free and
−
∆ . Then S is a densely defined, closed, symmetric operator with infinite
Σ,α
−
defectnumbers. Furthermore,ingeneralS containsaselfadjointpartwhichcanbe
split off. More precisely, consider the closed subspace
H =span (ran(S λ))
1 − ⊥
λ∈C[\[0,∞)
of L2(R3) and let H =H . Then S admits the orthogonalsum decomposition
2 ⊥1
S =S S
1 2
⊕
with respect to the space decomposition L2(R3) = H H , where the closed
1 2
⊕
symmetricoperatorS iscompletelynon-selfadjointorsimple(cf.[3,ChapterVII])
1
in H and S is a selfadjoint operator in H with purely absolutely continuous
1 2 2
spectrum. In the following let L2(R,dλ, ) be a spectral representation of the
λ
H
selfadjoint operator S in H ; cf. [6, Chapter 4].
2 2
Theorem 3.8. Fix η <0 such that 0 ρ(B α) and define the operator function
η
C [0, ) λ N(λ) by ∈ −
\ ∞ ∋ 7→
ei√λx y ei√ηx y
| − | | − |
(3.5) (N(λ)h)(x)= h(y) − dσ(y),
4π x y
ZΣ | − |
where h L2(Σ) and x Σ. Then the following assertions hold.
∈ ∈
(i) ImN(λ) S (L2(Σ)) for all λ C [0, ) and the limit
1
∈ ∈ \ ∞
ImN(λ+i0):= limImN(λ+iε)
ε 0
ց
exists in S (L2(Σ)) for a.e. λ [0, ).
1
(ii) The function λ N(λ), λ ∈C [∞0, ), is a Nevanlinna function such
7→ ∈ \ ∞
that the limit
N(λ+i0):= limN(λ+iε)
ε 0
ց
exists in the Hilbert–Schmidt norm for a.e. λ [0, ). Moreover, for a.e.
∈ ∞
λ [0, ) the operator N(λ+i0)+B α is boundedly invertible.
η
∈ ∞ −
9
(iii) The space L2(R,dλ, ), where
λ λ
G ⊕H
:=ran ImN(λ+i0) for a.e. λ [0, ),
λ
G ∈ ∞
forms a spectral rep(cid:0)resentation of(cid:1) ∆ .
free
−
(iv) Thescatteringmatrix S(λ) λ R ofthescatteringsystem ∆free, ∆Σ,α
acting in the space L2{(R,dλ}, ∈ ) admits the represe{n−tation − }
λ λ
G ⊕H
S (λ) 0
S(λ)= ′
0 I
(cid:18) Hλ(cid:19)
for a.e. λ [0, ), where
∈ ∞
1
S′(λ)=IGλ −2i ImN(λ+i0) N(λ+i0)+Bη−α − ImN(λ+i0).
p (cid:0) (cid:1) p
4. The operator B and the generalized trace
λ
In this section we discuss properties of the operator B in (2.7) and of the
λ
generalized trace defined in (2.8). We verify that the latter is well-defined and
independent ofλ. Our investigationofthe operatorB is split intotwo parts: first
λ
the special case of a circle Σ is treated, and afterwards the results are extended by
perturbation arguments to the general case.
4.1. Properties of B for a circle. Throughoutthis subsection we assume that
λ
Σ is a circle of radius R = L. Without loss of generality we assume that Σ lies
2π
in the xy-plane and is centered at the origin. We will make use of its arc length
parametrization
σ :[0,L] R3, σ(t)= Rcos(2πt/L),Rsin(2πt/L),0
→
and occasionally use the formula (cid:0) (cid:1)
π
(4.1) σ(s) σ(t) =2Rsin s t , s,t [0,L],
| − | | − |L ∈
(cid:16) (cid:17)
which holds for elementary geometric reasons. Furthermore, for x= σ(t) Σ and
∈
δ >0 let IΣ(x) be the open interval in Σ with center x and length 2δ as in (2.6).
