Topics From Spectral Theory of Differential Operators

50 Pages·2004·0.453 MB·English

by  Hinz A.M.

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,Schr¨odingeroperator,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 Schr¨odinger 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
“fatherofthemoderntheoryofSchr¨odingeroperators([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 Schr¨odinger 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 solu-
tion, 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 eigen-
solutions 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