CCoonntteemmppoorraarryyMMaatthheemmaattiiccss Volume340,2004 Topics from spectral theory of differential operators Andreas M. Hinz Dedicatedto Christine Contents Introduction 2 1. Self-adjointness of Schro¨dinger operators 3 1.0. Solving the Schro¨dinger equation 3 1.1. Linear operators in Hilbert space 7 1.2. Criteria for (essential) self-adjointness 13 1.3. Application to Schro¨dinger operators 18 2. Hardy-Rellich inequalities 21 2.0. Relative boundedness 21 2.1. Weighted estimates 24 2.2. Explicit bounds 26 3. Spectral properties of radially periodic Schro¨dinger operators 28 3.0. Spectra of self-adjoint operators 29 3.1. Asymptotic behavior of eigensolutions and the spectrum 38 3.2. Spherical symmetry 40 3.2.1. Some basic examples 40 3.2.2. Embedded eigenvalues 42 3.3. Radial periodicity 44 3.3.1. Dense point spectrum 44 3.3.2. Welsh eigenvalues 46 3.4. Numerical analysis 46 References 48 2000MathematicsSubjectClassification. Primary35J10,47B25;Secondary35P20,35Q40. Keywordsandphrases. self-adjointness,Schrödingeroperator,Rellichinequality,spectrum. MytraveltoMexicowassupportedbyIIMAS-UNAMandDeutscheForschungsgemeinschaft. (cid:1)c 2004AmericanMath(cid:1)cem20a0ti3caAl.SMo.cHieitnyz 11 22 ANDREASM.HINZ Introduction Spectral methods for (partial) differential equations emerged as early as the 18th and 19th centuries in the works of D. Bernoulli and J. Fourier ([22]). However, it took almost a hundred years to clarify the notions of function and integrability in ordertoestablish thesoundness andapplicability oftheseideas. Thisdevelopment culminated at the dawn of the 20th century with the integral of H. Lebesgue ([38, 39]), which triggered the accomplishment of the early theory of functional analysis in the works of D. Hilbert, E. Schmidt and F. Riesz, highlighted by the proof of E. Fischer ([21]) of completeness of L . 2 The mathematical ground was therefore well prepared, when E. Schro¨dinger ([58]) came up with his equation ∂ (0.1) i Ψ(t,x)=(−(cid:1)+V(x)) Ψ(t,x), ∂t which may well be considered as one of the most outstanding pieces of physics and mathematics of all time. With his spectral theorem, J. v. Neumann ([41, 42]) foundedthespectraltheoryofSchro¨dingeroperators,putintoamoregeneralframe by M. H. Stone ([64]). In the first part of Chapter 1, we will give a modern and comprehensive presentation of this theory which reduces the problem of existence anduniqueness of solutionsfor(0.1)withtheinitialstateΨ =Ψ(0,·)giventothe 0 problem of establishing self-adjointness of the corresponding linear operator. The latter question has attracted attention for more than half a century with an ever expandingclassofadmissiblepotentialfunctionsV andsupplyofmethodsevolving. InthesecondpartofChapter1,wewillprovidesomeabstractcriteriafor(essential) self-adjointnesswhichcanbeappliedinaveryelegantwaytoSchro¨dingeroperators using only some results from regularity theory of differential equations. ¿Fromthelate1930s,originatingintheworksofF.RellichandT.Kato,pertur- bation theorybecameamightytooltoinvestigatebothqualitativeandquantitative propertiesoflinearoperators(cf.