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Advances
in
Mathematics
www.elsevier.com/locate/aim
Spectral
analysis
of
selfadjoint
elliptic
differential
operators,
Dirichlet-to-Neumann
maps,
and
abstract
Weyl
functions
Jussi Behrndt∗, Jonathan Rohleder
TechnischeUniversitätGraz,InstitutfürNumerischeMathematik,Steyrergasse30, 8010Graz,Austria
a r t i c l e i n f o a b s t r a c t
Articlehistory: Received 11 June 2014
Received in revised form 13 July 2015
Accepted 23 August 2015 Available online 8 September 2015 Communicated by N.G. Makarov Keywords:
Elliptic differential operator Dirichlet-to-Neumann map Spectral analysis
Weyl function Boundary triple
Thespectrumofaselfadjointsecondorderellipticdifferential operator in L2(Rn) is described in terms of the limiting
behavior of Dirichlet-to-Neumann maps, which arise in a multi-dimensionalGlazmandecompositionandcorrespondto aninteriorandanexteriorboundaryvalueproblem.Thisleads toPDEanalogsofrenownedfactsinspectraltheoryofODEs. Themainresults inthispaper arefirstderived inthemore abstractcontextof extensiontheoryof symmetricoperators andcorrespondingWeylfunctions,andareappliedtothePDE settingafterwards.
© 2015TheAuthors.PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).
1. Introduction
TheTitchmarsh–Weylfunctionisanindispensabletoolindirectandinversespectral theoryofordinarydifferential operatorsand moregeneralsystemsofordinary differen-tialequations; seetheclassicalmonographs[15,60]and[9,16,28–30,37,41,47,56,57]fora smallselectionofmorerecentcontributions.ForasingularsecondorderSturm–Liouville
* Corresponding author.
E-mailaddresses:[email protected](J. Behrndt), [email protected](J. Rohleder).
http://dx.doi.org/10.1016/j.aim.2015.08.016
0001-8708/© 2015 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
differential operatorof the form L+ = −d
2
dx2 + q+ on R+ with a real-valued, bounded
potential q+theTitchmarsh–Weylfunctionm+ canbedefinedas
m+(λ) =
fλ(0) fλ(0)
, λ∈ C \ R, (1.1)
where fλ is asquare-integrable solutionof L+f = λf on R+; cf. [60,61]. The function
m+ : C\ R → C belongsto the class of Nevanlinna (or Riesz–Herglotz) functionsand
it isacelebratedfactthatit reflectsthecomplete spectral propertiesoftheselfadjoint realizations of L+ inL2(R+). E.g. the eigenvalues of the Dirichlet realization AD are precisely those λ∈ R,where limη0iηm+(λ+ iη)= 0, theisolated eigenvaluesamong
them coincide with the poles of m+, and the absolutely continuous spectrum of AD (roughlyspeaking)consistsofallλ withtheproperty 0< Im m+(λ+ i0)< +∞.
If L = −dxd22 + q is a singular Sturm–Liouville expression on R with q real-valued
and bounded, it is most natural to use decomposition methods of Glazman type for the analysisofthe correspondingselfadjoint operatorinL2(R);cf.[31].Moreprecisely, therestrictionof L toR+ givesrisetotheTitchmarsh–Weylfunctionm+ in(1.1),and
similarly a Titchmarsh–Weyl function m− associated to the restriction of L to R− is defined.Inthatcaseusually thefunctions
m(λ) =−m+(λ) + m−(λ) −1 and m(λ) = −m+(λ) 1 1 m−(λ)−1 −1 (1.2) areemployedforthedescriptionofthespectrum.Whereasthescalarfunctionm seemsto bemoreconvenientitwillingeneralnotcontainthecompletespectraldata,adrawback that is overcome when using the2× 2-matrix functionm. Some of these observations werealreadymadein[39,60],similarideascanalsobefoundin[36,38,42]forHamiltonian systems,andmorerecentlyinanabstractoperatortheoreticalframeworkin[17,19],see also [6].
One of the main objectives of this paper is to extend theclassical spectral analysis of ordinary differential operators via the Titchmarsh–Weyl functions in (1.2) to the multidimensionalsetting.Forthisconsiderthesecondorderpartialdifferentialexpression
L = − n j,k=1 ∂ ∂xj ajk ∂ ∂xk + n j=1 aj ∂ ∂xj − ∂ ∂xj aj + a (1.3)
with smooth, bounded coefficientsajk,aj :Rn → C and a:Rn → R bounded,and as-sumethatL isformallysymmetricanduniformlyellipticonRn.LetA betheselfadjoint operator associated to (1.3) inL2(Rn).Our main goal is to describethe spectral data ofA, thatis,isolatedandembeddedeigenvalues,continuous,absolutelycontinuousand singularcontinuousspectralpoints,intermsofthelimitingbehavior ofappropriate mul-tidimensionalcounterpartsofthefunctionsin(1.2).Notefirstthatthemultidimensional analogueof theTitchmarsh–Weylfunction(1.1)istheDirichlet-to-Neumannmap,and
inordertodefinesuitableanaloguesofthefunctionsin(1.2)weproceedasfollows:Split Rn into abounded domainΩ
i withsmooth boundaryΣ andits exteriorΩe =Rn\ Ωi.
Forλ∈ C\ R the Dirichlet-to-Neumann maps forL in Ωi and Ωe,respectively, onthe
compactinterfaceΣ aregivenby
Λi(λ)uλ,i|Σ:=
∂uλ,i
∂νLi
Σ and Λe(λ)uλ,e|Σ:=
∂uλ,e
∂νLe
Σ, λ∈ C \ R,
where uλ,j ∈ H2(Ωj) solve Luλ,j = λuλ,j, j = i,e, and uλ,j|Σ and ∂u∂νλ,jLj|Σ denote
the trace and the conormal derivative, respectively; cf. Section 4.1 for further details. Both λ→ Λi(λ) and λ→ Λe(λ) areviewed as functionswhose values areoperators in
L2(Σ) definedonthedensesubspaceH3/2(Σ).Themultidimensionalcounterpartsofthe
functionsin(1.2)are M (λ) =Λi(λ) + Λe(λ) −1 and M (λ) = Λi(λ) 1 1 −Λe(λ)−1 −1 (1.4) (thedifferencesinthe signsaredue to thedefinitionof theconormalderivative, where thenormalsofΩiandΩepointintooppositedirections).Observethat,incontrasttothe
one-dimensional situation described above, Rn is split into a bounded domain and an unboundeddomain.ThisyieldsthatΛi ismeromorphic,whichinturnessentiallyallows
ustogiveanalmostcompletecharacterizationofthespectrumofA withthefunctionM in(1.4)inTheorem 4.1; theonlypossiblespectral pointsthatcannot bedetected with M are eigenvalues of A with vanishing traces on Σ, and possible accumulation points of such eigenvalues. A complete picture of the spectrumof A in terms of the limiting behavior ofDirichlet-to-Neumannmapsisobtainedwithhelpofthe2×2-blockoperator matrixfunctionM in (1.4)inTheorem 4.7.
Wementionthatinconnection withSchrödingeroperators inR3 thefunction M in
(1.4)wasalreadyused in[2]fortheextension ofaclassicalconvergencepropertyofthe Titchmarsh–Weylfunctiontothethree-dimensionalcase,seealso[5,55].Wealsoremark thatforSchrödingeroperators onexteriordomains withC2-boundaries theconnection ofthespectrumtothelimitsoftheDirichlet-to-Neumannmapwasalreadyinvestigated by the authors in [8]. Furthermore, we refer to [11,13,25–27,33,34,48,52–54] for recent relatedworkonspectral problemsforellipticdifferentialoperators.
