On m-sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry

Comment.Math.Univ.Carolin. 45,1(2004)91 100 91 On m-sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry Ognjen Milatovic Abstract. WeconsideraSchrödinger-typedifferentialexpression H V = +V,where isac -boundedhermitianconnectiononahermitianvectorbundle Eofbounded geometryoveramanifoldofboundedgeometry(m, g)withmetric gandpositive C - bounded measure dµ, and V is a locally integrable section of the bundle of endomorphismsof E. Wegiveasufficientconditionfor m-sectorialityofarealizationof H V in L 2 (E). IntheproofweusegeneralizedKato sinequalityaswellasaresultonthe positivityof u L 2 (M)satisfyingtheequation( M + b)u=ν,where M isthescalar Laplacianon M, b >0isaconstantand ν 0isapositivedistributionon M. Keywords: Schrödinger operator, m-sectorial, manifold, bounded geometry, singular potential Classification: Primary 35P05, 58J50; Secondary 47B25, 81Q10 1. Introduction and the main result 1.1 The setting. Let(M, g)beac Riemannianmanifoldwithoutboundary,withmetric ganddimm = n. Wewillassumethat M isconnected. We will also assume that M has bounded geometry. Moreover, we will assume that wearegivenapositive C -boundedmeasure dµ,i.e.inanylocalcoordinates x 1, x 2,...,x n thereexistsastrictlypositive C -boundeddensity ρ(x)suchthat dµ=ρ(x)dx 1 dx 2... dx n. Let EbeaHermitianvectorbundleover M.Wewillassumethat Eisabundle of bounded geometry(i.e. it is supplied by an additional structure: trivializations of E on every canonical coordinate neighborhood U such that the corresponding matrixtransitionfunctions h U,U onallintersections U U ofsuchneighborhoodsare C -bounded,i.e.allderivatives y αh U,U (y),where αisamultiindex, withrespecttocanonicalcoordinatesareboundedwithbounds C α whichdonot dependonthechosenpair U, U ). Wedenoteby L 2 (E)theHilbertspaceofsquareintegrablesectionsof Ewith respect to the scalar product (1.1) (u, v) = u(x), v(x) dµ(x). M Here, denotesthefiberwiseinnerproductin E x.

92 O. Milatovic Inwhatfollows, C (E)denotessmoothsectionsof E,and C c (E)denotes smooth compactly supported sections of E. Let : C (E) C (T M E) beahermitianconnectionon E whichis C -boundedasalineardifferential operator, i.e. in any canonical coordinate system U(with the chosen trivializations of E U and(t M E) U ), iswrittenintheform = a α (y) y α, α 1 where αisamultiindex,andthecoefficients a α (y)arematrixfunctionswhose derivatives β y a α (y)foranymultiindex βareboundedbyaconstant C β which does not depend on the chosen canonical neighborhood. We will consider a Schrödinger type differential expression of the form H V = +V, where V isameasurablesectionofthebundleendeofendomorphismsof E. Here : C (T M E) C (E) isadifferentialoperatorwhichisformallyadjointto withrespecttothescalar product(1.1). Ifwetake =d,where d: C (M) Ω 1 (M)isthestandarddifferential, then d d: C (M) C (M)iscalledthescalarLaplacianandwillbedenoted by M. In what follows, we use the notations (1.2) (Re V)(x):= V(x)+(V(x)) 2, (Im V)(x):= V(x) (V(x)) 2i, x M, where i= 1and(V(x)) denotestheadjointofthelinearoperator V(x): E x E x (inthesenseoflinearalgebra). By(1.2),forall x M,(Re V)(x)and(ImV)(x)areself-adjointlinearoperators E x E x,andwehavethefollowingdecomposition: V(x)=(Re V)(x)+i(Im V)(x). Forevery x M,wehavethefollowingdecomposition: (1.3) (Re V)(x)=(Re V) + (x) (Re V) (x). Here(Re V) + (x)=p + (x)(re V)(x),where P + (x):= χ [0,+ ) ((Re V)(x)),and (Re V) (x)= P (x)(re V)(x),where P (x):= χ (,0) ((Re V)(x)). Here χ A denotes the characteristic function of the set A. Wemakethefollowingassumptionon V.