[36]). Itdependslargelyon,sometimesingenious, estimates to show that one part of the operator is subordinate to another one. As an example, we will discuss Hardy-Rellich inequalities in Chapter 2. With the basic properties of differential operators being established by 1970 (cf. [32, Chapter 3]), the last three decades of the 20th century were marked by a tremendous flow of diverse results about the spectra in a variety of cases like magnetic, random or one-dimensional Schro¨dinger operators and Dirac operators, a development led by the big promotor in the field, B.Simon (cf.[59]). To demon- stratethediversityofspectralphenomena,Chapter3willdiscuss,afteracondensed introductionintospectraofself-adjointoperators,thetechnicallyrathersimplecase ofsphericallysymmetricradiallyperiodicSchro¨dingeroperators. Here,wewillalso indicate that apart from analytical methods, numerical investigations are now vi- able, given the ever increasing power of electronic computing machinery. The following three chapters want to present a unified approach from the very beginnings of the theory to topics of current research. We do not attempt to give a comprehensive overview of the theory including appropriate recognition of all contributors, since this would be too enormous a task. . TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 33 1. Self-adjointness of Schrödinger operators In Section 1.0, we will show the necessity and naturalness to reduce the task of solvingtheinitialvalueproblemfortheSchro¨dingerequationtotheinvestigationof self-adjointnessofacorrespondinglinearoperatorinaHilbertspace. Thediscussion ofbasicpropertiesofsuchoperatorsinSection1.1leadstotheconclusionthatself- adjointness is also sufficient for a complete solution. It is therefore imperative to developcriteriaforself-adjointness,whichwillbedoneinSection1.2. Inparticular, we will give a straightforward argument for the fact that essential self-adjointness, i.e. the existence and uniqueness of a self-adjoint extension, is equivalent to self- adjointness of the closure of the operator, avoiding the (explicit) use of the Cayley transform and defect indices. The general theory is then applied in Section 1.3 to Schro¨dinger operators−(cid:1)+V withthe(local)Kato class (namedforT.Kato,the “fatherofthemoderntheoryofSchrödingeroperators([59,p.3523])”)emergingas themostnaturalandmostextensivehomefor(negativepartsof)potentialfunctions V. The Schro¨dinger equation (0.1) is rooted in the wave model of quantum me- chanics, starting from the idea that a free particle, i.e. subject to no outer force field,withmassm∈]0,∞[,totalenergyE ∈]0,∞[andvelocityv ∈Rd\{0}(d∈N) will behave like a plane wave and can therefore be described by a wave function Ψ of the form ∀t∈R∀x∈Rd : Ψ(t,x)=A·e2πi(k·x−νt), which propagates with constant speed ν/|k| ∈]0,∞[ into direction k ∈ Rd \{0}. (A∈C\{0}isanormalizationconstant.) Usingonlythemostfundamentalphysical lawsofquantumtheory,namelyEinstein’sequationE =hν(withPlanck’sconstant 1 h h) and de Broglie’s relation for the wave length λ:= = , one arrives at |k| m|v| (0.1) if