Fattorini H.O., Kerber A. The Cauchy Problem

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5.3. Perturbation Results for Operators in (2+(1,0) and in d

287

It seems natural to expect that if we assume that P itself or some

extension thereof belongs to C'+, then A + P will belong to C'+. However,

this is false in general and to place additional assumptions on A, P that will

make this true is far from trivial (see the comments in Section 5.9). We

present a result in this direction.

5.3.1 Theorem. Let A E (2+(1,0) (i.e., let A be m-dissipative) and

let P be a dissipative operator with D(P) 2 D(A) and

IIPuII<aIIAuII+bIIuII

(uED(A))

(5.3.1)

with 0 < a < 1, b 3 0. Then A + P (with domain D(A)) is m-dissipative.

Proof.

It is plain that A + P is dissipative, so that we only have to

prove that

(JAI-(A+P))D(A)=E

(5.3.2)

for some A > 0. We consider two cases.

(i) a < i.

Since (Al - (A + P))D(A) _ ((XI- A)- P)R(A)E _

(I - PR(A))E, (5.3.2) will hold if

IIPR(A)II <1.

(5.3.3)

But

IIPR(A)uII<aIIAR(A)uII+bIIR(A)uII=aII(AR(A)-I)uII+bIIR(A)utl

(a+)iiuii

(A > 0)

so that (5.3.3) (hence (5.3.2)) holds if A > b/(1- 2a).

(ii) a < 1. Let 0 < y < 1. We have

II(A+yP)ull- IlAull - yllPull % IlAull - IIPuII

(1- a)II Aull

- bIIuII

Let n be an integer so large that a/n < (1- a)/3. Then

11n1 Pull

< -IlAull+-llull

1 a

3 IlAull+ -llull

3ll(A+yP)ll+b(3 +-),lull.

Applying the result in (i) for y = 0, we see that A + (I /n) P is m-dissipative;

taking then y =1/n, y = 2/n ,...,y = (n -1)/n, and applying (i) in each

case, we prove at the last step that A + P is m-dissipative.b

60f

course, the argument proves (taking y =1) that A+(1 + n-')P is m-dissipative.

This is not unexpected since (1 + n-')P satisfies (5.3.1) with different constants a', b', a' < 1.

288

Perturbation and Approximation of Abstract Differential Equations

It is plain that Theorem 5.3.1 must fail if a = 1. To see this, take

A E (2+(1, 0) with D(A) * E and P = - A; here A + P is the restriction of 0

to D(A), which is not even closed. However, one may ask whether it is

always true that some extension of A + P is m-dissipative, as is the case in

the preceding example. The answer is in the affirmative but an additional

assumption is needed.

5.3.2

Corollary.

Let A be m-dissipative and let P be a dissipative

operator with D(P) 2 D(A) and

IlPull < IIAuII+ bllull

(5.3.4)

for some b >, 0. Assume in addition that P*, the adjoint of P is densely defined

in E*. Then A + P, the closure of A + P, is m-dissipative.

Proof Since A + P is densely defined and dissipative, it is closable

and A + P is dissipative with respect to some duality map 0: D(A + P) - E*

(Lemma 3.1.11). It only remains to prove that

(I -A+ P)D(A+ P) = E.

(5.3.5)

Since the range of a closed dissipative operator must necessarily be closed,

we only have to show that X = (I -A + P)D(A + P) is dense in E. Assume

this is not the case. Then there exists a nonzero functional u* E E* with

<u*,u)=0 (uEX).

Let v E E with

Ilu*II < (u*, v).

(5.3.6)

By virtue of Theorem 5.3.1 we know that A+ yP is m-dissipative for

0 < y < 1. Accordingly, the equation

u-(A+yP)u=v (5.3.7)

has a solution u = u. for every y, 0 < y < 1 with

IluYll < Ilvll

(5.3.8)

It follows from (5.3.4) that

IIPuYII <IIAuYII+blluyll

< II(A + YP) uyll+ YIIPuyli+ blluyll

= IIv - uyll+ YIIPuYII+ blluyll,

thus

(1-Y)IIPuYII<IIv-uyll+b11 uy11

(b +2)IIvIi.