On m-sectorial Schrödinger-type operators... 93 Assumption A. (i) (Re V) + L 1 loc (EndE),(Re V) L 1 loc (EndE)and(ImV) L1 loc (EndE). (ii) Thereexistsaconstant L >0suchthatforall u L 2 (E)andall x M, (1.4) (Im V)(x) u(x) 2 L (Re V) + (x)u(x), u(x), where (Im V)(x) denotesthenormoftheoperator(im V)(x): E x E x, u(x) denotesthenorminthefiber E x and, denotestheinnerproduct in E x. 1.2Sobolevspace W 1,2 (E). By W 1,2 (E)wewilldenotethecompletionofthe space C c (E)withrespecttothenorm 1definedbythescalarproduct (u, v) 1 := (u, v) + ( u, v) u, v C c (E). By W 1,2 (E)wewilldenotethedualof W 1,2 (E). 1.3 Quadratic forms. In what follows, all quadratic forms are considered in thehilbertspace L 2 (E). 1.By h 0 wedenotethequadraticform (1.5) h 0 (u)= u 2 dµ withthedomaind(h 0 )=W 1,2 (E) L 2 (E). Thequadraticform h 0 isnonnegative,denselydefined(since C c (E) D(h 0 ))andclosed(seesection1.2). 2.By h 1 wedenotethequadraticform (1.6) h 1 (u)= (Re V) + u, u +i (Im V)u, u dµ with the domain (1.7) D(h 1 )= { u L 2 (E): (Re V) + u, u +i (Im V)u, u dµ <+ }. Here, denotesthefiberwiseinnerproductin E x. Inwhatfollows,wewilldenoteby h 1 (, )thecorrespondingsesquilinearform obtainedviapolarizationidentityfrom h 1. Thequadraticform h 1 issectorial.indeed,bytheinequalities (1.8) (Im V)u(x), u(x) (Im V)(x)u(x) u(x) (Im V)(x) u(x) 2 andby(1.4),forall u D(h 1 ),thevaluesof h 1 (u)lieinasectorof Cwithvertex γ=0. Theform h 1 isdenselydefinedsince,by(i)ofassumptiona,wehave

94 O. Milatovic Cc (E) D(h 1). Theform h 1 isclosed. Indeed,byTheoremVI.1.11in[6],it sufficestoshowthatthepre-hilbertspaced(h 1 )withtheinnerproduct (u, v) h1 =(Re h 1 )(u, v)+(u, v)= (Re V) + u, v dµ+(u, v), iscomplete.here(, )denotestheinnerproductin L 2 (E)and(Re h 1 )(, )denotes therealpartofthesesquilinearform h 1 (, )(seethedefinitionbelowtheequation (1.9) in Section VI.1.1 of[6]). By(1.7),(1.8)and(1.4),itfollowsthatD(h 1 )isthesetofall u L 2 (E)such that u 2 h 1 <+,where h1 denotesthenormcorrespondingtotheinner product(, ) h1.byexamplevi.1.15in[6],itfollowsthatd(h 1 )iscomplete. 3.By h 2 wedenotethequadraticform (1.9) h 2 (u)= (Re V) u, u dµ with the domain (1.10) D(h 2 )= { u L 2 (E): (Re V) u, u dµ <+ }, where, denotesthefiberwiseinnerproductin E x. Theform h 2 isdenselydefinedbecause, by(i)ofassumptiona,wehave C c (E) D(h 2 ).Moreover,sinceforall x M,theoperator(Re V) (x): E x E x isself-adjoint,itfollowsthatthequadraticform h 2 issymmetric. Wemakethefollowingassumptionon h 2. AssumptionB. Assumethat h 2 is h 0 -boundedwithrelativebound b <1,i.e. (i) D(h 2 ) D(h 0 ), (ii) thereexistconstants a 0and0 b <1suchthat (1.11) h 2 (u) a u 2 + b h 0 (u), forall u D(h 0 ), where denotesthenormin L 2 (E). Remark1.4.If(M, g)isamanifoldofboundedgeometry,assumptionbholdsif (Re V) L p (EndE),where p=n/2for n 3, p >1for n=2,and p=1for n=1.fortheproof,see,forexample,theproofofremark2.1in[7]. 1.5Arealizationof H V in L 2 (E). Wedefineanoperator Sin L 2 (E)bythe formula Su=H V uonthedomain (1.12) { (Re u W 1,2 (E): V) + u, u +i (Im V)u, u dµ <+ and H V u L 2 (E) }.