one wants to determine the time evolution of Ψ starting from some initial state Ψ : Ψ has to fulfil a differential equation of first order in t, which is linear 0 because of the superposition principle for waves. Therefore it is also evident that wave functionshave tobecomplex valued. TheLaplacian−(cid:1), actingonthespace 1 variable x only, represents the kinetic energy m|v|2, while the so-called potential 2 (function) V in (0.1) embodies the potential energy induced by an external force field and has therefore to be real-valued. As an example, one might think of an electron subject to the coulombic force of a charged nucleus. 1.0. Solving the Schrödinger equation. The first attempt to find a non- trivialsolutionofequation(0.1)isbyseparationofvariables,i.e.theansatzΨ(t,x)= f(t)u(x). Then, for a (t ,x )∈R1+d with Ψ(t ,x )(cid:5)=0, we have 0 0 0 0 Ψ(t,x ) Ψ(t ,x) f(t)= 0 , u(x)= 0 , u(x ) f(t ) 0 0 (cid:1) (cid:2) ∂Ψ whence f ∈ C1(R), u ∈ C2 Rd and Ψ = f(cid:1)u, (cid:1)Ψ = f(cid:1)u (we write Ψ for t t ∂t etc.), such that (1.1) if(cid:1)u=f(−(cid:1)+V)u. 44 ANDREASM.HINZ Putting x=x , we see that f fulfils the ordinary differential equation 0 (cid:3) (cid:4) (cid:1)u(x ) f(cid:1)(t)=i 0 −V(x ) f(t)=:−iλf(t), u(x ) 0 0 whosegeneralsolutionisf(t)=cexp(−iλt)withsomec∈C\{0}. Ifweinsertthis into (1.1), we find that u has to fulfil the (time-independent) Schro¨dinger equation (1.2) ∀x∈Rd : −(cid:1)u(x)+V(x)u(x)=λu(x). Since |f(t)| = |c|exp(im(λ)t), the time evolution as given by f is bounded if and only if the eigenvalue λ is real; this is, of course, the physically relevant situation. There are only a few cases, like e.g. the harmonic oscillator (cf. infra, Exam- ple 3.26), where there are (sufficiently many) classical eigensolutions u of (1.2). In particular in view of possible singularities of the potential function V as in the Coulomb case (cf. infra, Example 3.27), we are forced to extend the notion of solution, based on the following observation: let (cid:1) (cid:2) (cid:5) (cid:1) (cid:2) (cid:6) C∞ Rd := ϕ∈C∞ Rd ; supp(ϕ)isbounded , 0 whereth(cid:1)esu(cid:2)pportofϕisdefinedbysupp(ϕ)={x∈Rd; ϕ(x)(cid:5)=0}. Thenforevery u ∈ C2 Rd for which V u is lo(cid:1)call(cid:2)y integrable, i.e. integrable after multiplication with any test function ϕ∈C∞ Rd , we get from integration by parts: (cid:7) 0 (cid:1) (cid:2) ∀ϕ∈C∞ Rd : {−(cid:1)u(x)+(V(x)−λ) u(x)}ϕ(x)dx 0 (cid:7) = u(x){−(cid:1)ϕ(x)+(V(x)−λ)ϕ(x)} dx, such that for these u, equation (1.2) is equivalent to (cid:7) (cid:1) (cid:2) (1.3) ∀ϕ∈C∞ Rd : u(x){−(cid:1)ϕ(x)+(V(x)−λ)ϕ(x)} dx=0. 0 As there is no regularity requirement on u in (1.3), we call every non-trivial locally integrableuforwhichV uislocallyintegrableaswellandwhichfulfills(1.3)aweak eigensolution of the Schro¨dinger equation for eigenvalue λ. In order to make use of functional analytic methods, we now have to find a suitable function space H, in which −(cid:1)+V acts as an operator. For the sake of linearity, H has to have the canonical algebraic structure of a vector space over C. We define (cid:8) (cid:1) (cid:2) D(S)= u∈H; ∃v ∈H ∀ϕ∈C∞ Rd : 0 (cid:9) (cid:7) (cid:7) u(x){−(cid:1)ϕ(x)+V(x)ϕ(x)} dx= v(x)ϕ(x)dx , ∀u∈D(S): S(u)=v. This defines a linear operator in H, i.e. D(S) is a linear subspace of H and ∀u,v ∈D(S), α∈C: S(u+αv)=Su+αSv. (For linear operators, the brackets for the argument are usually omitted, if no ambiguity is possible.) Let us suppose that we have sufficiently many weak eigensolutions e ∈ H \{0} for the eigenvalues λ ∈ R (they are then eigenfunctions n n of S, i.e. Se = λ e ), such that any u ∈ H can be written as a Fourier series n n n TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 55 (cid:10)∞ (cid:10)∞ u= αnen with α∈CN0 and accordingly Ψ(t,·)= exp(−iλnt)αnen exists n=0 n=0 in H for all t∈R. This presupposes a metric structure on H, compatible with the algebraic one, which can only be achieved by a norm (cid:7)·(cid:7) on H. Then we have (formally) for h(cid:5)=0 and setting f (t)=exp(−iλ t): n n (cid:10)∞ 1 {Ψ(t+h,x)−Ψ(t,x)}− {−iλ exp(−iλ t)}α e (x) h n n n n n=0 (cid:3) (cid:4) (cid:10)∞ f (t+h)−f (t) = n n −f(cid:1)(t) α e (x), h n n n n=0 anditwouldbedesirable tohave therighthandsidetendto0ash→0. Alas,this can not be expected in general! The way out is to assume further that the e are n mutually orthogonal. This necessitates the introduction of a compatible geometric structure in the form of an inner product (cid:9)·,∗(cid:11) in H, which thus will become a unitary space,where twovectorsxand y are calledorthogonal, iff(cid:9)x,y(cid:11)=0. Then there is something like the theorem of Pythagoras, namely (cid:10)∞ (cid:10)∞ u= α e ⇒(cid:7)u(cid:7)2 = |α |2, n n n n=0 n=0 where we have assumed that (cid:7)e (cid:7) = 1 for all n. By the properties of an inner n product we then have (cid:11) (cid:12) (cid:10)∞ (cid:10)∞ ∀m∈N : (cid:9)u,e (cid:11)= α e ,e = α (cid:9)e ,e (cid:11)=α . 0 m n n m n n m m n=0 n=0 We obtain (cid:13) (cid:13) (cid:13) (cid:10)∞ (cid:13)2 (cid:13)1 (cid:13) (1.4) (cid:13) {Ψ(t+h,·)−Ψ(t,·)}− {−iλ exp(−iλ t)} (cid:9)u,e (cid:11)e (cid:13) (cid:13)h n n n n(cid:13) n=0 (cid:14) (cid:14) = (cid:10)∞ (cid:14)(cid:14)(cid:14)fn(t+hh)−fn(t) −fn(cid:1)(t)(cid:14)(cid:14)(cid:14)2 |(cid:9)u,en(cid:11)|2. n=0 Now the convergence of the right hand side as h→0 can be investigated with the aid of the following dominated convergence theorem. (cid:1) (cid:2) Lemma 1.1. Let (a ) be a sequence of null sequences in C, with nm m∈N0 n∈N0 (cid:10)∞ ∀n∈N ∀m∈N : |a |≤b and b <∞. 0 0 nm n n n=0 (cid:10)∞ Then a →0 as m→∞. (cid:1) nm n=0 (cid:10)∞ ε Proof. For ε > 0 choose N ∈ N such that b < and then M ∈ N with 0 n 2 0 n=N+1 ε ∀m≥M ∀n∈{0,...,N}: |a |< . (cid:2) nm 2(N +1) 66 ANDREASM.HINZ (cid:14) (cid:14) As (cid:14)(cid:14)(cid:14)fn(t+hh)−fn(t) −fn(cid:1)(t)(cid:14)(cid:14)(cid:14)≤2|λn|, the right hand side of (1.4) tends to 0 (cid:10)∞ foreveryt∈R,if |λ |2|(cid:9)u,e (cid:11)|2 <∞. Thisisthecaseifandonlyiftheseriesin n n n=0 the left hand side of (1.4) converges for every t∈R to some Ψ (t,·)∈H, provided t that the unitary space (H,(cid:9)·,∗(cid:11)) is complete, i.e. H is a Hilbert space. (Ψ(t,·) will then be called differentiable in H with