In this paper our approach to Titchmarsh–Weyl functions and their connection to spectral properties of corresponding selfadjoint differential operators is more abstract andofgeneralnature,basedontheconceptsof(quasi)boundarytriplesandtheirWeyl functions.RecallfirstthatforasymmetricoperatorS inaHilbertspaceH aboundary triple {G,Γ0,Γ1} consists of a “boundary space” G and two linear mappings Γ0,Γ1 :
dom S∗→ G,whichsatisfyanabstractGreenidentity
and amaximalitycondition.ThecorrespondingWeylfunctionM isdefinedas
M (λ)Γ0fλ= Γ1fλ, λ∈ C \ R, (1.6) wherefλ∈ H solvestheequationS∗f = λf ;thevaluesM (λ) oftheWeylfunctionM are boundedoperatorsintheHilbertspaceG.TheexampleoftheSturm–Liouvilleexpression L+ inthebeginningof theintroduction fitsinto this scheme: There H = L2(R+), S is
the minimaloperatorassociated withthe differentialexpression L+ inL2(R+),G = C,
and themappingsΓ0,Γ1are givenby
Γ0f = f (0) and Γ1f = f(0), f ∈ dom S∗,
whereS∗isthemaximaloperatorassociatedwithL+inL2(R+).Thenthecorresponding
Weyl function is m+ in(1.1), theselfadjoint Dirichlet operatorAD coincides with the restrictionS∗ ker Γ0,and thespectrumcanbe describedwiththehelpofthelimitsof
theWeylfunction.Thecorrespondencebetweenthespectrumoftheparticularselfadjoint extension
A0:= S∗ ker Γ0
and thelimits of the Weyl function is nota special featureof the boundarytriple for the aboveSturm–Liouvilleequation. In fact,it holdsas soon as thesymmetric restric-tionS (and,thus,theboundarymappingsΓ0andΓ1)ischosenproperly.Moreabstract
considerations from [20,44–46] yield that the operator A0 (and hence its spectrum) is
determined uptounitaryequivalencebytheWeylfunctionifandonlyifthesymmetric operatorS issimpleorcompletelynon-selfadjoint,thatis,thereexistsnonontrivial sub-spaceofH whichreducesS toaselfadjointoperator.Thisconditioncanbereformulated equivalently as
H = clspγ(ν)g : ν∈ C \ R, g ∈ G , (1.7) where γ(ν)= (Γ0 ker(S∗− ν))−1 is the so-called γ-field and clsp denotes the closed
linearspan;cf.[43].UndertheassumptionthatS issimpleadescriptionoftheabsolutely continuous andsingularcontinuousspectrumintheframeworkofboundarytriplesand their Weyl functionswas givenin [10]; formore recentrelated worksee also [12,14,35, 49,51,58].
TheconceptofboundarytriplesandtheirWeylfunctionswasextendedin[3]insuch awaythatitisconvenientlyapplicabletoPDEproblems.Forthatonedefinesboundary mappings Γ0, Γ1 on asuitable, smaller subset of the domain of the maximal operator
andrequiresGreen’sidentity(1.5)onlytoholdonthissubset;thedefinitionoftheWeyl functionassociatedtosuchaquasiboundarytriple{G,Γ0,Γ1} isasin(1.6),exceptthat
only solutions inthe domain of the boundary maps are used; cf. Section 2.1. For the second order elliptic operatorL in (1.3)restricted to thesmooth domain Ωi ⊂ Rn one
Γ0u = u|Σ and Γ1u =− ∂u ∂νLi Σ, u∈ H 2(Ω i),
in which case the corresponding Weyl function is (minus) the Dirichlet-to-Neumann map−Λi.Basedonorthogonalcouplingsofsymmetricoperatorsandextendingabstract
ideasin[17] alsothefunctionsM andM in (1.4)canbe interpretedas Weylfunctions associatedtoproperlychosenquasiboundarytriples;e.g.,M correspondstothepairof boundarymappings Γ0u = ∂ui ∂νLi Σ+ ∂ue ∂νLe Σ, Γ1u = u|Σ, u = ui⊕ ue, ui|Σ= ue|Σ, (1.8)
where uj ∈ H2(Ωj), j = i,e. Moreover, ker Γ0 is the domain of the unique selfadjoint
operatorA associatedwithL inL2(Rn).WhentryingtolinkthespectralpropertiesofA tothelimitingbehaviorofthefunctionM itisnecessarytoextendtheknownresultsfor boundarytriplestothemoregeneralnotionofquasiboundarytriples.Moreover,asubtle difficultyarises:ThesymmetricoperatorS corresponding totheboundarymappingsin
(1.8)maypossesseigenvaluesand thusingeneralisnotsimple.
Intheabstractpartofthepresentpaperweshowhowthisdifficultycanbeovercome. Inthegeneralsetting ofquasiboundarytriplesand theirWeylfunctionswe showthat alocal simplicity conditiononanopen interval(or, moregenerally,aBorelset)Δ⊂ R sufficestocharacterizethespectrumofA0 inΔ.Tobemorespecific,weassumethat
E(Δ)H = clspE(Δ)γ(ν)g : ν ∈ C \ R, g ∈ ran Γ0
, (1.9)
where E(Δ) denotes the spectral projection of A0 = S∗ ker Γ0 on Δ; this is alocal
version of the condition (1.7). Under this assumption we provide characterizations of theisolatedandembeddedeigenvaluesandthecorrespondingeigenspaces,aswellasthe continuous,absolutelycontinuousandsingularcontinuousspectrumofA0in Δ interms
ofthelimits ofM (λ) whenλ approaches thereal axis.Forinstance,weprovethatthe eigenvaluesofA0inΔ arethoseλ,wherelimη0iηM (λ+ iη)g= 0 forsomeg∈ ran Γ0,
andthattheabsolutelycontinuousspectrumofA0canbecharacterizedbymeansofthe
pointsλ where0< Im(M (λ+ i0)g,g)G <∞.Moreover,weproveinclusionsandprovide conditions for the absence of singular continuous spectrum. Afterwards we apply the obtainedresultstotheselfadjointellipticdifferentialoperatorassociatedtoL in(1.3)in L2(Rn).Weprovethat,despitethefactthattheunderlyingsymmetricoperatorfailsto be simpleingeneral, thewholeabsolutely continuous spectrumof A0 canberecovered
from the mapping M in(1.4). Moreover, we provethat the eigenvaluesof A0 and the
correspondingeigenfunctionscanbecharacterized bylimitingpropertiesofM asfaras theeigenfunctionsdonotvanishontheinterfaceΣ.Acompletepictureofthespectrum ofA0 isobtainedwhenusingthefunctionM in (1.4).
Thispaperisorganizedinthefollowingway.InSection2werecallthebasicfactson quasiboundarytriplesandcorrespondingWeylfunctionsanddiscussthelocalsimplicity
property (1.9) indetail. InSection3the connection betweenthe spectra ofselfadjoint operators andcorresponding abstractWeylfunctionsisinvestigated.Section4contains theapplicationoftheabstractresultstothementionedPDEproblems.
Finally,letusfixsomenotation.ForaselfadjointoperatorA inaHilbertspaceH we denote byσ(A) (σp(A),σc(A),σac(A),σsc(A),σs(A),respectively)thespectrum(setof
eigenvalues, continuous, absolutely continuous, singularcontinuous, singular spectrum, respectively)ofA and byρ(A)=C\ σ(A) itsresolventset.
2. Quasiboundarytriples,associatedWeylfunctions,andalocalsimplicity condition Inthispreliminarysectionwefirstrecalltheconceptsofquasiboundarytriples,their γ-fields and their Weyl functions. Afterwards we discuss a local simplicity property of symmetric operators,whichwillbe assumedtohold inmostoftheresultsinSection3. 2.1. Quasiboundary triples
Thenotionof quasiboundarytripleswasintroduced in[3]as ageneralizationof the notions of boundarytriples andgeneralized boundarytriples, see [18,20,21,32,40]. The basicdefinitionisthefollowing.
Definition 2.1. Let S be a closed, densely defined, symmetric operator in a separable Hilbert space H and let T ⊂ S∗ be anoperator whose closure coincides with S∗, i.e., T = S∗. A triple {G,Γ0,Γ1} consisting of a Hilbert space G and two linear mappings
Γ0,Γ1 : dom T → G is called aquasi boundary triple for S∗ if the following conditions
are satisfied.
(i) TherangeofthemappingΓ:= (Γ0,Γ1): dom T → G × G isdense.
(ii) Theidentity
(T u, v)H− (u, T v)H= (Γ1u, Γ0v)G− (Γ0u, Γ1v)G (2.1) holdsforallu,v∈ dom T .
(iii) TheoperatorA0:= T ker Γ0 isselfadjointinH.
Inthefollowingwesuppresstheindicesinthescalarproductsandsimplywrite(·,·), when noconfusioncanarise.
Werecallsomefactsonquasiboundarytriples,whichcanbefoundin[3,4].LetS bea closed,denselydefined,symmetricoperatorinH.Aquasiboundarytriple{G,Γ0,Γ1} for
S∗existsifandonlyifthedefectnumbersofS areequal.Whatwewillusefrequentlyis thatif{G,Γ0,Γ1} isaquasiboundarytripleforS∗ thendom S = ker Γ0∩ ker Γ1.Recall
also that a quasi boundary triple with the additional property ran(Γ0,Γ1) = G × G
mappings Γ0, Γ1 are defined on dom S∗ and (2.1) holds with T replaced by S∗. In
particular,inthecaseoffinitedefectnumbersthenotionsofquasiboundarytriplesand (ordinary)boundarytriplescoincide.Formoredetailsonquasiboundarytripleswerefer to[3,4].