(5.3.9)

5.3. Perturbation Results for Operators in C+(1,0) and in d

289

Let now v* E D(P*). We have

I(v*,(1-Y)Pu,)I= (1-Y)1(P*v*,uY)I

(1-Y)IIP*u*IIIIvII-'0 as y-0.

Since D(P*) is dense in E* and ((1- y)Pu7; 0 < y < 1) is bounded in E, it

follows that

(1-y)PuY-0 asy-*l-

E*-weakly. Then we have

Ilu*II < (u*, v) = (u*, uY - AuY - yPu7)

=(u*,(1-y)Puy)+(u*,u7-AuY-Pu7)

=(u*,(1-y)PuY)-0 asy-*l-,

which is absurd since we have assumed u* $ 0. This ends the proof of

Corollary 5.3.2.

5.3.3

Remark.

Since P is assumed densely defined and dissipative it must

be closable by Lemma 3.1.11; thus by virtue of Theorem 4.4, if E is reflexive

(in particular if E is a Hilbert space), P* will always be densely defined.

Note also that we only need to assume P dissipative with respect to some

duality map 0: D(P) - E*; the same observation holds, of course, for

Theorem 5.3.1.

The assumptions on A in Corollary 5.3.2 can be relaxed in the

following way. Assume that A is only densely defined and dissipative, but

that A is m-dissipative, and that (5.3.4) holds for P dissipative with

D(P) 2 D(A). Let u E D(A), a sequence in D(A) with u -* u, Au,, -*

Au. Writing (5.3.4) for u,, - u,,,, we see that is as well convergent, thus

u e D(P) and

IlPull < IlAull+ bIlull

(5.3.10)

We can then apply Corollary 5.3.2 to A and F and deduce that T+ P is

m-dissipative. But if follows easily from the definition of closure and from

(5.3.10) that

A+P =A+P

(5.3.11)

so that the conclusion of Corollary 5.3.2 remains unchanged. We note an

important byproduct of the preceding arguments. Let u E D(A),

sequence in D(A) with u - u, Au,, - Au. We have shown that Pu is

convergent, so that u E D(A + P); in other words,

D(A) c D(A+P).

As the example P = - A shows, the opposite inclusion may not be true.

However, if P satisfies (5.3.1) with a < 1, we have II(A + P)uII % IlAull

- IlPull

290

Perturbation and Approximation of Abstract Differential Equations

(1- a)IIAuII- bllull for u E D(A) so that

IlPull

<a(1-a)-'Il(A+P)uII+bllull (uED(A)). (5.3.12)

Hence D(A + P) c D(A) results as well and

D(A)=D(A+P). (5.3.13)

We collect those observations.

5.3.4 Corollary.

Let A be densely defined and dissipative with A

m-dissipative, and let P be a dissipative operator with D(P)

D(A), D(P*)

dense in E* and

IlPull<aIIAull+bllull

(uED(A)) (5.3.14)

with 0 < a < 1, b >,0. Then A + P is m-dissipative. If a < 1, then (i) the

domain of A+ P coincides with the domain of A; (ii) if A= A is itself

m-dissipative, then A + P is m-dissipative and D(A + P) = D(A). The as-

sumption on D(P*) is unnecessary when a < 1.

Part (ii) of the case a < 1 is of course a restatement of Theorem 5.3.1.