respect to t ∈ R. Rules from classical calculus can be carried over, like e.g. the product rul(cid:1)e (cid:9)Φ(cid:2),Ψ(cid:11)(cid:1) =(cid:9)Φ(cid:1),Ψ(cid:11)+(cid:9)Φ,Ψ(cid:1)(cid:11).) Therefore, by Fischer’s theorem (cf. [21]), H = L Rd is the appropriate home 2 for the Schro¨dinger operator S. This also leads to the probabilistic interpretation of the wave function (cf. infra, Section 3.0). (cid:10)∞ TheoperatorT inH,whichassignstheimageTu= λ (cid:9)u,e (cid:11)e ∈H toev- n n n (cid:8) (cid:9) n=0 (cid:10)∞ ery u∈D(T)= v ∈H; λ2 |(cid:9)v,e (cid:11)|2 <∞ , is (formally) symmetric, because n n n=0(cid:11) (cid:12) (cid:10)∞ (cid:10)∞ ∀u,v ∈D(T): (cid:9)u,Tv(cid:11)= u, λ (cid:9)v,e (cid:11)e = λ (cid:9)v,e (cid:11)∗(cid:9)u,e (cid:11) n n n n n n n=0 (cid:11) n=0 (cid:12) (cid:10)∞ (cid:10)∞ = λ (cid:9)v,(cid:9)u,e (cid:11)e (cid:11)∗ = λ (cid:9)u,e (cid:11)e ,v =(cid:9)Tu,v(cid:11). n n n n n n n=0 n=0 So Ψ is the unique solution of the initial value problem Ψ(0,·)=u, ∀t∈R: iΨ (t,·)=TΨ(t,·), t where uniqueness follows from (cid:1) (cid:2) (cid:7)Ψ(cid:7)2 (cid:1) =(cid:9)Ψ,Ψ(cid:11)(cid:1) =(cid:9)Ψ(cid:1),Ψ(cid:11)+(cid:9)Ψ,Ψ(cid:1)(cid:11)=(cid:9)−iTΨ,Ψ(cid:11)+(cid:9)Ψ,−iTΨ(cid:11) =−i{(cid:9)TΨ,Ψ(cid:11)−(cid:9)Ψ,TΨ(cid:11)}=0, because then Ψ=0, if u=0. (cid:1) (cid:2) Finally, T ⊂S, because for any u∈D(T) and ϕ∈C∞ Rd , we have 0 (cid:15) (cid:16) (cid:7) (cid:7) (cid:10)∞ Tu(x)ϕ(x)dx= λ (cid:9)u,e (cid:11)e (x) ϕ(x)dx n n n n=0 (cid:7) (cid:10)∞ = (cid:9)u,e (cid:11) λ e (x)ϕ(x)dx n n n n=0 (cid:7) (cid:10)∞ = (cid:9)u,e (cid:11) e (x){−(cid:1)ϕ(x)+V(x)ϕ(x)} dx n n n(cid:7)=0 = u(x){−(cid:1)ϕ(x)+V(x)ϕ(x)} dx. For u ∈ D(T), the above Ψ is therefore also a solution of iΦ (t,·) = SΦ(t,·). t If D(S) (cid:5)= D(T), there could be other solutions for the corresponding initial value problem. Uniqueness, however, can be guaranteed, if S = T, which means in particularthatShastobe(formally)symmetrictoo. Unfortunately,thedomainsof bothS andT aregivenimplicitlyonlyanddependonpropertiesofV. Itisnoteven TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 77 clear,iftheycontainasubstantia(cid:1)lse(cid:2)t,namelyadensesubspaceofH. However,ifwe assume Vϕ∈H forallϕ∈C∞0(cid:1) R(cid:2)d (thisisequivalenttolocalsquareintegrability of V), it is obvious that C∞0 Rd ⊂ D(S) (cid:1)and(cid:2)that S0ϕ = −(cid:1)ϕ+Vϕ for the minimal Schro¨dinger operator S :=S (cid:3)C∞ Rd . Conversely, by the definition of 0 0 D(S), the maximal Schro¨dinger operator S is the adjoint operator of S : S = S∗. 0 0 The symmetry of S∗ means that S has exactly one self-adjoint extension, namely 0 0 S =S∗. (S isthencalledessentially self-adjoint.) Wewillnowmakethesenotions 0 more precise in Section 1.1 and show that essential self-adjointness of the minimal operator is also sufficient for the unique solvability of the initial value problem for the Schro¨dinger equation, i.e. we will be able to prove the following. Theorem 1.2. Let {e ; n∈N } be an orthonormal basis of H consisting of n 0 eigenfunctions e for eigenvalues λ of S and assume that S is essentially self- n n 0 adjoint. Then