Inorder to provethat atriple{G,Γ0,Γ1} is aquasi boundarytriple forthe adjoint
S∗ of a given symmetric operator S it is not necessary to know S∗ explicitly, as the followingusefulpropositionshows;cf. [3,Theorem 2.3]foraproof.
Proposition 2.2. Let T be a linear operator in a separable Hilbert space H, let G be a further Hilbert space, and let Γ0,Γ1 : dom T → G be linear mappings which satisfy the
followingconditions.
(i) Therange of themapΓ= (Γ0,Γ1) : dom T → G × G isdense inG × G and ker Γ
isdenseinH.
(ii) Theidentity (2.1)holds forallu,v∈ dom T .
(iii) Thereexistsaselfadjointrestriction A0 of T inH with dom A0⊂ ker Γ0.
ThenS := T ker Γ isaclosed,denselydefined,symmetricoperatorinH,T = S∗holds, and{G,Γ0,Γ1} isaquasi boundarytriple forS∗ with T ker Γ0= A0.
2.2. Weylfunctionsand γ-fields
LetS beaclosed,denselydefined,symmetricoperatorinH andlet{G,Γ0,Γ1} bea
quasiboundarytripleforT = S∗withA0= T ker Γ0.Inordertodefinetheγ-fieldand
theWeylfunctioncorresponding to{G,Γ0,Γ1} notethatthedirectsumdecomposition
dom T = dom A0 ker(T − λ) = ker Γ0 ker(T − λ)
holds foreach λ∈ ρ(A0) andthat, inparticular, therestriction of Γ0 to ker(T − λ) is
injective.The following definitionis formally thesame as forordinary andgeneralized boundarytriples.
Definition2.3. Let{G,Γ0,Γ1} be aquasiboundarytriplefor T = S∗ and letA0= T
ker Γ0.Thentheγ-field γ andtheWeylfunction M associatedwith{G,Γ0,Γ1} aregiven
by
γ(λ) =Γ0 ker(T − λ)
−1
and M (λ) = Γ1γ(λ), λ∈ ρ(A0),
respectively.
Itfollowsimmediatelyfromthedefinitionthatforeachλ∈ ρ(A0) theoperatorM (λ)
M (λ)Γ0uλ= Γ1uλ, uλ∈ ker(T − λ),
and thatran γ(λ)= ker(T − λ) holds.Wesummarizesomepropertiesoftheγ-fieldand the Weyl function. For the proofs of items (i)–(iv) in the next lemma we refer to [3, Proposition 2.6],item(v)is asimpleconsequenceof (ii)and(iii).
Lemma2.4.Let{G,Γ0,Γ1} beaquasiboundarytripleforT = S∗ withγ-fieldγ andWeyl
function M and let A0 = T ker Γ0.Then for λ,μ,ν ∈ ρ(A0) the followingassertions
hold.
(i) γ(λ) isaboundedoperatorfromG toH defined onthedense subspaceran Γ0.The
adjointγ(λ)∗:H → G iseverywhere definedon H andisbounded. Itisgivenby γ(λ)∗= Γ1(A0− λ)−1.
(ii) Theidentity
γ(λ)g =I + (λ− μ)(A0− λ)−1
γ(μ)g holds forallg∈ ran Γ0.
(iii) The γ-fieldandtheWeylfunctionare connectedvia
(λ− μ)γ(μ)∗γ(λ)g = M (λ)g− M(μ)∗g, g∈ ran Γ0,
andM (λ)⊂ M(λ)∗ holds.
(iv) M (λ) isan operatorinG defined onthedense subspaceran Γ0 andsatisfies
M (λ)g = Re M (μ)g
+ γ(μ)∗(λ− Re μ) + (λ − μ)(λ − μ)(A0− λ)−1
γ(μ)g (2.2) for allg ∈ ran Γ0.In particular,for every g ∈ ran Γ0 thefunction λ→ M(λ)g is
holomorphicon ρ(A0) andeachisolatedsingularityof λ→ M(λ)g is apoleof first
order.Moreover, limη0iηM (ζ + iη)g existsforallg∈ ran Γ0 and allζ∈ R.
(v) Theidentity γ(μ)∗(A0− λ)−1γ(ν)g = M (λ)g (λ− ν)(λ − μ) + M (μ)g (λ− μ)(ν − μ) + M (ν)g (ν− λ)(ν − μ) holds forallg∈ ran Γ0 if λ= ν,λ= μ andν = μ.
2.3. Simplesymmetricoperatorsandlocal simplicity
Let S be a closed, densely defined, symmetric operator in the separable Hilbert space H. Recall that S is said to be simple or completely non-selfadjoint if there is
nonontrivial S-invariantsubspaceH0ofH whichreduces S toaselfadjointoperatorin
H0, see [1, Chapter VII-81]. According to [43] the simplicity of S is equivalent to the
densityofthespanofthedefectspacesofS in H,i.e.,S issimpleifandonlyif H = clspker(S∗− ν) : ν ∈ C \ R (2.3) holds; here clsp stands for the closed linear span. Assume that {G,Γ0,Γ1} is a quasi
boundarytripleforT = S∗ withA0= T ker Γ0.ThenitfollowsthatS issimpleifand
onlyif(2.3)holds withker(S∗− ν) replacedbyker(T− ν).Moreover,ifγ istheγ-field corresponding to the quasiboundary triple{G,Γ0,Γ1} we conclude thatS issimple if
andonlyif
H = clspγ(ν)g : ν∈ C \ R, g ∈ ran Γ0
(2.4) holds. We also mention that the set C \ R in (2.4) can be replaced by any set G ⊂ ρ(A0) which has an accumulation point in each connected component of ρ(A0);
cf.Lemma 2.5(v)below.
Our aim is to generalize the notion of simplicity and to replace it by someweaker, localcondition, whichissatisfiedin,e.g.,theapplicationsinSection4.Instead of(2.4)
wewillassumethat
E(Δ)H = clspE(Δ)γ(ν)g : ν ∈ C \ R, g ∈ ran Γ0
(2.5) holdsonaBorelset(lateronusuallyanopeninterval)Δ;hereE(·) denotesthespectral measureof A0.ThisconditionwillbeimposedinmanyofthegeneralresultsinSection3.
Inthenextlemma wediscussthisconditionandsomeconsequencesofit.
Lemma 2.5. Let S be a closed, densely defined, symmetric operator in H and let {G,Γ0,Γ1} be a quasi boundary triple for T = S∗ with A0 = T ker Γ0. Then the
followingholds.
(i) IfS is simple then(2.5)issatisfiedforevery BorelsetΔ⊂ R. (ii) If(2.5) holdsforsomeBorelsetΔ⊂ R then
E(Δ)H = clspE(Δ)γ(ν)g : ν ∈ C \ R, g ∈ ran Γ0
(2.6) holdsforevery Borelset Δ ⊂ Δ.
(iii) Ifδ1,δ2,. . . are disjointopenintervalssuchthat
E(δj)H = clsp
E(δj)γ(ν)g : ν∈ C \ R, g ∈ ran Γ0
for all j (2.7) then(2.5)holdsforΔ= jδj.
(v) If(2.5) holdsandG is asubset of ρ(A0) which has anaccumulation pointin each
connectedcomponent of ρ(A0) then
E(Δ)H = clspE(Δ)γ(ν)g : ν∈ G, g ∈ ran Γ0
. (2.8)
Proof. Assertion (i)isaconsequenceof item(ii)since (2.5)holdswith Δ=R whenS is simple.
For (ii)notethattheinclusion⊃ in(2.6)clearlyholds. Fortheconverseinclusionlet u ∈ E(Δ)H.As Δ ⊂ Δ wehave u∈ E(Δ)H and hence there exists asequence (vn) in thelinearspan of {E(Δ)γ(ν)g : ν ∈ C\ R,g ∈ ran Γ0} which converges to u.Then
(E(Δ)vn) isasequenceinthelinearspanof{E(Δ)γ(ν)g : ν∈ C\R,g∈ ran Γ0} which
converges toE(Δ)u= u.