Corollary 5.3.4 has important applications to self-adjoint operators

in a Hilbert space H. Call an operator A in H essentially self-adjoint if (a) A

is densely defined and symmetric. (b) X is self-adjoint 7 Using the facts,

proved in Section 3.6, that A is self-adjoint if and only if both iA and - iA

are m-dissipative (that is, if iA E C (1, 0)) and that P is symmetric if and only

if iP and - iP are dissipative, we obtain the following result:

5.3.5 Corollary. Let A be essentially self-adjoint, P a symmetric

operator with D(P)

D(A) satisfying (5.3.14) with 0 < a < 1, b > 0. Then

A + P is self-adjoint, that is, A + P is essentially self-adjoint. If a < 1, then (i)

D(A) = D(A + P), (ii) if A is self-adjoint, then A + P is self-adjoint and

D(A+ P) = D(A).

The following result applies to operators in d and is roughly of the

same type as Theorem 5.3.1; although P is not assumed here to belong to C'

or any subclass thereof, we must use a reinforced version of (5.3.1).

5.3.6

Theorem.

Let A E d, and let P be an operator with D(P) C

D(A) and such that, for every a > 0, no matter how small, there exists

b = b(a) with

IlPull < aIIAull+ bllull

(u E D(A)). (5.3.15)

Then A + P (with domain D(A)) belongs to 9.

71t was observed in Section 6 that if A is symmetric, iA is dissipative, hence !exists by

Lemma 3.1.11.

5.3. Perturbation Results for Operators in e+(l,0) and in & 291

Proof.

If A E e, then (Theorem 4.2.1) there exists a such that R(X)

exists for Re A > a and

IIR(A)II <

IA - al

(5.3.16)

there. Clearly we have

AI-A-P= (I-PR(A))(AI-A)

in D(A); accordingly, if IIPR(A)II < 1, A E p(A + P), and

R(X; A+P)=R(A)(I-PR(A)) '. (5.3.17)

We estimate IIPR(A)II using (5.3.15):

IIPR(A)II < aIIAR(A)II+ bIIR(A)II

0(1+

IA IA«I

)+IXb Cl

(5.3.18)

Take now a'> a. Then J X /(A - a) I is bounded in Re A > a'; therefore we

may choose a so small that

a(l+IA

),<-y<2

(ReA>a').

-al

Taking a' larger if necessary, we may also assume that

IAbal

<Y (ReA>, a').

Combining these two estimates with (5.3.18), we see that I I PR (A )I I < 2Y < 1

in Re A >,a';

therefore, (I -PR(A))-' exists and II(I -PR(A))-'ll

(1- 2y) -' there. In view of (5.3.16) and (5.3.17), we obtain

IIR(A;A+P)II<IAC'al <IACa'l

(ReA,a,),

which, using Lemma 4.2.3, shows that A + P E Q as claimed.

Note that we only need to assume the existence of an estimate of the

type of (5.3.15) for some a < 1/(2+2C).

5.3.7 Corollary.

Let A E Q and let P be bounded. Then A + P E Q.

Proof

(5.3.15) holds for any a and b = IIPII.

5.3.8 Remark. The comments on Remark 5.3.3 apply here with some

modifications. Assume that A is closable and that X E-= d, all the other

hypotheses in Theorem 5.3.5 remaining unchanged except that we assume

now that P is closable. Hence (5.3.15) holds for A and P. Theorem 5.3.5

then implies that X+ P E Q. If A + P were closable, we would have

A+P=A +P =A+P

(5.3.19)

292 Perturbation and Approximation of Abstract Differential Equations

(the first equality follows from the fact that X+ P, which belongs to d,

must be closed). It is remarkable that (5.3.19) will always be true, as (5.3.15)

and the fact that A and P are closable imply that A + P is closable. To see

this, we take a < 1 in (5.3.15) and use its consequence (5.3.12). Let be a

sequence in D(A) with u - 0, (A + P)u -> v. Then it follows from (5.3.12)

applied to u,, - Um that Pu - w E E. Since P is closable, w = 0; then

Au -> v, and we deduce that v = 0.

_ 5.3.9 Corollary. Let A be a closable, densely defined operator with

A E Q, P a closable operator with D(P) 2 D(A) satisfying (5.3.15) for a

arbitrarily small. Then A + P (with domain D(A)) is closable and q -+P E C9',.

