Radial positive definite functions and spectral theory of the Schr\"odinger operators with point interactions

0.28 MB·

by  N. Goloshchapova

MathematischeNachrichten,24January2017
Radial positive definite functions and spectral theory of the Schro¨dinger
operators with point interactions
N.Goloshchapova1, ,M.Malamud1, ,andV.Zastavnyi2,
∗ ∗∗ ∗∗∗
1 R.Luxemburgstr.74, 83114Donetsk,Ukraine
7 2 Univesitetskajastr.24, 83001Donetsk,Ukraine
1
0
2
Keywords Schro¨dingeroperator,pointinteractions,self-adjointextension,spectrum,positivedefinitefunction

n MSC(2000) 47A10,47B25
a
J Dedicatedtothe75thanniversaryofEduardTsekanovskii.

3
WecompletetheclassicalSchoenbergrepresentationtheoremforradialpositivedefinitefunctions. Weapply
2
thisresulttostudy spectral propertiesof self-adjoint realizations of two-and three-dimensional Schro¨dinger

operatorswithpointinteractionsonafiniteset.Inparticular,weprovethatanyrealizationhaspurelyabsolutely
]
P continuousnon-negativespectrum.
S
Copyrightlinewillbeprovidedbythepublisher
.
h
t
a
m
1 Introduction
[
An importanttopic in quantummechanicsis the spectral theoryof Schro¨dingeroperatorson the Hilbertspace
1 L2(Rd), d 1,2,3 ,withpotentialssupportedonadiscrete(finiteorcountable)setofpointsofRd. Thereis
v ∈ { }
anextensiveliteratureonsuchoperators(see[1,3–5,8,10,20,22,26,28]andreferencestherein).Thefirstmath-
6
3 ematicalproblemistoassociateaself-adjointoperator(Hamiltonian)onL2(Rd)withthedifferentialexpression
4
6 m
0 Ld := ∆+ αjδ( xj), αj R, m N. (1.1)
− ·− ∈ ∈
1. Xj=1
0 Thereareatleasttwonaturalwaystoassociateself-adjointoperatorHX,αwiththefollowingdifferentialexpres-
7 sioninL2(R1)
1
: d2 m
v L := + α δ( x ), m N
i 1 −dx2 j ·− j ∈
X j=1
X
r for any fixed set α := α m R. The first one is based on the quadratic forms method. Another way to
a { j}j=1 ⊂
introduce local interactionson X := x m R is to consider maximal operator correspondingto L and
{ j}j=1 ⊂ 1
imposeboundaryconditionsatx , j 1,..,m (see[3]),i.e.,
j
∈{ }
dom(H )= f W2,2(R X) W1,2(R):f (x +) f (x )=α f(x ) .
X,α ′ j ′ j j j
{ ∈ \ ∩ − − }
Incontrasttoone-dimensionalcase,thedifferentialexpression(1.1)doesnotdefineanoperatorinL2(Rd),
d 2,bymeansofthequadraticformssincethelinearfunctionalδ : f f(x)isnotcontinuousinW1,2(Rd)
x
≥ →
ford 2.However,itisstillpossibletoapplytheextensiontheoryapproach.Namely,F.BerezinandL.Faddeev
≥
intheirpioneeringpaper[8]proposedtoconsider(1.1)(withm = 1andd = 3)intheframeworkofextension
theory.TheyassociatedwithL thefamilyofallself-adjointextensionsofthefollowingsymmetricoperator
d
H := ∆, dom(H):= f W2,2(Rd):f(x )=0, j 1,..,m , m N. (1.2)
j
− ∈ ∈{ } ∈
∗ Correspondingauthor E-mail:nataliia@i(cid:8)me.usp.br, (cid:9)
∗∗ E-mail:[email protected].
∗∗∗ E-mail:[email protected].
Copyrightlinewillbeprovidedbythepublisher
2 N.Goloshchapova,M.Malamud,andV.Zastavnyi:SpectraltheoryoftheSchro¨dingeroperatorswithpointinteractions
It is wellknownthatH is closed non-negativesymmetricoperatorwith equaldeficiencyindicesn (H) = m
(see[3]). In[3],theauthorsproposedtoassociatewiththeHamiltonian(1.1)certainm-parametricfa±milyH(d)
X,α
describinglocalpointinteractions. Theyparameterizedthefamilyintermsoftheresolvents. Thelatterenabled
theauthorstoobtainanexplicitdescriptionofthespectrumforanyoperatorfromthefamilyH(d) .
X,α
Intherecentpublications[5,10,20], boundarytripletsandthecorrespondingWeylfunctionstechnique(see
[12,19] and also Section 2.1) was involved to investigate multi-dimensionalSchro¨dinger operators with point
interactions(thecasesd 2,3 ). Inthepresentpaper,weapplyboundarytripletsapproachtoparametrizeall
∈ { }
self-adjointextensionsofH.Besides,usingWeylfunctionstechnique,weinvestigatetheirspectra.Moreover,we
substantiallyinvolvethetheoryofradialpositivedefinitefunctions[2,chapterV]inourapproach. Inparticular,
weemploystrictpositivedefiniteness(seeDefinition3.1)ofthefunctionssinss|·| andtheBesselfunctionsJ0(s|·|)
with anys > 0 onR3 andR2, respectively,toprovepureabsolutecontinu|i·t|yofnon-negativespectrumofany
self-adjointrealizationofL . ForthispurposewecompletetheclassicalSchoenbergtheorem[31]regardingthe
d
integralrepresentationofradialpositivedefinitefunctions.