Inordertoprove(iii)letδj beasintheassumptionsandletΔ=
jδj.Theinclusion
⊃ in (2.5)again isobvious.Fortheconverseinclusionletu∈ E(Δ)H anddefine H := clspE(Δ)γ(ν)g : ν∈ C \ R, g ∈ ran Γ0 . (2.9) Since u = E(Δ)u = j E(δj)u
itissufficienttoshowE(δj)u∈ H forallj.Note firstthatbyassumption(2.7)wehave
E(δj)u∈ clsp
E(δj)γ(μ)h : μ∈ C \ R, h ∈ ran Γ0
and hencetheassertionfollowsifweverify
E(δj)γ(μ)h∈ H (2.10)
forallμ∈ C\ R,h∈ ran Γ0,andallj.ForthispurposeconsidersomefixedE(δj)γ(μ)h. Accordingto Lemma 2.4(ii)wehave
γ(ν)g = γ(μ)g + (ν− μ)(A0− ν)−1γ(μ)g
forallν ∈ C\ R andallg∈ ran Γ0,and henceH in (2.9)canberewrittenintheform
H = clsp
E(Δ)γ(μ)g, E(Δ)(A0− ν)−1γ(μ)g : ν∈ C \ R, g ∈ ran Γ0
. It followsthatforη,ε> 0 theelement
βj−η
αj+η
E(Δ)(A0− (λ + iε))−1− (A0− (λ − iε))−1
belongs to H, where we have written δj = (αj,βj). From this and Stone’s formula it follows
E(δj)γ(μ)h = E(δj)E(Δ)γ(μ)h∈ H,
whichproves(2.10)and, hence,yieldstheinclusion⊂ in(2.5).Item (iii)isproved. Inorder to verify(iv), assumethatSu= λu for someu∈ dom S andλ∈ Δ. Then A0u= λu andhenceu∈ E(Δ)H. On theother hand,for g ∈ ran Γ0 and ν ∈ C\ R it
followstogetherwithLemma 2.4(i)that
(u, E(Δ)γ(ν)g) = (γ(ν)∗u, g) =Γ1(A0− ν)−1u, g
= (λ− ν)−1(Γ1u, g) = 0,
as u ∈ dom S ⊂ ker Γ1. Hence, u ∈ E(Δ)H is orthogonal to the linear span of the
elements E(Δ)γ(ν)g, ν ∈ C\ R, g ∈ ran Γ0, which is dense in E(Δ)H by (2.5). This
impliesu= 0 andthusS doesnotpossesseigenvaluesinΔ.
It remains to show (v). The inclusion ⊃ in (2.8) is obvious. In order to prove the inclusion⊂ it suffices toverify thatthevectors E(Δ)γ(ν)g,g ∈ ran Γ0, ν ∈ G, spana
densesetinE(Δ)H.SupposethatE(Δ)u isorthogonaltothis set,thatis,
0 = (E(Δ)γ(ν)g, E(Δ)u) (2.11)
holds for all g ∈ ran Γ0 and all ν ∈ G. Since ρ(A0) ν → γ(ν)g is analytic for each
g ∈ ran Γ0 (see Lemma 2.4 (ii)) it follows that for each g ∈ ran Γ0 the function ν →
(E(Δ)γ(ν)g,E(Δ)u) isanalyticonρ(A0),andhence(2.11)impliesthatthisfunctionis
identicallyequaltozero.Now(2.5)yieldsE(Δ)u= 0 and(v)follows. 2 3. Spectralproperties ofselfadjointoperatorsandcorrespondingWeylfunctions
Thissectioncontainsthemainabstractresultsofthispaper.Wedescribethespectral properties of agiven selfadjoint operator by means of acorresponding Weyl function. Forthiswefixthefollowing setting.
Assumption3.1.LetS beaclosed,denselydefined,symmetricoperatorintheseparable Hilbertspace H andlet {G,Γ0,Γ1} be aquasiboundarytriple for T = S∗ with
corre-spondingγ-field γ andWeylfunction M .Moreover, letA0= T ker Γ0 anddenote by
E(·) thespectral measureofA0.
3.1. Eigenvaluesandcorresponding eigenspaces
Letus startwithacharacterizationoftheisolatedandembeddedeigenvaluesas well as the corresponding eigenspaces of a selfadjoint operator by means of an associated Weylfunction.Wewrites-lim forthestronglimitofanoperatorfunction.
Theorem 3.2. LetAssumption 3.1 be satisfied. Thenλ∈ R isan eigenvalue of A0 such
thatK := ker(A0−λ) ker(S−λ)= {0} ifandonlyifRλM := s- limη0iηM (λ+iη)= 0.
If dimK < ∞ thenthemapping
τ :K → ran RλM, u→ Γ1u, (3.1)
is bijective;if dimK = ∞ then themapping τ :K → clτ
ran RλM
, u→ Γ1u, (3.2)
is bijective, where clτ denotes theclosure in the normed spaceran τ , equipped with the
norm inG.
Remark3.3. Recall thatthelimit(RλM )g = limη0iηM (λ+ iη)g existsfor allλ∈ R and all g ∈ ran Γ0 by Lemma 2.4 (iv). Moreover, if λ is an isolated singularity of M ,
thatis,there existsanopen neighborhoodO of λ suchthatM isstrongly holomorphic on O \ {λ} then RλM = 0 if and only if for some g ∈ ran Γ0 the G-valued function
ζ → M(ζ)g hasapole atλ.Inthis caseRλM coincideswiththeresidue ResλM of M at λ inthestrongsense,i.e.,
(RλM )g = (ResλM )g = 1 2πi C M (z)g dz, g∈ ran Γ0,
where C denotesthe boundaryofan openball B such thatM isstronglyholomorphic in a neighborhood of B except the point λ. We also remark that without additional assumptionstheWeylfunctionisnotabletodistinguishbetweenisolatedandembedded eigenvaluesofA0;cf. Proposition 3.6below.
ProofofTheorem 3.2. Letλ∈ R befixed.NotefirstthatthemappingΓ1 K isinjective.
Indeed,foru∈ K = ker(A0− λ) ker(S − λ) withΓ1u= 0 wehaveu∈ ker Γ0∩ ker Γ1=
dom S and Su= λu; henceu= 0.Itisouraimtoprovetheinclusions
ran RλM⊂ ran(Γ1 K) ⊂ ran RλM . (3.3) Note that the closure clτ(ran RλM ) of ran RλM in the normed space ran τ , equipped withthenormofG,coincideswithran RλM∩ran τ.Hence(3.3)impliesthatthemapping
τ in(3.1)and(3.2)iswell-definedandbijective.
Inordertoverify(3.3)letg∈ ran Γ0anddenote byE(·) thespectralmeasureofA0.
Then iη(A0− (λ + iη))−1γ(ν)g + E({λ})γ(ν)g2 = R t− (λ + iη)iη + 1{λ}(t) 2 d(E(t)γ(ν)g, γ(ν)g)→ 0 as η 0 (3.4)
holdsforallν∈ C\ R.SincebyLemma 2.4(i)theoperatorγ(ν)∗ isbounded,itfollows from(3.4)that lim η0iηγ(ν) ∗(A 0− (λ + iη))−1γ(ν)g =−γ(ν)∗E({λ})γ(ν)g (3.5)
holdsforallν∈ C\ R.TogetherwithLemma 2.4(v)weconcludethatthelimitonthe lefthandsideof(3.5)coincideswith
lim η0iη M (λ + iη)g ((λ + iη)− ν)((λ + iη) − ν) = (RλM )g (λ− ν)(λ − ν). (3.6) WiththehelpofLemma 2.4(i),(3.5)and(3.6)weobtain
Γ1E({λ})γ(ν)g = Γ1(A0− ν)−1(A0− ν)E({λ})γ(ν)g = (λ− ν)γ(ν)∗E({λ})γ(ν)g =−(λ − ν) lim η0iηγ(ν) ∗(A 0− (λ + iη))−1γ(ν)g = 1 ν− λ(RλM )g
forallν∈ C\ R.DenotingbyP theorthogonalprojectioninH ontoK = ker(A0− λ)
ker(S− λ) itfollows
Γ1P γ(ν)g =
1
ν− λ(RλM )g, (3.7)
where wehaveused Γ1(ker(S− λ))={0}.From thisthefirst inclusionin(3.3)follows
immediately.
For the second inclusion in (3.3) note that the mapping Γ1 K is continuous as
Γ1u= γ(μ)∗(A0− μ)u= (λ− μ)γ(μ)∗u holdsforallu∈ K byLemma 2.4(i).Moreover,
for each ν ∈ C\ R the linear space {P γ(ν)g : g ∈ ran Γ0} is dense in K. In fact, fix
ν∈ C\ R andletu∈ K beorthogonalto P γ(ν)g forallg∈ ran Γ0.Then
0 = (u, P γ(ν)g) = (γ(ν)∗u, g) = (Γ1(A0− ν)−1u, g) = (λ− ν)−1(Γ1u, g)
byLemma 2.4(i),whichimpliesΓ1u= 0 asran Γ0isdense.Hencewehaveu∈ ker Γ0∩
ker Γ1 = dom S and this implies u∈ K ∩ ker(S − λ), so thatu = 0. Now the second
inclusionin(3.3)followstogetherwith(3.7)andthefactthatΓ1 K iscontinuous.Hence
themappingτ in(3.2)iswell-definedandbijective.IfK isfinite-dimensionalthenclearly theclosurein(3.2)canbeomittedandweend upwiththebijectivityof(3.1). 2
As an immediate consequence of Theorem 3.2 all eigenvalues of A0 which are not
Corollary 3.4. Let Assumption 3.1 be satisfied, and assume that λ ∈ R is not an eigenvalue of the symmetric operator S. Then λ is an eigenvalue of A0 if and only
if RλM := s- limη0iηM (λ+ iη) = 0. If the multiplicity of the eigenvalue λ is finite
then themapping
τ : ker(A0− λ) → ran RλM, u→ Γ1u,
is bijective;if themultiplicityoftheeigenvalue λ isinfinitethen themapping τ : ker(A0− λ) → clτ
ran RλM
, u→ Γ1u,
is bijective, where clτ denotes theclosure in the normed spaceran τ , equipped with the
norm inG.