On m-sectorial Schrödinger-type operators... 95 Wewilldenotethesetin(1.12)byDom(S). Remark1.6.Forall u D(h 0 )=W 1,2 (E)wehave u W 1,2 (E). From Corollary2.11belowitfollowsthatforall u W 1,2 (E) D(h 1 ),wehave V u L 1 loc (E). Thus H V uin(1.12)isadistributionalsectionof E,andthecondition H V u L 2 (E)makessense. Remark1.7.By(1.4)andby(1.8),thesetDom(S)in(1.12)isequalto { u W 1,2 (E): (Re V) + u, u dµ <+ and H V u L 2 (E) }. Wenowstatethemainresult. Theorem 1.8. Assumethat(M, g)isamanifoldofboundedgeometrywith positive C -boundedmeasure dµ, EisaHermitianvectorbundleofbounded geometryover M,and isac -boundedhermitianconnectionon E.Suppose thatassumptionsaandbhold.thentheoperator Sis m-sectorial. Remark1.9.Theorem1.8extendsaresultofT.Kato;seeTheoremVI.4.6(a)in[6] (seealsoremark5(a)in[5])whichwasprovenfortheoperator +V,where isthestandardlaplacianon R n withthestandardmetricandmeasure,and V L 1 loc (Rn )isasinassumptionsaandbabove(withimv =0).Theorem1.8 also extends the result in[7] which establishes the self-adjointness of a realization in L 2 (E)of H V = +V onmanifold(m, g)with dµ, E,and asinthe hypothesesoftheorem1.8,and V = V 1 + V 2,where0 V 1 L 1 loc (EndE)and 0 V 2 L 1 loc (EndE)arelinearself-adjointbundleendomorphismssatisfying AssumptionsAandB(withImV =0). 2. ProofofTheorem1.8 WeadopttheargumentsfromSectionVI.4.3in[6]tooursettingwiththehelp of a more general version of Kato s inequality(2.1). 2.1 Kato s inequality. We begin with the following variant of Kato s inequality for Bochner Laplacian(for the proof, see Theorem 5.7 in[2]). Lemma2.2. Assumethat(M, g)isariemannianmanifold.assumethat Eisa Hermitianvectorbundleover Mand isahermitianconnectionon E.Assume that w L 1 loc (E)and w L 1 loc (E).Then (2.1) M w Re w,sign w, where sign w(x) = { w(x) if w(x) 0, w(x) 0 otherwise. Remark 2.3. The original version of Kato s inequality was proven in Kato[3].

96 O. Milatovic 2.4 Positivity. In what follows, we will use the following lemma whose proof is giveninappendixbof[2]. Lemma 2.5. Assumethat(M, g)isamanifoldofboundedgeometrywitha smooth positive measure dµ. Assume that ( b+ M ) u = ν 0, u L 2 (M), where b >0, M = d disthescalarlaplacianon M,andtheinequality ν 0 meansthat ν isapositivedistributionon M,i.e.(ν, φ) 0forany0 φ C c (M). Then u 0(almost everywhere or, equivalently, as a distribution). Remark2.6.ItisnotknownwhetherLemma2.5holdsif Misanarbitrarycomplete Riemannian manifold. For more details about difficulties in the case of arbitrary complete Riemannian manifolds, see Appendix B of[2]. Lemma2.7. Thequadraticform h:=(h 0 +h 1 )+h 2 isdenselydefined,sectorial and closed. Proof:Since h 0 and h 1 aresectorialandclosed,itfollowsbytheoremvi.1.31 from[6]that h 0 +h 1 issectorialandclosed.by(i)ofassumptionbitfollowsthat D(h 2 ) D(h 0 ) D(h 1 ),andby(1.11),(1.5),and(1.6),thefollowinginequality holds: h 2 (u) a u 2 + b h 0 (u)+h 1 (u), forall u D(h 0 ) D(h 1 ), where denotesthenormin L 2 (E),and a 0and0 b <1areasin(1.11). Thusthequadraticform h 2 is(h 0 +h 1 )-boundedwithrelativebound b <1.Since h 0 + h 1 isaclosedsectorialform,bytheoremvi.1.33from[6],itfollowsthat h=(h 0 + h 1 )+h 2 isaclosedsectorialform. Since Cc (E) D(h 0 ) D(h 1 ) D(h 2 ),itfollowsthat hisdenselydefined. In what follows, h(, ) will denote the corresponding sesquilinear form obtained from h via polarization identity. 2.8 m-sectorial operator H associated with h. Since h is a densely defined, closedandsectorialformin L 2 (E),byTheoremVI.2.1from[6]thereexistsan m-sectorialoperator Hin L 2 (E)suchthat (i) Dom(H) D(h)and h(u, v)=(hu, v), (ii) Dom(H)isacoreof h, (iii) if u D(h), w L 2 (E),and forall u Dom(H)and v D(h), h(u, v)=(w, v) holdsforevery vbelongingtoacoreof h,then u Dom(H)and Hu=w. The operator H is uniquely determined by condition(i). We will also use the following lemma.