The paper is organized as follows. Section 2 is introductory. It contains definitions of a boundary triplet
and the corresponding Weyl function [12,19] and also facts about the Weyl functions [9,23]. In Section 3,
wecompleteSchoenbergtheorembyestablishingstrictpositivedefinitenessofanynon-constantradialpositive
definitefunctiononRn, n 2.InSections4and5,weinvestigate3D-and2D-Schro¨dingeroperatorswithpoint
≥
interactions,respectively.
Namely,inSubsection4.1(resp.,5.1),wedefineboundarytripletΠ = ,Γ ,Γ forH andcomputethe
0 1 ∗
{H }
correspondingWeylfunction. ItappearstobeclosetothatcontainedinKrein’sresolventformulaforthefamily
H(d) in[3]. Inparticular,fortheproofofthesurjectivityofthemappingΓ = (Γ ,Γ ) weemploythestrict
X,α 0 1 ⊤
positivedefinitenessofthefunctione onRnforany n N.
−|·|
∈
Subsections4.2and 5.2 are devotedto the spectralanalysis of the self-adjointrealizationsof L . To inves-
d
tigate the absolutely continuousspectrum we apply technique elaborated in [9,23]. For this purpose we need
invertibilityofthematrices
m
δ + sin(√x|xk−xj|) and J (√xx x ) m , x R ,
kj √xx x +δ 0 | j − k| j,k=1 ∈ +
(cid:18) | k− j| kj(cid:19)j,k=1
(cid:0) (cid:1)
whichweextractfromthestrictpositivedefinitenessofthefunctions sinss|·| andJ0(s|·|), s>0,onR3andR2,
respectively.WeemphasizethatintheproofofTheorems4.7and5.7ou|r·|complementtoSchoenbergtheoremis
usedinfullgenerality.Indeed,itfollowsfromtheintegralrepresentation(3.2)thatthestrictpositivedefiniteness
of sinss|·| andJ0(s|·|)foralls > 0yieldsthestrictpositivedefinitenessofanyradialpositivedefinitefunction
f onR|·|3andR2,respectively.
Finally, descriptionof the non-negativeself-adjointextensionsof H (d = 3) is providedin Subsection 4.3.
For suitable choice of a boundarytripletΠ strongresolventlimit of the correspondingWeyl functionM(x) at
x = 0appearstobepositivedefinitematrixinaviewofstrictpositivedefinitenessofthefunction 1−e−|·| on
Rnforany n N. e |·|f
∈
Notation.LetHand denoteseparableHilbertspaces;[H, ]denotesthespaceofboundedlinearoperators
H H
from H to , [ ] := [ , ]; the set of closedoperatorsin is denotedby ( ). Let A be a linear operator
H H H H H C H
inaHilbertspaceH. Inwhatfollowsdomain,kernel,andrangeofAaredenotedbydom(A),ker(A),ran(A),
respectively;σ(A)andρ(A)denotethespectrumandtheresolventsetofA;N denotesthedefectsubspaceof
z
A; C[0, )denotestheBanachspaceoffunctionscontinuousandboundedon[0, ).
∞ ∞
2 Extension theory ofsymmetricoperators
2.1 Boundarytripletsandproperextensions
In this subsection, we recall basic notions and facts of the theory of boundary triplets (we refer the reader to
[12,13,19]foradetailedexposition).InwhatfollowsAalwaysdenotesaclosedsymmetricoperatorinseparable
HilbertspaceHwithequaldeficiencyindicesn (A)=n (A) .
+
− ≤∞
Copyrightlinewillbeprovidedbythepublisher
mnheaderwillbeprovidedbythepublisher 3
Definition2.1 [19] AtotalityΠ = ,Γ ,Γ iscalledaboundarytripletfortheadjointoperatorA ofA
0 1 ∗
{H }
if isanauxiliaryHilbertspaceandΓ ,Γ : dom(A ) arelinearmappingssuchthat
0 1 ∗
H →H
(i)thefollowingabstractsecondGreenidentityholds
(A∗f,g)H (f,A∗g)H =(Γ1f,Γ0g) (Γ0f,Γ1g) , f,g dom(A∗); (2.1)
− H− H ∈
(ii)themappingΓ:=(Γ ,Γ ) :dom(A ) issurjective.
0 1 ⊤ ∗
→H⊕H
WithaboundarytripletΠ= ,Γ ,Γ forA oneassociatestwoself-adjointextensionsofAdefinedby
0 1 ∗
{H }
A0 :=A∗↾ker(Γ0) and A1 :=A∗↾ker(Γ1).
Definition2.2 (i)AclosedextensionAofAiscalledproperifA A A . Thesetofallproperextensions
∗
⊆ ⊆
ofAisdenotedbyExt .
A
(ii)TwoproperextensionsA andA eofAarecalleddisjointifdome(A ) dom(A )=dom(A).
1 2 1 2
∩
Remark2.3 (i)ForanysymmetricoperatorAwithn (A) = n (A), aboundarytripletΠ = ,Γ ,Γ
+ 0 1
forA existsandisnotunique[1e9].Itiseknownalsothatdim =n −(A)aendkerΓ=keer(Γ ,Γ ) {=Hdom(A)}.