for every u ∈ D(S) the unique solution of the initial value problem for the Schro¨dinger equation Ψ(0,·)=u, ∀t∈R: iΨ (t,·)=SΨ(t,·), t is given by (cid:10)∞ ∀t∈R∀x∈Rd : Ψ(t,x)= exp(−iλ t)(cid:9)u,e (cid:11)e (x). n n n n=0 (cid:1) Self-adjointness will also prove to be the key to solve the problem even if there is no orthonormal basis of H consisting of eigenfunctions of S. 1.1. Linearoperators in Hilbert space. Propertiesofoperatorsreflectthe algebraic, metric and geometric structure of a Hilbert space H. Linearity is associ- ated withthe vectorspace, boundedness, closability and closedness with the norm, symmetry with the inner product and finally self-adjointness with completeness. Although these properties can be characterized in the corresponding more general settings, we will, for simplicity, concentrate on operators T ⊂ H2, i.e. defined on some subset D(T) of H and with values in H. In view of our application to the solutionoftheSchro¨dingerequation,wewillalsolimitourconsiderationstoanon- trivial H over the field C. (Many of the results in this and the next section are valid for real Hilbert spaces, but some of the proofs involve subtleties, which we do not want to address here.) We assume familiarity with the basic properties of Hilbert spaces (see, e.g., [65]). Definition 1.3. T is a linear operator in H, iff T is a linear subspace of H2 and T ∩({0}×H)={(0,0)}. The domain of T is D(T)={u∈H; ∃v ∈H : (u,v)∈T}, and we write Tu for v. For λ ∈ C, the subspace (T −λ)−1({0}) is called the eigenspace of λ (and T); if this eigenspace is non-trivial, λ is called an eigenvalue of T and every non- trivial element of the eigenspace is called an eigenvector (or eigenfunction, if H is a function space) for λ (and T). (cid:1) A linear operator T is a function from D(T) to H with the property ∀u,v ∈ D(T)∀κ∈C: T(u+κv)=Tu+κTv. 88 ANDREASM.HINZ The most handsome operators are those which are bounded. Lemma 1.4. If T is a linear operator in H, then T is continuous, iff T is bounded, i.e. (cid:7)T(cid:7):=sup{(cid:7)Tu(cid:7); u∈D(T), (cid:7)u(cid:7)=1}<∞. (cid:1) Proof. If T is continuous, then there is a δ > 0 such that (cid:7)Tv(cid:7) < 1 for every v ∈ 1 1 D(T) with (cid:7)v(cid:7) < 2δ and consequently (cid:7)Tu(cid:7)= (cid:7)T(δu)(cid:7)< for every u ∈ D(T) δ δ with (cid:7)u(cid:7)=1. Conversely,ifT isboundedandu,v ∈D(T)with(cid:7)u−v(cid:7)<δ,then(cid:7)Tu−Tv(cid:7)= (cid:7)T(u−v)(cid:7)≤(cid:7)T(cid:7)(cid:7)u−v(cid:7)≤(cid:7)T(cid:7)δ. (cid:2) (cid:1) (cid:2) Unfortunately,differentialoperatorsinL Rd arenotboundedingeneraland 2 therefore we have to resign ourselves to closedness or even closability. Definition 1.5. A linear operator T in H is closable, iff its closure T is a linear operator. D(T) is then called a core of T. T is called closed, iff T =T. (cid:1) Corollary 1.6. Let T be a bounded linear operator in H. Then T is closable, and T is the only bounded extension of T with domain D(T). In particular, T is closed if and only if D(T) is closed. (cid:1) Proof. LetD(T)⊃(u ) →u∈D(T). Then u(cid:18)→ lim Tu defines the operator n n∈N n→∞ n T on