5.4.

PERTURBATION RESULTS FOR THE SCHRODINGER

AND DIRAC EQUATIONS

We retake here the study of the Schrodinger equation (1.4.1), this time for a

potential U * 0. Setting q = - h -'U, we may write (1.4.1) as an equation in

the Hilbert space E = L2(R3)c in the form

u'(t)=(& (5.4.1)

where d =

is the operator described in Section 1.4 with K = h/2m and 2

is the multiplication operator

(2u)(x) = iq(x)u(x) (5.4.2)

(the domain of 2 will be precised later). It was proved in Section 1.4 that the

operator d belongs to (2(1,0), thus by the results in Section 3.6, ids is

self-adjoint in E. (This was also shown directly in Example 1.4.6.) In view of

the interpretation of the solution of (1.4.1) sketched in Section 1.4, it is of

great importance to find conditions on 2 (that is, on q) that make i (d + 2)

a self-adjoint, or at least an essentially self-adjoint operator. For one result

in this direction, we use (part of) Corollary 5.3.5 with P = 2.

5.4.1 Theorem. Let

q(x)=qo(x)+q,(x), (5.4.3)

where q0 E L°°(R3)R and q, E L2(R3)R, and let the domain of the operator 2

in (5.4.2) consist of all u E L 2(1

3) such that qu E L2(083). Then D(2) 2 D(Q)

and the operator i(? + 2) is self-adjoint.

Proof.

To show that

Q D(d), it is obviously sufficient to

prove that every u E D(l) is bounded in R 3. This follows from the Fourier

analysis manipulations at the end of Section 1.6, but a slightly improved

5.4. Perturbation Results for the Schrodinger and Dirac Equations

293

version is essential here. Using the notations in Section 8 for Fourier

transforms, let u E D(fl and p > 0. Then

qu(a) =

1612+p2(IaI2+p2)u(a)

= IoI2 +

(- r(ou)(a)+ p2 u(a)). (5.4.4)

Making use of the Schwarz inequality, we see that q u E L' (R 3) and

II'uIILa1<

77P-1/2II(-0+p2)uIIL2.

(5.4.5)

Accordingly, after eventual modification in a null set, u E L2 (R 3 ) n Co (R 3)

c D(.) and'

IIuIIca<C(P 1/2IIouII+P3/211u1I),

where C does not depend on p. Moreover, I I k u I I = K - 'II d u I I, hence

II2uII < IIgoIIL°IIuII+Ilgil1L2IIUIIco

<C'(p '/ZIIl uII+(p3/2+1)IIuII)

(5.4.6)

so that (5.3.14) holds for any a > 0 if p is sufficiently large and Corollary

5.3.5 applies (note that i2 is symmetric since q is real-valued). This com-

pletes the proof.

Observe that Theorem 5.4.1 applies to the Coulomb potential U(x) _

c I x I - ' for any value of c. Potentials of the form U( x) = V(I x - xo I )

correspond to motion of particles in central attractive (or repulsive) fields

(see Schiff [1955:1]). Generalizing the previous observation, Theorem 5.4.1

applies if V is bounded at infinity and r2V(r)2 is integrable near zero (for

instance, if U(r) = cr with $ < z ). Also, it should be pointed out that

even if q is complex-valued, inequality (5.4.6) holds although fails of

course to be symmetric. Nevertheless, if we assume that

Ilmq(x)I < w a.e. in 183,

it is easy to see that Corollary 5.3.4 applies both to d, 2 - wI and to

- d, - 2 - wI showing that t

. E e (1, w).

5.4.2 Example. The Schrodinger equation of a system of n particles. The

quantum mechanical description of the motion of a particle sketched in

Section 1.4 extends to the case of n particles. The Schrodinger equation

8Here II'II, without subindex, indicates the L2 norm. The same observation holds in

forthcoming inequalities.