∗ 0 1 ⊤
H ±
(ii)Moreover,foreachself-adjointextensionAofA thereexistsa boundarytripletΠ = ,Γ ,Γ such
0 1
{H }
thatA=A ↾ker(Γ )=:A .
∗ 0 0
(iii) For each boundarytriplet Π = ,Γ ,Γe for A and each bounded self-adjoint operator B in a
0 1 ∗
{H } H
tripleetΠB ={H,ΓB0,ΓB1}withΓB1 :=Γ0andΓB0 :=BΓ0−Γ1isalsoaboundarytripletforA∗.
AroleofaboundarytripletforA intheextensiontheoryissimilartothatofcoordinatesystemintheanalytic
∗
geometry.Namely,itallowsonetoparameterizethesetExt bymeansoflinearrelationsin inplaceofJ.von
A
H
Neumannformulas.Toexplainthiswerecallthefollowingdefinitions.
Definition2.4 (i)AclosedlinearrelationΘin isaclosedsubspaceof .
H H⊕H
(ii)AlinearrelationΘissymmetricif(g ,f ) (f ,g )=0forall f ,g , f ,g Θ.
1 2 1 2 1 1 2 2
− { } { }∈
(iii)TheadjointrelationΘ isdefinedby
∗
Θ∗ = k,k′ :(h′,k)=(h,k′) forall h,h′ Θ .
{{ } { }∈ }
(iv)AclosedlinearrelationΘiscalledself-adjointifbothΘandΘ aremaximalsymmetric,i.e.,theydonot
∗
admitsymmetricextensions.
For the symmetric relation Θ Θ in the multivalued part mul(Θ) is the orthogonal complement of
∗
⊆ H
dom(Θ)in . Setting := dom(Θ)and = mul(Θ),oneverifiesthatΘcanberewrittenasthedirect
op
H H H∞
orthogonalsumofaself-adjointoperatorΘ inthesubspace anda“pure”relationΘ = 0,f : f
op op ′ ′
H ∞ { } ∈
mul(Θ) inthesubspace .
H∞ (cid:8)
Proposition2.5 [12,19]LetΠ= ,Γ ,Γ beaboundarytripletforA . Thenthemapping
(cid:9) {H 0 1} ∗
Ext A:=A Θ:=Γ(dom(A))= Γ f,Γ f : f dom(A) (2.2)
A Θ 0 1
∋ → {{ } ∈ }
establishesabijectivecorrespondencebetweenthesetofallclosedproperextensionsExt ofAandthesetof
e e e A
allclosedlinearrelations ( )in . Furthermore,thefollowingassertionshold.
C H H
(i)Theequality(A ) =A holdsforanyΘ ( ).
Θ ∗ Θ
∗ ∈C H
(ii)TheextensionA ien(2.2)issymmetric(self-adjoint)ifandonlyif Θissymmetric(self-adjoint). More-
Θ
over,n (A )=n (Θ). e
Θ
± ±
(iii)If,inaddition,theclosedextensionsA andA aredisjoint,then(2.2)takestheform
Θ 0
A =A =A ↾dom(A ), dom(A )=dom(A )↾ker Γ BΓ , B ( ).
Θ B ∗ B B ∗ 1 0
− ∈C H
(cid:0) (cid:1)
2.2 Weylfunction,γ-fieldandspectraofproperextensions
ItisknownthatWeylfunctionisanimportanttoolinthespectraltheoryofsingularSturm-Liouvilleoperators.In
[12,13]theconceptofWeylfunctionwasgeneralizedtoanarbitrarysymmetricoperatorAwithequaldeficiency
indices. InthissubsectionwerecallbasicfactsaboutWeylfunctions.
Copyrightlinewillbeprovidedbythepublisher
4 N.Goloshchapova,M.Malamud,andV.Zastavnyi:SpectraltheoryoftheSchro¨dingeroperatorswithpointinteractions
Definition2.6 [12]LetΠ= ,Γ ,Γ beaboundarytripletforA .TheoperatorvaluedfunctionM():
0 1 ∗
{H } ·
ρ(A ) [ ]definedby
0
→ H
M(z)Γ f =Γ f , f N , z ρ(A ), (2.3)
0 z 1 z z z 0
∈ ∈
iscalledtheWeylfunction,correspondingtothetripletΠ.
ThedefinitionoftheWeylfunctioniscorrectandtheWeylfunctionM()isNevanlinnaorR-function.
·
Inthefollowingwewillbeconcernedwithasimplesymmetricoperators.RecallthatasymmetricoperatorA
issaidtobesimpleifthereisnonontrivialsubspacewhichreducesittoself-adjointoperator.
Thespectrumandtheresolventsetoftheclosed(notnecessarilyself-adjoint)extensionsofsimplesymmetric
operatorAcanbedescribedwiththehelpofthefunctionM(). Namely,thefollowingpropositionholds.
·
Proposition2.7 LetAbeadenselydefinedsimple symmetricoperatorinH, Θ ( ), A Ext , and
Θ A
∈ C H ∈
z ρ(A ). Thenthefollowingequivalenceshold.
0
∈
(i)z ρ(A ) 0 ρ(Θ M(z)); e
Θ
∈ ⇐⇒ ∈ −
(ii) z σ (A ) 0 σ (Θ M(z)), τ p, c, r ;
τ Θ τ
∈ ⇐⇒ ∈ − ∈{ }
(iii)f ker(A z) Γ f ker(Θ M(z)) and dimker(A z)=dimker(Θ M(z)).
z Θ 0 z Θ
∈ − ⇐⇒ ∈ − − −
The followingpropositiongivesus quantitativecharacterizationof the negativespectrumof self-adjointex-
tensionsoftheoperatorA.