3.2. Continuous,absolutely continuous,andsingular continuousspectra
In this subsection we describe the continuous, absolutely continuous, and singular continuousspectrumofaselfadjointoperatorA0bymeansofthelimitsofanassociated
Weyl function M . Again we fix the setting in Assumption 3.1. It is clear thatan ad-ditional minimalityor simplicityconditionmust beimposed. Usuallyoneassumesthat the underlyingsymmetric operatorS is simple;cf. [10]. However, forour purposes the weaker assumption of local simplicity in Section 2.3 is more appropriate: in order to characterize thespectrumofA0 inanopen intervalΔ⊂ R weassumethat
E(Δ)H = clspE(Δ)γ(ν)g : ν∈ C \ R, g ∈ ran Γ0
. (3.8)
For instance, in Theorem 3.2 it turned out that an eigenvalue λ of A0 with its full
multiplicity can only be detected by the Weyl function if λ ∈ σ/ p(S). This condition
corresponds totheidentity(3.8)withΔ replacedby{λ};cf.Lemma 2.5(iv).
In the next theorem we agree to say that the Weyl function M can be continued analyticallyto somepointλ∈ R ifthereexists anopenneighborhoodO ofλ in C such thatζ→ M(ζ)g canbe continuedanalyticallytoO forallg∈ ran Γ0.Wementionthat
theproof of(i)is similartotheproofof[23, Theorem1.1].
Theorem 3.5. LetAssumption 3.1 be satisfied, and let Δ ⊂ R bean open interval such that thecondition (3.8)issatisfied. Thenthefollowingassertionsholdforeach λ∈ Δ.
(i) λ∈ ρ(A0) if andonly ifM canbe continuedanalytically intoλ.
(ii) λ ∈ σc(A0) if and only if s- limη0iηM (λ+ iη) = 0 and M cannot be continued
analyticallyintoλ.
Proof. (i) Recall first thatby Lemma 2.4 (iv)the function λ→ M(λ)g is analytic on ρ(A0) for each g ∈ ran Γ0, which proves the implication (⇒). In order to verify the
implication (⇐) in (i), let us assume that M can be continued analytically to some λ∈ Δ, that is, there exists anopen neighborhood O of λ in C withO ∩ R ⊂ Δ such thatζ→ M(ζ)g canbecontinuedanalyticallytoO foreachg∈ ran Γ0.Choosea,b∈ R
with λ ∈ (a,b), [a,b]⊂ O, and a,b ∈ σ/ p(A0). The spectral projection E((a,b)) of A0
correspondingto theinterval (a,b) isgiven byStone’sformula
E((a, b)) = s- lim δ0 1 2πi b a (A0− (t + iδ))−1− (A0− (t − iδ))−1 dt, (3.9)
where the integralon theright-hand side is understoodinthe strong sense.Using the identityinLemma 2.4(v)and(3.9)astraightforwardcalculationleadsto
E((a, b))γ(ν)g2 =γ(ν)∗E((a, b))γ(ν)g, g = lim δ0 1 2πi b a γ(ν)∗(A0− (t + iδ))−1γ(ν)g, g −γ(ν)∗(A0− (t − iδ))−1γ(ν)g, g dt = 0 forallg∈ ran Γ0andallν ∈ C\R,sinceζ→ (M(ζ)g,g) admitsananalyticcontinuation
into O for all g ∈ ran Γ0. Thus the assumption (3.8) and [a,b] ⊂ Δ together with
Lemma 2.5(ii)implyE((a,b))= 0.Inparticular, λ∈ ρ(A0).
(ii) According to Lemma 2.5 (iv) the condition (3.8) implies thatS does not have eigenvaluesinΔ.Henceitem (ii)follows immediatelyfromitem (i)and Corollary 3.4.
If S is simple then by Lemma 2.5 (i) the assumption (3.8) is satisfied for Δ = R. Hence (i)and (ii)hold forallλ∈ R. 2
Nowwereturntothecharacterizationofeigenvalues.Weformulateasufficient condi-tionunderwhichtheWeylfunctionisabletodistinguishbetweenisolatedandembedded eigenvalues.
Proposition 3.6. Let Assumption 3.1 be satisfied and let Δ ⊂ R be an open interval. Assumethat thecondition (3.8)issatisfiedandletλ∈ Δ.Thenallassertionsof Corol-lary 3.4holdfor λ.Moreover,λ isanisolatedeigenvalueof A0if andonly ifλ isapole
inthestrongsenseofM .InthiscaseRλM istheresidueofM inthestrongsenseat λ;
cf.Remark 3.3.
Proof. Letλ∈ R andletΔ⊂ R bean openinterval withλ∈ Δ such that(3.8)holds. Thenλ∈ σ/ p(S) byLemma 2.5(iv)andhencetheassertionsinCorollary 3.4holdfor λ.
neighborhoodO ofλ suchthatζ→ M(ζ)g isholomorphiconO \ {λ} forallg∈ ran Γ0.
From Corollary 3.4weconcludethatthereexistsg∈ ran Γ0 suchthat
lim
η0iηM (λ + iη)g = 0. (3.10)
HenceLemma 2.4(iv)impliesthatM hasapoleoffirstorderinthestrongsenseat λ. Conversely, if M has a pole (of first order) in the strong sense at λ then there exists g ∈ ran Γ0 suchthat(3.10) holds. AccordingtoLemma 2.4(iv)theorder ofthepoleis
oneand,hence,
lim
η0iηM (λ + iη)g = (ResλM )g= 0.
It follows with thehelpof Corollary 3.4 thatλ is aneigenvalue ofA0. Moreover,
The-orem 3.5 (i) implies that there exists an open neighborhood O of λ in C such that O\{λ}⊂ ρ(A0).Henceλ isisolatedinthespectrumofA0.Thiscompletestheproof. 2
Next we discuss the relation of the function M to the absolutely continuous and singularcontinuousspectrumofA0.Inthespecialcaseofordinaryboundarytriplesand
Δ=R thefollowing resultsreducetothosein[10].Forourpurposesalocalizedversion and an extensionto quasi boundarytriplesis necessary. Theproofs presentedhereare somewhat more direct than those in [10]; in particular, the integral representation of Nevanlinna functionsandthecorrespondingmeasuresareavoided.
In the following for afinite Borel measure μ on R we denote the set of all growth points ofμ bysupp μ,thatis,
supp μ =x∈ R : μ((x − ε, x + ε)) > 0 for all ε > 0 .
Note thatsupp μ isclosed withμ(R\ supp μ)= 0 andthatsupp μ isminimal withthis property,thatis,eachclosedsetS ⊂ R withμ(R\S)= 0 satisfiessupp μ⊂ S.Moreover, for aBorel set χ⊂ R we define the absolutely continuous closure (also called essential closure) by
clac(χ) :=
x∈ R : |(x − ε, x + ε) ∩ χ| > 0 for all ε > 0 , where |· | denotestheLebesguemeasure,andthecontinuousclosure by
clc(χ) :=
x∈ R : (x − ε, x + ε) ∩ χ is not countable for all ε > 0 . (3.11) Observethatclac(χ) andclc(χ) bothareclosedandthatclac(χ)⊂ clc(χ)⊂ χ holds,but
ingeneraltheconverseinclusionsarenottrue.Infact,clac(χ)=∅ ifandonlyif|χ|= 0,
and clc(χ)=∅ ifandonlyifχ iscountable.
Lemma3.7.Letμ beafiniteBorelmeasureonR anddenotebyF itsStieltjestransform, F (λ) = R 1 t− λdμ(t), λ∈ C \ R.
ThenthelimitIm F (x+i0)= limy0Im F (x+iy) existsandisfiniteforLebesguealmost
all x∈ R. Let μac and μs be the absolutely continuous and singular part, respectively,
of μ in the Lebesgue decomposition μ = μac+ μs, and decompose μs into the singular
continuouspart μsc andthepurepointpart. Thenthefollowingassertions hold.
(i) supp μac= clac({x∈ R: 0< Im F (x+ i0)< +∞}).
(ii) supp μs⊂ {x ∈ R : Im F (x + i0) = +∞}.