On m-sectorial Schrödinger-type operators... 97 Lemma2.9. Assumethat0 T L 1 loc (EndE)isalinearself-adjointbundle map.assumealsothat u Q(T),where Q(T)={u L 2 (E): Tu, u L 1 (M)}. Then Tu L 1 loc (E). Proof:Byaddingaconstantwecanassumethat T 1(inoperatorsense). Assumethat u Q(T). Wechoose(inameasurableway)anorthogonal basisin eachfiber E x anddiagonalize1 T(x) End(E x ) toget T(x) = diag(c 1 (x), c 2 (x),..., c m (x)), where 0 < c j L 1 loc (M), j = 1,2,..., m and m=dime x. Let u j (x)(j=1,2,..., m)bethecomponentsof u(x) E x withrespectto thechosenorthogonalbasisof E x.thenforall x M Tu, u = m c j (x) u j (x) 2. j=1 Since u Q(T),weknowthat0< Tu, u dµ <+. Since c j >0,itfollows that c j u j 2 L 1 (M),forall j=1,2,...,m. Now,forall x Mand j=1,2,, m (2.2) 2 c j u j =2 c j u j c j + c j u j 2. Therighthandsideof(2.2)isclearlyin L 1 loc (M).Therefore c ju j L 1 loc (M). But(Tu)(x)hascomponents c j (x)u j (x)(j=1,2,...,m)withrespecttochosenbasesof E x.therefore Tu L 1 loc (E),andthelemmaisproven. Corollary2.10. If u D(h 1 ),then((re V) + + i(im V))u L 1 loc (E). Proof:Let u D(h 1 ).Then (Re V) + u, u L 1 (M),and,hence,byLemma2.9 weget(re V) + u L 1 loc (E).By(1.4)weobtain (Im V) u 2 L 1 (M).Sincefor all x Mwehave 2 (Im V)(x)u(x) 2 (Im V)(x) u(x) (Im V)(x) + (Im V)(x) u(x) 2, and since, by AssumptionA, (Im V) L 1 loc (M), it followsthat (ImV)u L 1 loc (E),andthecorollaryisproven. Corollary2.11. If u D(h),then V u L 1 loc (E). Proof: Let u D(h) = D(h 0 ) D(h 1 ). ByCorollary2.10it followsthat ((Re V) + + i(im V))u L 1 loc (E).SinceD(h) D(h 2)andsince(Re V) (x) 0 asanoperator E x E x,bylemma2.9wehave(re V) u L 1 loc (E). Thus V u L 1 loc (E),andthecorollaryisproven.

98 O. Milatovic Lemma 2.12. The following operator relation holds: H S. Proof:Wewillshowthatforall u Dom(H),wehave Hu=H V u. Let u Dom(H).Byproperty(i)ofSection2.8wehave u D(h);hence,by Corollary2.11weget V u L 1 loc (E).Then,forany v C c (E),wehave (2.3) (Hu, v)=h(u, v)=( u, v)+ V u, v dµ, where(, )denotesthe L 2 -innerproduct. Thefirstequalityin(2.3)holdsbyproperty(i)fromSection2.8,andthesecond equality holds by definition of h. Hence,usingintegrationbypartsinthefirsttermontherighthandsideof thesecondequalityin(2.3)(see,forexamplelemma8.8from[2]),weget (2.4) (u, v)= Hu V u, v dµ, forall v C c (E). Since V u L 1 loc (E)and Hu L2 (E),itfollowsthat(Hu V u) L 1 loc (E),and (2.4)implies u=hu V u(asdistributionalsectionsof E).Therefore, u+vu=hu, andthisshowsthat Hu=H V uforall u Dom(H). Nowbydefinitionof SitfollowsthatDom(H) Dom(S)and Hu=Sufor all u Dom(H).Therefore H S,andthelemmaisproven. Lemma2.13. C c (E)isacoreofthequadraticform h 0+ h 1. Proof:Itsufficestoshow(seeTheoremVI.1.21in[6]andtheparagraphabove theequation(1.31)insectionvi.1.3of[6])that C c (E)isdenseintheHilbert spaced(h 0 + h 1 )=D(h 0 ) D(h 1 )withtheinnerproduct (u, v) h0 +h 1 := h 0 (u, v)+(re h 1 )(u, v)+(u, v), where h 0 (, )denotesthesesquilinearformcorrespondingto h 0 viapolarization identityand(re h 1 )denotestherealpartofthesesquilinearform h 1 (, ). Let u D(h 0 + h 1 )and(u, v) h0 +h 1 =0forall v C c (E).Wewillshowthat u=0. We have (2.5) 0 = h 0 (u, v)+(re h 1 )(u, v)+(u, v) = (u, v)+ (Re V) + u, v dµ+(u, v).