Proposition 2.8 [12] Let A be a densely defined non-negative symmetric operator in H, and let Π =
,Γ ,Γ be a boundarytriplet for A . Let also M() be the correspondingWeyl function and A = A ,
0 1 ∗ 0 F
{H } ·
whereA istheFriedrichsextensionofA. Thenthefollowingassertionshold.
F
(i) The strong resolvent limit M(0) := s R limM(x) exists and is self-adjoint linear relation semi-
− − x 0
boundedfrombelow. ↑
(ii) If, in addition, M(0) [ ], then the number of negative squares of A = A equals the number of
∈ H Θ ∗Θ
negativesquaresoftherelationΘ M(0),i.e.,κ (A )=κ (Θ M(0)).
Θ
− − − −
Inparticular,theself-adjointextensionA ofAisnon-negativeifandonlyifthelinearrelationΘ M(0)is
Θ
−
non-negative.
Denote
M(x+i0):=s limM(x+iy), d (x):=rank(Im(M(x+i0))),
M
−y 0
↓
M (z):=(M(z)h,h), Ω (M ):= x R: 0<Im(M (x+i0))<+ , z C , h ,
h ac h h +
{ ∈ ∞} ∈ ∈H
whereM (x+i0):=lim (M(x+iy)h,h). SinceIm(M (z))>0, z C ,thelimitM (x+i0)existsand
h y 0 h + h
isfinitefora.e. x R. ↓ ∈
∈
Tostatethenextpropositionweneedaconceptoftheabsolutelycontinuousclosurecl (δ)ofaBorelsubset
ac
δ Rintroducedin[9]and[16]. Wereferto[16,23]forthedefinitionandbasicproperties.
⊂
Proposition2.9 [9,23]LetAbeasimpledenselydefinedclosedsymmetricoperatorwithequaldeficiency
indices in separable Hilbert space H. Let Π = ,Γ ,Γ be a boundary triplet for A and M() the cor-
0 1 ∗
{H } ·
responding Weyl function. Assume also that τ = h N , 1 N is a total set in .Let also
{ k}k=1 ≤ ≤ ∞ H
A =A ↾ker(Γ BΓ ),withB =B ( ). Thenthefollowingassertionshold.
B ∗ 1 0 ∗
− ∈C H
(i)TheoperatorA hasnosingularcontinuousspectrumwithintheinterval(a,b)ifforeachk 1,2,..,N
0
∈{ }
theset(a,b) Ω (M )iscountable.
(ii)Ifthe\limiatcM(xhk+i0)existsfora.e.x R,thenσ (A )=cl (supp(d (x))).
ac 0 ac M
(iii)ForanyBorelsubset Rtheabsolu∈telycontinuouspartsA Eac( )andA Eac ( )oftheoperators
D ⊂ 0 A0 D B AB D
A E ( )andA E ( )areunitarilyequivalentifandonlyif d (x)=d (x)fora.e. x .
0 A0 D B AB D M MB ∈D
3 Positivedefinite functions. Complement oftheSchoenberg theorem
Let(u,v)=u v +...+u v beascalarproductoftwovectorsu=(u ,...,u )andv =(v ,...,v )from
1 1 n n 1 n 1 n
Rn, n N, and let u = (u,u) be Euclidean norm. Recall some basic facts and notions of the theory of
∈ | |
positivedefinitefunctions[2].
p
Copyrightlinewillbeprovidedbythepublisher
mnheaderwillbeprovidedbythepublisher 5
Definition3.1 [2,33](i)Functiong() : Rn CissaidtobepositivedefiniteonRn andisreferredtothe
classΦ(Rn)ifitiscontinuousat0andfor·anyfin→itesubsetsX := x m Rnandξ := ξ m C, m
Nthefollowinginequalityholds { k}k=1 ⊂ { k}k=1 ⊂ ∈
m
ξ ξ g(x x ) 0. (3.1)
k j k− j ≥
k,j=1
X
(ii)Moreover,g() issaid tobe strictlypositiveonRn ifthe inequality(3.1) isstrictforanysubsetofdistinct
pointsX = x m· Rnandforanysubsetξ = ξ m Csatisfyingcondition m ξ >0.
{ k}k=1 ⊂ { k}k=1 ⊂ k=1| k|
Clearly, positivedefinitenessofthe functiong()isequivalentto thenon-negativedefinitenessofthematrix
G(X) = (g )m with g = g(x x ) for· any subset X = x m RnP, while its strict positive
kj k,j=1 kj k − j { k}k=1 ⊂
definiteness is equivalent to (strict) positive definiteness of the matrix G(X) for any subset of distinct points
X = x m Rn.
Th{efko}llko=w1in⊂gclassicalBochnertheoremgivesthedescriptionoftheclassΦ(Rn).
Theorem3.2 [11]Afunctiong()ispositivedefiniteonRnifandonlyif
·
g(x)= ei(u,x)dµ(u),
RZn
whereµisafinitenon-negativeBorelmeasureonRn.