(iii) supp μsc⊂ clc({x∈ R: Im F (x+ i0)= +∞,limy0yF (x+ iy)= 0}).
Proof. From[59,Lemma 3.14andTheorem 3.23]itfollowsimmediatelythatassertion (i) istrue,thatthelimitIm F (x+ i0) existsandisfiniteforLebesguealmostallx∈ R,and that
μs
R \ {x ∈ R : Im F (x + i0) = +∞}= 0, (3.12) which implies (ii). In order to verify (iii) note first thatlimy0yF (x+ iy) = iμ({x}) holdsforallx∈ R since
yF (x + iy)− iμ({x}) ≤ R y t− (x + iy)− i1{x}(t) dμ(t) → 0, y 0.
Inparticular,μ({x})= 0 ifandonlyiflimy0yF (x+iy)= 0.Henceitfollowsfrom(3.12) andthedefinitionofμscthat
μsc(R \ Msc) = 0, (3.13) where Msc:= x∈ R : Im F (x + i0) = +∞, lim y0yF (x + iy) = 0 .
For x∈ R\ clc(Msc) bydefinition there exists ε> 0 such that(x− ε,x+ ε)∩ Msc is
countable;thusμsc((x− ε,x+ ε)∩ Msc)= 0.Withthehelpof(3.13) itfollows
μsc((x− ε, x + ε)) ≤ μsc((x− ε, x + ε) ∩ Msc) + μsc(R \ Msc) = 0,
thatis, x∈ supp μ/ sc. 2
Theabsolutelycontinuousspectrumofaselfadjointoperatorinsomeinterval Δ can becharacterized inthefollowingway.
Theorem 3.8. Let Assumption 3.1 be satisfied and let Δ ⊂ R be an open interval such that thecondition
E(δ)H = clspE(δ)γ(ν)g : ν∈ C \ R, g ∈ ran Γ0
(3.14) issatisfiedforeach openintervalδ⊂ Δ withδ∩ σp(S)=∅. Thentheabsolutely
contin-uous spectrumofA0 inΔ isgivenby
σac(A0)∩ Δ = g∈ran Γ0 clac x∈ Δ : 0 < Im(M(x + i0)g, g) < +∞ . (3.15)
If S is simple then(3.15) holdsforeach openintervalΔ,including thecaseΔ=R. Proof. The proofof Theorem 3.8consistsof twoseparate stepsinwhichtheassertions
(3.17) and (3.19)belowwill beshown.Theidentity (3.15) isthen animmediate conse-quenceof(3.17) and(3.19)(notethattherighthandsidein(3.19)doesnotdepend on ζ∈ C\ R).Wefixsomenotationfirst.Letusset
DΔ:=
E(Δ)γ(ζ)g : ζ∈ C \ R, g ∈ ran Γ0
(3.16) and define the measures μu := (E(·)u,u) for u ∈ H. Denote by Pac the orthogonal
projection in H onto the absolutely continuous subspace Hac of A0. Observe that the
spectral measure of theabsolutely continuouspartof A0 isE(·)Pac and thatthe
abso-lutely continuousmeasuresμu,ac aregivenbyμu,ac= (E(·)Pacu,Pacu)= μPacu.
Step1. Inthis steptheidentity
σac(A0)∩ Δ =
u∈DΔ
supp μu,ac (3.17)
will be verified. First of all the open set Δ := Δ\σp(S) is the disjoint union of open
intervals δj,1≤ j < N,N ∈ N∪ {∞},andforeachδj wehave
E(δj)H = clsp
E(δj)γ(ν)g : ν ∈ C \ R, g ∈ ran Γ0
byassumption.WiththehelpofLemma 2.5(iii)weconclude
E(Δ)H = clspE(Δ)γ(ν)g : ν ∈ C \ R, g ∈ ran Γ0
.
Since Δ ⊂ Δ itfollowsimmediately thatE(Δ)H ⊂ clsp DΔ.Moreover,wehave
PacE(Δ)H = PacE(Δ)H ⊂ Pac clspDΔ ⊂ clsp PacDΔ⊂ PacE(Δ)H and therefore
PacE(Δ)H = clsp PacDΔ. (3.18)
Inorderto verify(3.17),assumefirstthatx doesnotbelongtotheleft handside of
(3.17), that is, x ∈ σ/ ac(A0)∩ Δ. Then there exists > 0 such that (x− ,x+ )∩ Δ
containsnoabsolutelycontinuousspectrumofA0.Thisyields
E((x− , x + ) ∩ Δ)Pac= 0
andforu∈ E(Δ)H oneobtains μu,ac((x− , x + )) = E((x− , x + ))Pacu, Pacu =E((x− , x + ))PacE(Δ)u, Pacu =E((x− , x + ) ∩ Δ)Pacu, Pacu = 0.
Therefore(x− ,x+ )∩ supp μu,ac=∅ forallu∈ E(Δ)H,inparticular,forallu∈ DΔ.
Thus
x /∈
u∈DΔ
supp μu,ac
andtheinclusion⊃ in(3.17)follows.Fortheconverseinclusionassumethatx doesnot belongtotherighthandsideof(3.17).Thenthereexists> 0 suchthat(x− ,x+ )⊂ R\ supp μu,ac forallu∈ DΔ,thatis,
E((x − , x + ))Pacu2= μu,ac((x− , x + )) = 0
forallu∈ DΔ,andhencealsoforallu∈ clsp DΔ.Withthehelpof(3.18)itfollows
E((x− , x + ) ∩ Δ)Pacu = E((x− , x + ))PacE(Δ)u = 0
forallu∈ H.Thisshowsthat(x− ,x+ )∩ Δ doesnotcontainabsolutely continuous spectrumofA0, inparticular, x∈ σ/ ac(A0)∩ Δ andtheinclusion⊂ in (3.17)follows.
Step2. In thisstepweshowthattheidentity supp μu,ac= clac
x∈ Δ : 0 < ImM (x + i0)g, g< +∞ (3.19) holdsforallu= E(Δ)γ(ζ)g∈ DΔ.Indeed,withthehelpoftheformula(2.2)wecompute
Im(M (x + iy)g, g) = yγ(ζ)g2+|x − ζ|2− y2Im(A0− (x + iy))−1γ(ζ)g, γ(ζ)g + 2(x− Re ζ)y Re(A0− (x + iy))−1γ(ζ)g, γ(ζ)g , (3.20)
forallx∈ R,y > 0,g∈ ran Γ0,andζ∈ C\R.Moreover,dominatedconvergenceimplies that y Re(A0− (x + iy))−1γ(ζ)g, γ(ζ)g = R y(t− x) (t− x)2+ y2d(E(t)γ(ζ)g, γ(ζ)g)
converges tozeroasy 0.Thereforeforx∈ R(3.20)implies
Im(M (x + i0)g, g) =|x − ζ|2Im(A0− (x + i0))−1γ(ζ)g, γ(ζ)g
, (3.21) inthesensethatoneofthelimitsexistsifandonlyiftheotherlimitexists,where +∞ is allowedas(improper) limit.
For u∈ H,x∈ R,and y > 0 theimaginary partoftheStieltjes transformFu of the measure μu= (E(·)u,u) isgivenby
Im Fu(x + iy) = Im R 1 t− (x + iy)d(E(t)u, u) = Im(A0− (x + iy))−1u, u , (3.22)
and foru∈ E(Δ)H we obtain
Im Fu(x + i0) = Im(A0− (x + i0))−1u, u if x∈ Δ, 0 if x /∈ Δ,
inparticular,ifu= E(Δ)γ(ζ)g∈ DΔ then
Im Fu(x + i0) = Im(A0− (x + i0))−1γ(ζ)g, γ(ζ)g if x∈ Δ, 0 if x /∈ Δ.
Takinginto account(3.21) wethenfind
Im Fu(x + i0) =
|x − ζ|−2Im(M (x + i0)g, g) if x∈ Δ,
0 if x /∈ Δ, (3.23)
foru= E(Δ)γ(ζ)g∈ DΔ.From Lemma 3.7(i)weconcludetogetherwith (3.23)that
supp μu,ac= clac x∈ Δ : 0 < Im Fu(x + i0) < +∞ = clac x∈ Δ : 0 < Im(M(x + i0)g, g) < +∞ holds foru= E(Δ)γ(ζ)g∈ DΔ,whichshows(3.19). 2
Corollary3.9.LetAssumption 3.1besatisfiedandassumethat(3.14)holdsforeachopen intervalδ⊂ R suchthat δ∩ σp(S)=∅.Then
σac(A0) = g∈ran Γ0 clac x∈ R : 0 < Im(M(x + i0)g, g) < +∞ .