On m-sectorial Schrödinger-type operators... 99 Hereweusedintegrationbypartsinthefirsttermontherighthandsideofthe second equality. Since u D(h 0 +h 1 ) D(h 1 ),itfollowsthat (Re V) + u, u +i (ImV)u, u L 1 (M). Hence (Re V) + u, u L 1 (M). ByLemma2.9weget(Re V) + u L 1 loc (E).From(2.5)wegetthefollowingdistributionalequality: (2.6) u=( (Re V) + 1)u. From(2.6)wehave u L 1 loc (E).ByLemma2.2andby(2.6),weobtain (2.7) M u Re u,signu = ((Re V) + +1)u,signu u. Thelastinequalityin(2.7)holdssince(Re V) + (x) 0asanoperator E x E x. Therefore, (2.8) ( M +1) u 0. ByLemma2.5,itfollowsthat u 0.So u=0,andthelemmaisproven. Lemma2.14. C c (E)isacoreofthequadraticform h=(h 0+ h 1 )+h 2. Proof: Sincethequadraticform h 2 is(h 0 + h 1 )-bounded,thelemmafollows immediately from Lemma 2.13. 3. ProofofTheorem1.8 ByLemma2.12wehave H S,soitisenoughtoshowthatDom(S) Dom(H). Let u Dom(S).BydefinitionofDom(S)inSection1.5,wehave u D(h 0 ) D(h 2 )and u D(h 1 ).Hence u D(h). Forall v C c (E)wehave h(u, v)=h 0 (u, v)+h 1 (u, v)+h 2 (u, v)=(u, v)+ V u, v dµ=(h V u, v). Thelastequalityholdssince H V u=su L 2 (E). ByLemma2.14itfollows that C c (E)isaformcoreof h. Nowfromproperty(iii)ofSection2.8wehave u Dom(H)with Hu=H V u.thisconcludestheproofofthetheorem. References [1] Aubin T., Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, Berlin, 1998. [2] Braverman M., Milatovic O., Shubin M., Essential self-adjointness of Schrödinger type operators on manifolds, Russian Math. Surveys 57(2002), no. 4, 641 692. [3] Kato T., Schrödinger operators with singular potentials, Israel J. Math. 13(1972), 135 148.

100 O. Milatovic [4] Kato T., A second look at the essential selfadjointness of the Schrödinger operators, Physical reality and mathematical description, Reidel, Dordrecht, 1974, pp. 193 201. [5] Kato T., On some Schrödinger operators with a singular complex potential, Ann. Scuola Norm.Sup.PisaCl.Sci(4)5(1978),105 114. [6] Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1980. [7] Milatovic O., Self-adjointness of Schrödinger-type operators with singular potentials on manifolds of bounded geometry, Electron. J. Differential Equations, No. 64(2003), 8pp (electronic). [8] Reed M., Simon B., Methods of Modern Mathematical Physics I, II: Functional Analysis. Fourier Analysis, Self-adjointness, Academic Press, New York e.a., 1975. [9] Shubin M.A., Spectral theory of elliptic operators on noncompact manifolds, Astérisque no. 207(1992), 35 108. [10] Taylor M., Partial Differential Equations II: Qualitative Studies of Linear Equations, Springer-Verlag, New York e.a., 1996. Department of Mathematics, University of Toledo, Toledo, OH 43606-3390, USA E-mail: Ognjen.Milatovic@utoledo.edu (Received September 9, 2003)