Definition 3.3 A function f() C[0,+ ) is said to be radial positive definite function of the class Φ ,
n
n N,iff( )ispositivedefini·te∈onRn,i.e∞.,iff( ) Φ(Rn).
∈ |·| |·| ∈
AcharacterizationoftheclassΦ isgivenbythefollowingSchoenbergtheorem[30,31](seealso[2,Theo-
n
rem5.4.2]).
Theorem3.4 Functionf()belongstotheclassΦ ifandonlyif
n
·
+
∞
f(t)= Ω (st)dµ(s), t 0, (3.2)
n
≥
Z0
whereµisanon-negativefiniteBorelmeasureon[0, ),and
∞
n 2 n−22 ∞ t2 p Γ n
Ωn(t)=Γ 2 t Jn−22(t)= −4 p!Γ n2+p , (3.3)
(cid:16) (cid:17) (cid:18) (cid:19) Xp=0(cid:18) (cid:19) (cid:0)2 (cid:1)
(cid:0) (cid:1)
Ω (x)= ei(u,x)dν (u), x Rn. (3.4)
n n
| | ∈
ZSn
Hereν istheBorelmeasureuniformlydistributedovertheunitsphereS centeredattheoriginandν (S )=1.
n n n n
Remark3.5 Itisnotdifficulttoshowthatforanyn NthestrictinclusionΦ Φ takesplace. Indeed,
n+1 n
∈ ⊂
itisknown[18],[34,Theorem5],[35,Example1]that(1 t)δ Φ ifandonlyifδ n+1. EarlierAskey[6]
−| | + ∈ n ≥ 2
andTrigub[32]consideredthecaseofnaturalδandprovedthenecessityforoddn.
Ontheotherhand,thestrictinclusionΦ Φ followsfromanotherexample. Namely,Re(e zt) Φ
n+1 n − n
ifandonlyif argz π/(2n), z C(see[34⊂,Theorem3]).ThereforeΩ Φ ,butΩ Φ ,n N∈.
n n n n+1
| |≤ ∈ ∈ 6∈ ∈
OurcomplementtotheSchoenbergtheoremreadsasfollows.
Theorem3.6 Iff() Φ ,n 2,andf() conston[0,+ ),thenthefunctionf( )isstrictlypositive
n
definiteonRn. · ∈ ≥ · 6≡ ∞ |·|
Westartwiththefollowingauxiliarylemmaandpresenttwodifferentproofs.
Lemma 3.7 LetSr(y)beasphereofradiusr inRn centeredaty andn 2, letalsoX = x m bea
subsetofRnsuchthatnx =x asp=j andξ = ξ m C. If ≥ { k}k=1
p 6 j 6 { k}k=1 ⊂
m
ξ ei(u,xp) =0, forall u Sr(y), (3.5)
p ∈ n
p=1
X
thenξ =0forp 1,..,m .
p
∈{ }
Copyrightlinewillbeprovidedbythepublisher
6 N.Goloshchapova,M.Malamud,andV.Zastavnyi:SpectraltheoryoftheSchro¨dingeroperatorswithpointinteractions
Thefirstproof. Withoutlossofgeneralitywemayassumethaty = 0andr = 1. Letalsofordefiniteness
ξ = 0. For m = 1 the statement is obvious. Put m 2. Denote max x = R > 0. Let e n be
1 6 ≥ 1 p m| p| 0 { j}j=1
thestandardorthogonalbasisinRn. We mayassumethatx = R e .≤L≤etP betheorthogonalprojectoronto
1 0 1
span e ,e . Then Px = r (cosϕ e +sinϕ e ) with 0 r = Px x R , 0 ϕ < 2π.
1 2 p p p 1 p 2 p p p 0 p
{ } ≤ | | ≤ | | ≤ ≤
Assumethatr = r = ... = r = R andr < R asp > m (ifm < m). ThenPx = x , 1 p m.
1 2 m 0 p 0 ′ ′ p p ′
′ ≤ ≤
Evidently,wemayalso assumethat0 = ϕ < ϕ < ... < ϕ < 2π. Putin(3.5)u = cosϕe +sinϕe
1 2 m 1 2
S1(0), ϕ R.Thereforetheequality(3.5)takestheform ′ ∈
n ∈
m
ξ exp(ir cos(ϕ ϕ ))=0, ϕ R. (3.6)
p p p
− ∈
p=1
X
It is well known that generating function for the Bessel functions admits the following representation [15,
chapter19, 3]
§
ea2(z−z−1) = ∞ Jk(a)zk, z =0, a C. (3.7)
6 ∈
k=
X−∞
Puttingin(3.7)z = iei(ϕ−ϕp) anda = rp, p 1,..,m ,wearriveatthefollowingexpansionintoFourier
∈ { }
seriesforthefunctionsf (ϕ)=exp(ir cos(ϕ ϕ ))
p p p
−
∞
exp(irpcos(ϕ ϕp))= J(rp)ike−ikϕpeikϕ. (3.8)
−
k=
X−∞
Itiseasilyseenthatfrom(3.6)and(3.8)followstheequality
m
∞
ξ exp( ikϕ )J (r )ik eikϕ =0.
p p k p
" − #
k= p=1
X−∞ X
Therefore
m
ξ exp( ikϕ )J (r )=0, k N. (3.9)
p p k p
− ∈
p=1
X
UsingthefollowingexpansionintoseriesforJ (x)([27,Section2])
k
J (x)= x k ∞ x2 p 1 ,
k 2 − 4 p!Γ(k+p+1)
(cid:0) (cid:1) Xp=0(cid:16) (cid:17)
wegetk!2kJ (x)=xk[1+a (x)],where
k k
a (x) (k+1) 1 exp x2 1 , x R, k N.