Corollary3.10. LetAssumption 3.1be satisfiedand letΔ⊂ R be anopen intervalsuch that the condition (3.8) holds. Then the absolutely continuous spectrum of A0 in Δ is
givenby σac(A0)∩ Δ = g∈ran Γ0 clac x∈ Δ : 0 < Im(M(x + i0)g, g) < +∞ .
Inthenextcorollaryanecessaryandsufficientconditionfortheabsenceofabsolutely continuousspectrumisgiven.
Corollary 3.11. Let Assumption 3.1 be satisfied and let Δ ⊂ R be an open interval. Assumethatthecondition(3.14) holdsforeachopenintervalδ⊂ Δ withδ∩ σp(S)=∅.
Thenσac(A0)∩ Δ=∅ if andonly ifforeach g∈ ran Γ0 onehas Im(M (x+ i0)g,g)= 0
foralmostallx∈ Δ.
Proof. Wemakeuseofthefactthatforg∈ ran Γ0
clac
{x ∈ Δ : 0 < Im(M(x + i0)g, g) < +∞}=∅ (3.24) ifandonlyif
{x∈ Δ : 0 < Im(M(x + i0)g, g) < +∞}= 0. (3.25) Assumefirstthatσac(A0)∩Δ=∅.Then(3.15)yields(3.24)forallg∈ ran Γ0,andhence
(3.25)holds forallg ∈ ran Γ0.Moreover,for u= γ(ζ)g andx∈ R by(3.21) and(3.22)
wehave
Im(M (x + i0)g, g) =|x − ζ|2Im Fu(x + i0),
and by Lemma 3.7 this limit exists and is finite for Lebesgue almost all x ∈ R. Hence (3.25) implies that for all g ∈ ran Γ0 one has Im(M (x+ i0)g,g) = 0 for
al-mostallx∈ Δ.Forthe converseimplication assumethatfor everyg∈ ran Γ0 onehas
Im(M (x+ i0)g,g)= 0 foralmostallx∈ Δ.Then(3.25)andhencealso(3.24) holdsfor allg∈ ran Γ0.Thus(3.15) yieldsσac(A0)∩ Δ=∅. 2
Letus provenext inclusionsfor thesingular and singularcontinuous spectra of A0.
Theorem3.12.LetAssumption 3.1besatisfied,andletΔ⊂ R beanopeninterval.Then thefollowingassertions hold.
(i) Ifthecondition (3.8)holdsthen thesingularspectrumof A0 inΔ satisfies
σs(A0)∩ Δ ⊂ g∈ran Γ0 x∈ Δ : Im(M(x + i0)g, g) = +∞ .
(ii) If thecondition (3.14) issatisfied foreach open intervalδ⊂ Δ with δ∩ σp(S)=∅
thenthesingularcontinuous spectrumofA0 inΔ,σsc(A0)∩ Δ,iscontainedin the
set g∈ran Γ0
clc
x∈ Δ : Im(M(x + i0)g, g) = +∞, lim
y0y(M (x + iy)g, g) = 0
.
If S issimple then (i)and (ii)holdforeachopenintervalΔ,including thecaseΔ=R. Proof. We showthestatements(i)and(ii)at once.Letusdefine
DΔ:=
E(Δ)γ(ζ)g : ζ∈ C \ R, g ∈ ran Γ0
.
Note firstthatthesameargumentsasinStep 1oftheproofof Theorem 3.8imply σi(A0)∩ Δ =
u∈DΔ
supp μu,i, i = s, sc. (3.26)
In order to apply Lemma 3.7 (ii) and (iii), respectively, we calculate the limits that appearthere.Infact,itfollowsfrom (2.2)thatforeachg∈ ran Γ0and eachζ∈ C\ R
lim
y0Im(M (x + iy)g, g) =|x − ζ| 2lim y0Im (A0− (x + iy))−1γ(ζ)g, γ(ζ)g (3.27) and lim y0y(M (x + iy)g, g) =|x − ζ| 2lim y0y (A0− (x + iy))−1γ(ζ)g, γ(ζ)g (3.28) hold;cf.(3.21) forthefirstidentityandthetextbelow(3.21)foritsinterpretationas a possiblyimproperlimit.Letu= E(Δ)γ(ζ)g∈ DΔ andlet
Fu(x + iy) = R 1 t− (x + iy)d(E(t)u, u) = (A0− (x + iy))−1u, u
be theStieltjestransformofμu= (E(·)u,u).Then Im Fu(x + i0) = Im
(A0− (x + i0))−1E(Δ)γ(ζ)g, E(Δ)γ(ζ)g
forallx∈ R. Fromthisweconcludewiththehelpof(3.27)that Im Fu(x + i0) =
|x − ζ|−2Im(M (x + i0)g, g) if x∈ Δ,
0 if x /∈ Δ. (3.29)
Similarly,from(3.28)weobtain lim
y0yFu(x + iy) =
|x − ζ|−2limy0y(M (x + iy)g, g) if x∈ Δ,
0 if x /∈ Δ. (3.30)
Itfollowsfrom (3.29),(3.30),andLemma 3.7that supp μu,s⊂ x∈ Δ : Im(M(x + i0)g, g) = +∞ and supp μu,sc⊂ clc
x∈ Δ : Im(M(x + i0)g, g) = +∞, lim
y0y(M (x + iy)g, g) = 0 foru= E(Δ)γ(ζ)g∈ DΔ.Thustheassertionsofthetheorem followfrom(3.26). 2
Weformulatetwo immediatecorollaries whichconcernthesingularcontinuous spec-trum.
Corollary 3.13. Let Assumption 3.1 be satisfied and assume that (3.14) holds for each open interval δ ⊂ R such that δ∩ σp(S) = ∅. Then the singular continuous spectrum
σsc(A0) of A0 iscontainedin theset
g∈ran Γ0
clc
x∈ R : Im(M(x + i0)g, g) = +∞, lim
y0y(M (x + iy)g, g) = 0
.
Corollary 3.14. Let Assumption 3.1 be satisfied, let Δ ⊂ R be an open interval, and assume that the condition (3.8)holds. Then thesingular continuous spectrum of A0 in
Δ,σsc(A0)∩ Δ,is containedintheset
g∈ran Γ0
clc
x∈ Δ : Im(M(x + i0)g, g) = +∞, lim
y0y(M (x + iy)g, g) = 0
.
Asafurtherimmediatecorollary ofTheorem 3.12we formulateasufficient criterion fortheabsenceofsingularcontinuousspectrumintermsofthelimitingbehavior ofthe function M .Thecorrespondingresultforordinaryboundarytriples(in thespecialcase Δ=R)canbefoundin[10].
Corollary3.15. LetAssumption 3.1be satisfiedand letΔ⊂ R be anopen intervalsuch that thecondition(3.14) issatisfiedforeach openintervalδ⊂ Δ withδ∩ σp(S)=∅.If
Im(M (x + iy)g, g)→ +∞ and y(M(x + iy)g, g) → 0 as y 0
then σsc(A0)∩ Δ = ∅. If S is simple the assertion holds for each open interval Δ,
including thecaseΔ=R.
Asafurthercorollaryofthetheorems ofthissectionweprovidesufficientcriteriafor the spectrumof theoperator A0 to be purely absolutely continuous or purely singular
continuous, respectively,insomeset.
Corollary 3.16.LetAssumption 3.1 besatisfied, letΔ⊂ R beanopenintervalsuchthat thecondition (3.8)issatisfied, andassume that
lim
y0yM (x + iy)g = 0 (3.31)
forallg∈ ran Γ0 andallx∈ Δ.Then thefollowingassertionshold.
(i) Ifforeachg∈ ran Γ0thereexistatmostcountablymanyx∈ Δ suchthatIm(M (x+
i0)g,g)= +∞ thenσ(A0)∩ Δ= σac(A0)∩ Δ.
(ii) If for each g ∈ ran Γ0 one has Im(M (x+ i0)g,g) = 0 for almost all x∈ Δ then
σ(A0)∩ Δ= σsc(A0)∩ Δ.
In particular,if S issimple andΔ is an arbitrary open intervalsuch that (3.31) holds forallg∈ ran Γ0 andallx∈ Δ then (i)and (ii) aresatisfied.
4. SecondorderellipticdifferentialoperatorsonRRRn
In this section we show how the spectrumof aselfadjoint second order elliptic dif-ferential operatoron Rn,n≥ 2, canbe described with thehelpofaTitchmarsh–Weyl function acting on an n− 1-dimensional compact interface Σ which splits Rn into a bounded domainΩi andanunboundeddomainΩe withcommonboundaryΣ.