D(T)= D(T): since (cid:7)Tu −Tu (cid:7) ≤(cid:7)T(cid:7)(cid:7)u −u (cid:7), (Tu ) is a Cauchy n N n N n n∈N sequence; moreover, the limit is independent of the choice of the sequence (u ) n n∈N approximatingu,ascanbeseenbyobservingthattheimagesofthemixedsequence built from two such sequences converge as well. Linearity of T is obvious. Furthermore, (cid:7)Tu(cid:7)= lim (cid:7)Tu (cid:7)≤ lim (cid:7)T(cid:7)(cid:7)u (cid:7)=(cid:7)T(cid:7)(cid:7)u(cid:7), n n n→∞ n→∞ such that (cid:7)T(cid:7)≤(cid:7)T(cid:7); (cid:7)T(cid:7)≤(cid:7)T(cid:7) is trivial since T ⊂T. (cid:18) (cid:19) (cid:18) (cid:19) IfT(cid:17)isaboundedextensionofT withD T(cid:17) =D(T),thenforeveryu∈D T(cid:17) , thereisasequenceD(T)⊃(u ) →u,suchthatTu =T(cid:17)u →T(cid:17)ubycontinuity n n∈N n n of T(cid:17), granted by Lemma 1.4. (cid:2) The operator S of Section 1.0 is closed, as can be seen directly or by recourse to the fact that S =S∗. 0 Lemma 1.7. Let T be a linear operator in H. Then (cid:5) (cid:6) T∗ := (u,v)∈H2; ∀ϕ∈D(T): (cid:9)u,Tϕ(cid:11)=(cid:9)v,ϕ(cid:11) defines a (closed) linear operator in H, called the adjoint of T, if and only if T is densely defined, i.e. D(T)=H. (cid:1) Proof. Obviously, T∗ isalinearsubspace ofH2,anditsclosedness followsfromthe continuity of theinnerproduct. T∗ isalinearoperatorifandonly ifD(T)⊥ ={0}. (cid:2) Corollary 1.8. Let T be a densely defined linear operator in H. Then T = T(cid:1)∗∗(cid:2); in particular, T is closable if and only if T∗ is densely defined, in which case T ∗ =T∗. (cid:1) TOPICS FROM SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 99 Proof. We have (cid:5) (cid:6) T =(cid:5)T⊥⊥ = (T∗ϕ,−ϕ)∈H2; ϕ∈D(T∗) ⊥ (cid:6) = (u,v)∈H2; ∀ϕ∈D(T∗): (cid:9)u,T∗ϕ(cid:11)=(cid:9)v,ϕ(cid:11) =T∗∗, and the equiv(cid:1)ale(cid:2)nce follows from Lemma 1.7. If we apply this to T∗, we get T∗ = T∗ =T∗∗∗ = T ∗. (cid:2) Another type of closable operators are symmetric operators. Definition1.9. AlinearoperatorT inH iscalledsymmetric,iffT isdensely defined and T ⊂T∗. (cid:1) Corollary 1.10. Let T be a symmetric operator in H. Then T is closable, and T is symmetric. (cid:1) (cid:1) (cid:2) Proof. ByCorollary1.8, T isclosable and T ∗ =T∗. AsT ⊂T∗, andT∗ isclosed (cid:1) (cid:2) ∗ by Lemma 1.7, we get T ⊂ T . (cid:2) Symmetric operators have other nice features. Lemma 1.11. Let T be a symmetric operator in H. Then a) ∀u∈D(T): (cid:9)Tu,u(cid:11)∈R; in particular all eigenvalues of T are real. b) Eigenspaces for different eigenvalues of T are orthogonal. c) ∀λ∈C∀u∈D(T): (cid:7)(T −λ)u(cid:7)≥|im(λ)|(cid:7)u(cid:7). (cid:1) Proof. a) (cid:9)Tu,u(cid:11)∗ =(cid:9)u,Tu(cid:11)=(cid:9)Tu,u(cid:11). If u is an eigenvector for the eigenvalue λ, then λ(cid:7)u(cid:7)2 =(cid:9)λu,u(cid:11)=(cid:9)Tu,u(cid:11)∈R and consequently λ∈R. b) Let λ and λ be eigenvalues with eigenvectors e and f, respectively. Then e f (λ −λ∗)(cid:9)e,f(cid:11)=(cid:9)Te,f(cid:11)−(cid:9)e,Tf(cid:11)=0, e f whence from λ (cid:5)=λ =λ∗, we obtain (cid:9)e,f(cid:11)=0. e f f c) We may assume (cid:7)u(cid:7)=1. Then (cid:7)(T −λ)u(cid:7)≥|(cid:9)(T −λ)u,u(cid:11)|=|(cid:9)T −re(λ)u,u(cid:11)−iim(λ)|≥|im(λ)|, the latter since (cid:9)Tu,u(cid:11)∈R from (a). (cid:2) We are now ready for the decisive step to prove Theorem 1.2. Theorem 1.12. Let M be an orthonormal basis of H and λ∈RM. Then (cid:8) (cid:9) (cid:10) (cid:10) D(T)= u∈H; λ2|(cid:9)u,e(cid:11)|2 <∞ , Tu= λ (cid:9)u,e(cid:11)e, e e e∈M e∈M defines a self-adjoint operator, i.e. T =T∗. (cid:1) Proof. Obviously, D(T) is a subspace of H, and D(T)=H since span(M)⊂D(T). The existence of Tu is guaranteed by Fourier expansion in H, which also yields linearity of T. Moreover, by Parseval’s identity, (cid:10) (cid:10) ∀u,v ∈D(T): (cid:9)u,Tv(cid:11)= (cid:9)u,e(cid:11)(cid:9)λ v,e(cid:11)∗ = (cid:9)λ u,e(cid:11)(cid:9)v,e(cid:11)∗ =(cid:9)Tu,v(cid:11), e e e∈M e∈M such that T is symmetric. 1100 ANDREASM.HINZ If u ∈ D(T∗), then λ (cid:9)u,e(cid:11) = (cid:9)u,Te(cid:11) = (cid:9)T∗u,e(cid:11) for every e ∈ M ⊂ D(T) and e consequently u∈D(T), whence T =T∗. (cid:2) ThistheoremnowcompletestheproofofTheorem1.2,becausetheself-adjoint operator T cannot have a strict extension S which is symmetric, by virtue of T∗ = T ⊂S ⊂S∗ ⊂T∗. We now try to liberate ourselves from the assumption of the existence of an orthonormal basis consisting of eigenfunctions. We observe that the operator T in Theorem 1.12 can be rewritten as (cid:8) (cid:9) (cid:10) (cid:10) D(T)= u∈H; λ2(cid:7)P u(cid:7)2 <∞ , Tu= λP u, λ λ λ∈R λ∈R where Pλu is the projection of u to the eigenspace of λ. (A proj(cid:10)ector P is a symmetric operator with D(P) = H and P2 = P.) Here, the sum λ2(cid:7)P u(cid:7)2 λ λ∈R is built up in a monoton increasing way in countably many positive steps, if we let λ grow from −∞ to ∞. This suggests a generalization by replacing the sum with an integral. To this end, we observe further that the family (P ) , being λ λ∈R ortho(cid:10)gonal, i.e. PλPµ = 0 for λ (cid:5)= µ, by Lemma 1.11b, generates, by putting E = P , a spectral family (E ) . λ µ λ λ∈R µ≤λ Definition 1.13. A family (E ) of projectors in H is called a spectral λ λ∈R family, iff it is • non-decreasing, i.e. ∀µ≤λ∀u∈H : (cid:9)E u,u(cid:11)≤(cid:9)E u,u(cid:11), µ λ (Note that this implies ∀µ≤λ: E E =E =E E .) µ λ µ λ µ • right-continuous, i.e. ∀λ∈R: E = lim E =:E , λ n→∞ λ+n1 λ+ and E−∞ := lim E−n =0, E∞ := lim En =1. (cid:1) n→∞ n→∞ Then(cid:8), with Eλ− :=nl→im∞Eλ−n1, (cid:9) (cid:10) (cid:1) (cid:2) (cid:10) D(T)= u∈H; λ2 (cid:7)Eλu(cid:7)2−(cid:7)Eλ−u(cid:7)2 <∞ , Tu= λ(Eλ−Eλ−)u, λ∈R λ∈R or, using the Cauchy-Stieltjes integral, (cid:3) (cid:7) (cid:4) (cid:7) D(T)= u∈H; λ2d(cid:7)E u(cid:7)2 <∞ , ∀v ∈H : (cid:9)Tu,v(cid:11)= λd(cid:9)E u,v(cid:11). λ λ The famous spectral theorem of von Neumann [41, Satz 3.6] states that the connectionbetweenspectralfamiliesandself-adjointoperatorsinH isnotrestricted to those with a complete set of eigenvectors. Theorem 1.14. Let (E ) be a spectral family in H and f ∈C(R). Then (cid:3)λ λ∈R (cid:7) (cid:4) D= u∈H; |f(λ)|2d(cid:7)E u(cid:7)2 <∞ λ is dense in H and (cid:7) ∀v ∈H : (cid:9)f(T)u,v(cid:11)= f(λ)d(cid:9)E u,v(cid:11) λ

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