| k |≤ − 4 − ∈ ∈
h (cid:16) (cid:17) i
Multiplyingtheequality(3.9)byk!2kR0−k,weobtain
m′ m
ξpexp(−ikϕp)[1+ak(R0)]=− ξpexp(−ikϕp)(rpR0−1)k[1+ak(rp)],
p=1 p=m+1
X X′
whereright-handsideequals0ifm′ =m. Ifm′ <m,thenrpR0−1 <1asp>m′. Thereby,
m′
lim ξ exp( ikϕ )=0. (3.10)
p p
k −
→∞p=1
X
Copyrightlinewillbeprovidedbythepublisher
mnheaderwillbeprovidedbythepublisher 7
Since arithmeticmeansofthe sequencein (3.10) convergesto ξ (ϕ = 0), thenξ = 0. Thus, the theoremis
1 1 1
proved.
The second proof. As in the first proof, we reduce considerations to investigation of equality (3.6). By
uniquenesstheoremforanalyticfunctions,equality(3.6)remainsvalidforanyz C
∈
m
ξ exp(ir cos(z ϕ ))=0, z =x+iy C. (3.11)
p p p
− ∈
p=1
X
Since,byEulerformula,cos(z ϕp)=(ei(z−ϕp)+e−i(z−ϕp))/2,wehave
−
r
exp(irpcos(z ϕp)) =exp( p Im(ei(z−ϕp)+e−i(z−ϕp)))=exp(rpshysin(x ϕp)).
| − | − 2 −
Denoteψ (z)=arg(exp(ir cos(z ϕ ))) [0,2π).Thus,by(3.11),
p p p
− ∈
m
ξ [exp(r shysin(x ϕ ))]eiψp(z) =0. (3.12)
p p p
−
p=1
X
Multiplying(3.12)byexp( R shy),wearriveat
0
−
m′ m
ξ [exp(R shy(sin(x ϕ ) 1))]eiψp(z)+ ξ [exp(shy(r sin(x ϕ ) R ))]eiψp(z) =0.
p 0 p p p p 0
− − − −
Xp=1 p=Xm′+1
(3.13)
Settingx=ϕ + π in(3.13)andpassingtothelimitasy + ,weobtainξ =0.
1 2 → ∞ 1
Remark 3.8 The first proof of Lemma 3.7 belongs to Viktor Zastavnyi. Chronologically it was obtained
earlierthenthesecondproofproposedbytheothertwoauthors.
Remark3.9 ItmightbeeasilyshownthatLemma3.7isnotvalidforanymanifoldinRn. Belowwegivethe
explanatoryexample. Itisobviousthatanyhyperplaneπ in Rn, n 2, whichdoesnotcontaintheorigin,is
a
≥
givenby
π = u Rn : (u,a)=1, where a Rn, a=0 .
a
{ ∈ ∈ 6 }
Thenonsuchhyperplanetheexpression1+exp(i(u,πa))isidenticallyzero. Thus,foranyfinitesetofhyper-
planeswiththeabovepropertythereexistsasetofpointsy Rn,y =0,k 1,..,q suchthat
k k
∈ 6 ∈{ }
q m q
0 1+ei(u,πyk) = ξ ei(u,xp), u π .
≡ p ∈ yk
kY=1(cid:16) (cid:17) Xp=1 k[=1
Herem 2,ξ >0, p 1,..,m andX = x m isasubsetofRnsuchthatx =x asp=j.
≥ p ∈{ } { p}p=1 p 6 j 6
NowwearereadytoprovethecomplementofTheorem3.4. Belowwepresenttwoslightlydifferentproofs.
ThefirstproofofTheorem3.6. Letµbenon-negativefiniteBorelmeasureon[0,+ )fromtherepresen-
∞
tation(3.2)forthefunctionf. Itisobviousthatµ((0,+ ))>0(otherwise,f(t) f(0)).
LetX = x m Rn andξ = ξ m Cbes∞ubsetssuchthatx = x ≡asp = j and m ξ > 0.
{ k}k=1 ⊂ { k}k=1 ⊂ p 6 j 6 k=1| k|
FromLemma3.7withy =0,r =1itfollowsthat
P
m 2
ξ ei(u,sxk) dν (u)>0 forany s>0,
k n
ZSn(cid:12)(cid:12)kX=1 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) Copyrightlinewillbeprovidedbythepublisher
(cid:12) (cid:12)
8 N.Goloshchapova,M.Malamud,andV.Zastavnyi:SpectraltheoryoftheSchro¨dingeroperatorswithpointinteractions
whereν isthemeasurefromrepresentation(3.4). Sinceµ((0,+ ))>0,weget
n
∞
m + m 2
ξ ξ f(x x )= ∞ ξ ei(u,sxk) dν (u) dµ(s)>0.