Weconsider thedifferentialexpression
L = − n j,k=1 ∂ ∂xj ajk ∂ ∂xk + n j=1 aj ∂ ∂xj − ∂ ∂xj aj + a,
whereajk,aj ∈ C∞(Rn) togetherwiththeirderivativesareboundedandsatisfyajk(x)=
akj(x) forallx∈ Rn,1≤ j,k≤ n,anda∈ L∞(Rn) isrealvalued.Moreover,weassume thatL isuniformly ellipticonRn,thatis, thereexistsE > 0 with
n j,k=1 ajk(x)ξjξk ≥ E n k=1 ξk2, x∈ Rn, ξ = (ξ1, . . . , ξn)∈ Rn. (4.1)
Theselfadjointoperatorassociated withL inL2(Rn) isgiven by
A0u =Lu, dom A0= H2(Rn), (4.2)
whereH2(Rn) istheusualL2-basedSobolevspaceoforder2 onRn.InSections4.1and
4.2two differentchoicesofTitchmarsh–Weyl functionsforthedifferentialexpression L arestudied.
4.1. A Weylfunctioncorresponding toatransmissionproblem
WefirstconsideraWeylfunctionfortheoperatorA0whichappearsintransmission
problemsinconnectionwithsinglelayerpotentials(see,e.g.[50,Chapter 6])andwhich was also used in [2] to generalize the classical limit point/limit circle analysis from singularSturm–Liouvilletheory toSchrödingeroperatorsinR3.
Let Σ be the boundary of a bounded C∞-domain Ωi ⊂ Rn and denote by Ωe the
corresponding exteriordomain,that is,Ωe =Rn\ Ωi.In thefollowing we make use of
operators induced by L in L2(Ω
i) and L2(Ωe), respectively. For j = i, e we write Lj for the restrictionof the differential expression L to functions on Ωj. For functionsin
L2(Ω
j) weuse theindex j and we write u= ui⊕ ue for u∈ L2(Rn).As Σ issmooth,
theselfadjointDirichletoperatorassociated withLj inL2(Ωj) isgivenby
AD,juj=Ljuj, dom AD,j=
uj∈ H2(Ωj) : uj|Σ= 0
, j = i, e,
where uj|Σ denotes the trace of uj at Σ = ∂Ωj. Let Hs(Σ) be the Sobolev spaces of orders s ≥ 0 onΣ. We recall that for each λ ∈ ρ(AD,j) and each g ∈ H3/2(Σ) there
exists a unique solution uλ,j ∈ H2(Ωj) of the boundary value problem Ljuj = λuj,
uj|Σ= g.Thisimpliesthatforeachλ∈ ρ(AD,j) theDirichlet-to-Neumannmap
Λj(λ) : H3/2(Σ)→ H1/2(Σ), uλ,j|Σ→ ∂uλ,j ∂νLj Σ, (4.3)
is well-defined; herethe conormal derivativewith respect to Lj inthe directionof the outerunitnormalνj= (νj,1,. . . ,νj,n) atΣ= ∂Ωj isdefinedby
∂u ∂νLj Σ= n k,l=1 aklνj,k ∂u ∂xl Σ+ n k=1 akνj,ku|Σ.
Notethattheouterunitnormalsat∂Ωi and∂Ωecoincideuptoaminussign.Itwillturn
outbelowthattheoperatorΛi(λ)+Λe(λ) isinvertibleforallλ∈ ρ(A0)∩ρ(AD,i)∩ρ(AD,e)
and,hence,theoperatorfunction
λ→ M(λ) =Λi(λ) + Λe(λ)
−1
iswell-definedonρ(A0)∩ρ(AD,i)∩ρ(AD,e).WeremarkthatthevaluesM (λ) arebounded
operators inL2(Σ) withdomainH1/2(Σ);cf.Lemma 4.2belowforthedetails.
The following theorem is the main result of this section. It states that the abso-lutelycontinuousspectrumofA0canberecoveredcompletelyfromtheknowledgeofthe
function M in (4.4), whilethe eigenvaluesand corresponding eigenspaces maybe only partially visible for thefunctionM . This dependsonthe choiceofthe interfaceΣ and thefactthatthesymmetricoperator
Su =Lu, dom S =u∈ H2(Rn) : u|Σ= 0
, (4.5)
may have eigenvalues. In particular, in general S is not simple; cf. Example 4.5 and
Example 4.6 below.
Theorem 4.1. Let A0, Σ, S, and M be as above, let λ,μ ∈ R such that λ ∈ σ/ p(S),
μ∈ σ/ p(S), andletΔ⊂ R be anopen interval.Thenthefollowingassertions hold.
(i) μ∈ σp(A0) if andonly ifRμM := s- limη0iηM (μ+ iη)= 0;ifthemultiplicityof
theeigenvalue μ isfinite thenthemapping
τ : ker(A0− μ) → ran RμM, u→ u|Σ, (4.6)
isbijective; ifthemultiplicityof theeigenvalueμ is infinite thenthemapping τ : ker(A0− μ) → clτ
ran RμM
, u→ u|Σ, (4.7)
isbijective, whereclτ denotestheclosure inthenormed spaceran τ , equippedwith
thenormin L2(Σ).
(ii) λ isanisolatedeigenvalueofA0 ifandonlyif λ isapoleinthestrongsense of M .
Inthiscase(4.6)and(4.7)withμ= λ arebijectivemappingsandRλM = ResλM . (iii) λ∈ ρ(A0) if andonlyif M canbe continuedanalytically intoλ.
(iv) λ ∈ σc(A0) if and only if s- limη0iηM (λ+ iη)= 0 and M cannot be continued
analytically intoλ.
(v) Theabsolutely continuousspectrumσac(A0) ofA0 inΔ isgivenby
σac(A0)∩ Δ = g∈H1/2(Σ) clac x∈ Δ : 0 < Im(M(x + i0)g, g) < +∞
and, in particular, σac(A0)∩ Δ =∅ if and only if for each g ∈ H1/2(Σ) one has
Im(M (x+ i0)g,g)= 0 foralmostallx∈ Δ.
(vi) Thesingularcontinuous spectrum ofA0 inΔ, σsc(A0)∩ Δ, iscontainedin
g∈H1/2(Σ)
clc
x∈ Δ : Im(M(x + i0)g, g) = +∞, lim
y0y(M (x + iy)g, g) = 0
and, in particular, if for each g ∈ H1/2(Σ) there exist at most countably many x∈ Δ suchthatIm(M (x+ iy)g,g)→ +∞ andy(M (x+ iy)g,g)→ 0 asy 0 then σsc(A0)∩ Δ=∅.
TheproofofTheorem 4.1makesuseofthefollowingtwolemmas andisgivenatthe endofthis subsection.
Lemma4.2. LetS be definedasin (4.5)andlet T u =Liui⊕ Leue, dom T =ui⊕ ue ∈ H2(Ωi)⊕ H2(Ωe) : ui|Σ= ue|Σ . (4.8) Then{L2(Σ),Γ 0,Γ1}, where Γ0, Γ1: dom T → L2(Σ), Γ0u = ∂ui ∂νLi Σ+ ∂ue ∂νLe Σ, Γ1u = u|Σ,
isa quasiboundary triple for S∗ suchthat A0= T ker Γ0 andran Γ0= H1/2(Σ). For
all λ ∈ ρ(A0)∩ ρ(AD,i)∩ ρ(AD,e) the corresponding Weyl function coincides with the
functionM in(4.4),anddom M (λ)= H1/2(Σ).
Proof. Theproofissimilartotheproofof[5,Proposition3.2].Fortheconvenienceofthe readerweprovidethedetails.Inordertoshowthat{L2(Σ),Γ
0,Γ1} isaquasiboundary
tripleforS∗weverify(i)–(iii)intheassumptionsofProposition 2.2.Recallfirstthatby theclassicaltracetheoremthemapping
H2(Ωj)→ H3/2(Σ)× H1/2(Σ), uj → uj|Σ, ∂uj ∂νLj Σ , j = i, e,
is onto.Hence, forgiven ϕ∈ H1/2(Σ) and ψ∈ H3/2(Σ) thereexist uj ∈ H2(Ωj) such that ∂ui ∂νLi Σ= ϕ, ∂ue ∂νLe Σ= 0, and ui|Σ= ψ = ue|Σ,
and it follows ui⊕ ue ∈ dom T , Γ0(ui⊕ ue) = ϕ, and Γ1(ui⊕ ue) = ψ. This implies
thatran(Γ0,Γ1) = H1/2(Σ)× H3/2(Σ).Inparticular,ran(Γ0,Γ1) isdenseinL2(Σ)×
L2(Σ). Furthermore, C0∞(Rn\Σ) is adense subspace of L2(Rn) which is contained in ker Γ0∩ ker Γ1. Thus(i) inProposition 2.2 holds. Nextwe verify theidentity (2.1) for
u= ui ⊕ ue,v = vi⊕ ve ∈ dom T . With the helpof Green’s identity and u|Σ = uj|Σ,