δ
Let us first prove the following preliminary lemma. Its proof is partly inspired
by [60, Lemma 1].
Lemma 4.1. Let λ 0 and x Σ. Then the limit
≤ ∈
e √ λx y lnδ
− − | − |
k := lim dσ(y)+
λ
δց0(cid:20)ZΣ\IδΣ(x) 4π|x−y| 2π (cid:21)
exists in R, is independent of x and equals
π2 e−√−λ·2Rsin(s) 1 ln(4R)
k = − ds+ .
λ
2πsin(s) 2π
Z0
In particular, k as λ .
λ
→−∞ →−∞
Proof. First of all, it follows from the symmetry of the circle Σ that k is indeed
λ
independent of x (if it exists). Hence, without loss of generality, we can choose
10 J.BEHRNDT,R.L.FRANK,C.KU¨HN,V.LOTOREICHIK,ANDJ.ROHLEDER
x=σ(0). Using (4.1) and the substitution s= πt we obtain
L
e−√−λ|x−y| L−δ e−√−λ·2Rsin(Lπt)
dσ(y)= dt
4π x y 4π 2Rsin(πt)
ZΣ\IδΣ(x) | − | Zδ · L
π−Lπδ e−√−λ·2Rsin(s)
= ds,
4πsin(s)
ZLπδ
where we have used π = 1 in the last equality. As sin(π s)=sin(π +s) for all
s R it follows L 2R 2 − 2
∈
e √ λx y lnδ
− − | − |
dσ(y)+
4π x y 2π
ZΣ\IδΣ(x) | − |
= π2 e−√−λ·2Rsin(s) ds+ ln(2δR)−ln(π2)+ln(πR)
2πsin(s) 2π
δ
(4.2) Z2R
1 π2 e−√−λ·2Rsin(s) π2 1
= ds ds+ln(πR)
2π(cid:20) Z2δR sin(s) −Z2δR s (cid:21)
1 π2 e−√−λ·2Rsin(s) 1 π2 1 1
= − ds+ ds+ln(πR) .
2π(cid:20) Z2δR sin(s) Z2δR sin(s) − s (cid:21)
With d ln(sin(s/2)) ln(cos(s/2)) = 1 , s (0,π), we get
ds − sins ∈ 2
π
(cid:0) 2 1 (cid:1) 1 4
ds=ln .
sin(s) − s π
Z0 (cid:18) (cid:19) (cid:18) (cid:19)
Hence in the limit δ 0 the equation (4.2) becomes
ց
π2 e−√−λ·2Rsin(s) 1 ln(4R)
k = − ds+ .
λ
2πsin(s) 2π
Z0
In particular, k exists and is finite. By monotone convergence we have
λ
π2 1 e−√−λ·2Rsin(s) π2 1 π2 1
− ds ds ds=+
sin(s) → sin(s) ≥ s ∞
Z0 Z0 Z0
as λ , and hence k as λ . (cid:3)
λ
→−∞ →−∞ →−∞
As a first step towards the study of the operator B on the circle we show
λ
properties of B in the following lemma.
0
Lemma 4.2. Consider the operator B in (2.7), i.e.,
0
1 lnδ
(B h)(x)= lim h(y) dσ(y)+h(x) , h C0,1(Σ).
0
δց0(cid:20)ZΣ\IδΣ(x) 4π|x−y| 2π (cid:21) ∈
Then the following assertions hold.
(i) B is a well-defined, essentially selfadjoint operator in L2(Σ).
0
(ii) B is bounded from above, has a compact resolvent, and its eigenvalues
0
ν (0), k = 1,2,..., ordered nonincreasingly and counted with multiplici-
k
ties, are given by
k
ln(4R) ln(4R) 1 1
ν (0)= , ν (0)=ν (0)= .
1 2k 2k+1
2π 2π − π 2j 1
j=1 −
X