kX,j=1 k j | k− j| Z0 ZSn(cid:12)(cid:12)Xk=1 k (cid:12)(cid:12) n 
(cid:12) (cid:12)
 (cid:12) (cid:12) 
Thus,thetheoremisproved. (cid:12) (cid:12)
Thesecondproof.ByTheorem3.2,wehave
f(x)= ei(u,x)dµ(u), x Rn,
| | ZRn ∈
where µ is non-negativefinite Borel measure on Rn. It is easily seen that suppµ is a radial set, i.e., if x
0
∈
suppµ, thenthesupportcontainssphereSr(0)ofradiusr = x 0 centeredattheorigin. Iff(t) f(0),
n | 0| ≥ 6≡
thensuppµcontainsasphereSr(0).
n
Letf(t) const, m N, andthe set X = x m Rn is suchthatx = x as k = j. Considerthe
followingqu6≡adraticformi∈nCm { k}k=1 ⊂ k 6 j 6
m m 2
Q(ξ):= ξ ξ f(x x )= ξ ei(u,xk) dµ(u) 0, ξ = ξ m C.
kX,j=1 k j | k− j| ZRn(cid:12)(cid:12)Xk=1 k (cid:12)(cid:12) ≥ { k}k=1 ⊂
(cid:12) (cid:12)
(cid:12) (cid:12)
If Q(ξ) = 0, then the functiong(u) := mk=1ξkei((cid:12)u,xk) equals0(cid:12)on suppµ and thereforeequals0 on Snr(0).
Finally,Lemma3.7yieldsthatξ =0, k 1,..,m .
k
P∈{ }
Definition3.10 Afunctionf() C[0, ) C (0,+ )iscalledcompletelymonotonicfunctionon[0, )
∞
oftheclassM[0, )iftheinequa·lit∈y( 1)∞kf(k∩)(t) 0ho∞ldsforallk Z andt>0. ∞
+
∞ − ≥ ∈
Schoenbergnoted in [30,31] that the function f() Φ if and only if f(√) M[0, ). Thus, it is
n
· ∈ n N · ∈ ∞
easilyimpliedbySchoenbergtheoremthatf() M[0, )T∈yieldsf() Φ .
n
· ∈ ∞ · ∈n N
∈
T
Corollary 3.11 [33, Theorem 7.14] If f() M[0, ) and f() const on [0,+ ), then the function
f( )isstrictlypositivedefiniteonRnforan·yn∈ N. ∞ · 6≡ ∞
|·| ∈
Remark 3.12 (i) In [33, Lemma 6.7] assertion of Lemma 3.7 was proven provided that the equality (3.5)
holdsonacertainopensubsetofR.
(ii)Theorem3.6wasformulatedin[33,Theorem6.18]and[14,Theorem3.7]undertheadditionalcondition
tn 1f(t) L1[0, ).
−
∈ ∞
Example3.13 Letuspresentsomeexamplesofstrictlypositivefunctions.
(1)UsingtheequalityΓ(2p)= 22p−1Γ(p)Γ(p+1/2),weobtain
√π
Ω (t)=cost, Ω (t)=J (t), Ω (t)=sint/t.
1 2 0 3
ByTheorem3.6,thefunctionsΩ (sx)arestrictlypositivedefiniteonRnforanys>0andn 2.
n
| | ≥
(2)Itiseasilyseenthatthefunctionse tand(1 e t)/t= 1e tsdsarecompletelymonotonicon[0,+ ).
− − − 0 − ∞
Thus,byCorollary3.11,thefunctionse x and(1 e x)/x arestrictlypositivedefiniteonRnforalln N.
−| | − −| | | |R ∈
4 Three-dimensional Schro¨dinger operator withpointinteractions
4.1 BoundarytripletandWeylfunction
FirstwedefineaboundarytripletfortheoperatorH . Denoter := x x , x R3,letalso√ standsfor the
∗ j j
| − | ∈ ·
branchofthecorrespondingmultifunctiondefinedonC R bythecondition√1=1.
+
\
Copyrightlinewillbeprovidedbythepublisher
mnheaderwillbeprovidedbythepublisher 9
Proposition4.1 LetH betheminimalSchro¨dingeroperatordefinedby(1.2)andletξ := ξ m , ξ :=
ξ m Cm.Thenthefollowingassertionshold 0 { 0j}j=1 1
{ 1j}j=1 ∈
(i) The operator H is closed and symmetric. The deficiency indices of H are n (H) = m. The defect
subspaceN :=N (H)is ±
z z
m ei√zrj
N = c : c C, j 1,...,m , z C [0, ). (4.1)
z j j
{ 4πr ∈ ∈{ }} ∈ \ ∞
j
j=1
X
(ii)TheadjointoperatorH isgivenby
∗
dom(H∗)=f = m ξ0je−rrj +ξ1je−rj +fH : ξ0,ξ1 ∈Cm, fH ∈dom(H), (4.2)
j
 Xj=1(cid:0) (cid:1) 
H∗f = mξ0je−rj +ξ1j(e−rj 2e−rj) ∆fH.  (4.3)
− r − r −
j j
j=1
X(cid:0) (cid:1)
(iii)AtotalityΠ= ,Γ ,Γ ,where
0 1
{H }
=Cm, Γ f := Γ f m =4π lim f(x)x x m =4π ξ m , (4.4)
H 0 { 0j }j=1 {x→xj | − j|}j=1 { 0j}j=1
Γ f := Γ f m = lim f(x) ξ0j m , (4.5)
1 { 1j }j=1 {x→xj − |x−xj| }j=1
(cid:0) (cid:1)
formsaboundarytripletforH .
∗
(iv)TheoperatorH =H ↾ker(Γ )(=H )coincideswiththefreeHamiltonian,
0 ∗ 0 0∗
H = ∆, dom(H )=dom( ∆)=W2,2(R3).
0 0
− −
Proof. (i)Thestatement(i)ofProposition4.1iswell-known. Itwasobtainedintheclassicalbook[3](see
Theorem1.1.2).
(ii)Clearly,e−rj W2,2(R3) dom(H∗)forj 1,..,m .
∈ ⊂ ∈{ }
Since the function e−|x| is strictly positive definite on R3 (Example 3.13), the matrix (e−|xk−xj|)mk,j=1 is
positivedefinite. Thereforethefunctionse−rj, j 1,..,m arelinearlyindependent. Considertheoperator
∈ { }
H definedby
e H :=H∗ ↾dom(H), dom(H)=dom(H)+˙ span{e−rj}mj=1 =W2,2(R3). (4.6)
Sincedom(He) = dom( ∆) =eW2,2(R3)eandbothoperatorsH and ∆areproperextensionsofH,wehave
− −
H = ∆andtheoperatorH isself-adjoint.
−
Further,seincethefunctions e−rj N , j 1,..,m areelinearlyindependent,and,bythesecondJ.von
Neeumannformula, e 4πrj ∈ −1 ∈ { }
dim(dom(H∗)/dom(H))=n (H)=m,
±
representation(4.2)isproved. e
(iii)Letf,g dom(H ). By(4.2),wehave
∗
∈
f = m fk+fH, fk =ξ0k e−rk +ξ1ke−rk, g = m gk+gH, gk =η0k e−rk +η1ke−rk,
r r
k k
k=1 k=1
X X
wheref ,g dom(H),andξ , ξ ,η , η C, k 1,..,m .
H H 0k 1k 0k 1k
∈ ∈ ∈{ }
Copyrightlinewillbeprovidedbythepublisher
10 N.Goloshchapova,M.Malamud,andV.Zastavnyi:SpectraltheoryoftheSchro¨dingeroperatorswithpointinteractions
Applying(4.4)-(4.5)tof,g dom(H ),weobtain
∗
∈
m
Γ0f =4π{ξ0j}mj=1, Γ1f =−ξ0j + ξ0kex−|xj−xxk| + m ξ1ke−|xj−xk| ,
j k
 Xk6=j | − | kX=1 j=1
m
Γ0g =4π{η0j}mj=1, Γ1g =−η0j + η0kex−|xj−xxk| + m η1ke−|xj−xk| .
j k
 Xk6=j | − | Xk=1 j=1
(4.7)
 
Itiseasilyseenthat
(H∗f,g) (f,H∗g)= m ξ0jH∗ e−rj ,η1ke−rk ξ0j e−rj,η1kH∗(e−rk)
− r − r
k,j=1(cid:18)(cid:18) (cid:18) j (cid:19) (cid:19) (cid:18) j (cid:19)
X
+ ξ1jH∗(e−rj),η0k e−rk ξ1je−rj,η0kH∗ e−rk .
r − r
(cid:18) k (cid:19) (cid:18) (cid:18) k (cid:19)(cid:19)(cid:19)
UsingthesecondGreenformula,weget
H∗ e−rj ,e−rk e−rj,H∗(e−rk)
r − r
(cid:18) (cid:18) j (cid:19) (cid:19) (cid:18) j (cid:19)
= lim ∆ e−rj e−rkdx+ e−rj∆(e−rk)dx
r→∞(cid:18) Z − (cid:18) rj (cid:19) Z rj (cid:19)
Br(xj)\B1(xj) Br(xj)\B1(xj)
r r
= lim ∂ e−rj e−rk + e−rj ∂(e−rk) ds
r→∞ Z (cid:18)−∂n(cid:18) rj (cid:19) rj ∂n (cid:19)
Sr(xj)
+ lim ∂ e−rj e−rk e−rj ∂(e−rk) ds= 4πe−|xj−xk|, (4.8)
r→∞ Z (cid:18)∂n(cid:18) rj (cid:19) − rj ∂n (cid:19) −
S1(xj)
r
wherenstandsfortheexteriornormalvectortoS (x )andS (x ),respectively.
r j 1 j
r
Indeed,noticingthat ∂ e−rj = e−rj(1+ 1), wecaneasilyshowthatthefirstintegralintheformula
∂n rj − rj rj
(4.8)tendstozeroasrtends(cid:16)toinfi(cid:17)nity.Further,
lim ∂ e−rj e−rkds= lim ∂ e−|y| e−|y+xj−xk|ds
r→∞ Z ∂n(cid:18) rj (cid:19) r→∞ Z ∂n(cid:18) |y| (cid:19)
S1(xj) S1(0)
r r
4π
= lim e−1/rr(1+r)e−|y∗+xj−xk| = 4π lim e−|y∗+xj−xk| = 4πe−|xj−xk|,
−r→∞(cid:20)r2 (cid:21) − y∗→0 −
and
e−rj ∂(e−rk) e−|y|∂(e−|y+xj−xk|)
lim ds= lim ds
r→∞ Z rj ∂n r→∞ Z |y| ∂n
S1(xj) S1(0)
r r
=rl→im∞(cid:20)4rπ2e−1/rr∂(e−|y∂+nxj−xk|)|y=y∗∗(cid:21)= 0, y∗ ∈S1r(0), y∗∗ ∈Sr1(0).
Copyrightlinewillbeprovidedbythepublisher