294

Perturbation and Approximation of Abstract Differential Equations

takes the form

n 2

=- E

Oj + Up (5.4.7)

i at

j=1

where the function 4' depends on the 3n-dimensional vector x = (x',...,x")

(xj = (x{, x

x j

1 < j < n), Aj denotes the Laplacian in the variables xj

and the potential U depends on x; m j is the mass of the j th particle. Again

14' 12 is interpreted as a probability density in R 3n, hence the L2 (08 3") norm

of ty must remain constant. The Fourier analysis of the unperturbed case

U = 0 is essentially covered by the results in Section 1.4; we define, for

K1 = h/2at t,...,K" = h/2mn,

(Tu=l(KI A,+

+KnQn)

with domain u consisting of all u E L2(R3n) such that EKjLI j (understood in

the sense of distributions) belongs to L2. An element u E L2 (08 3n) belongs to

D(d) if and only if its Fourier transform f u satisfies 10 12' u E L2 (OB 3n)

(here a = (a',... , an), aj = (a{, a", as'),

1 < j < n)

and 5d u =

(- i EK j I aj 12)6j-u. The operator id is self-adjoint (this follows from Example

1.4.6). It can be shown arguing as in Remark 1.4.3 that D(d) = H2(R3n)

and that the norm of H 2 (08 3n) is equivalent to the graph norm I I u I I + I I du I

I or

to its Hilbert version

(11U112

+ II

guII2 )'/2 (which

are instantly seen to be

equivalent themselves).

We examine the perturbation of d by an operator 2 given by (5.4.2),

with q = - h -' U. An important instance is that where the particles move

under the influence of Coulomb forces, both due to a central field and to

the other particles; here

q(x)=E

+ EE

CJk

(5.4.8)

IxJI

j<k IxJ -

xkI

where cj, cjk are real constants. Theorem 5.4.1 does not apply now: for

instance, if n >, 2, the function 1/ Ix' I does not admit a decomposition of

the form (5.4.3); moreover the elements of D(d) are no longer continuous

or even bounded in 08 3" since 2 < 3n /2 (see Section 1.6).

We define a set Il,...,In of (nonlinear) operators from L2(R3n) into

L2(R3) as follows:

1/2

(Iju)(xj) _ fR

3(n - 1)

Iu(x',...,x")12dx' ... dxj-tdxj+' ... dx"

Clearly IlIjull = lull for all u E L2. We shall show that if u E D(d) and a > 0

is arbitrary, there exists b >, 0 such that

I (Iju)(x3)12 < all( u112 + bIIuI12

(xi E 083, u E D(¢)). (5.4.9)

To prove (5.4.9) we may obviously assume that j = 1. Denote (as before) by

5.4. Perturbation Results for the Schrodinger and Dirac Equations

295

6S the Fourier transform in Ran and by IF, (resp. F, _) the Fourier transform

in x' (resp. in

x2'...' x"). If u e 6D (R 3n ), we have

III U(x1)12 =

f l U(x)I2 dx2 ... dx"

= f l

j1_u(x1,a2,...,a")12da2...do"

13 fj f I

U(a',...Ian)I day}

do2... don.

(5.4.10)

We apply now (5.4.5) to 1J,

u(x', a2,...'a") for p > 0, obtaining

II6JU(a)IIL;i1rp-1/211(1x112+P2)JU(a)IIL2.

Replacing in (5.4.10), we obtain

IIiu(x')IZ <C(p-IIIdUII2+p311U112),

(5.4.11)

therefore (5.4.9) results for arbitrarily small a. Using the fact that 6D is dense

in D(d)=H2 in the norm of H2, hence in the graph norm of CT (see

Remark 1.4.3), the inequality is seen to hold for any u E D((?) (note that

lIju(xj)- Ijv(xj)I < IIj(u - v)(xj)I, hence passage to the limit is licit).

Inequality (5.4.9) is used for the potential (5.4.8) as follows. The domain of

2 is defined again as the set of all u E L2 such that qu E L22; 2 is symmetric

since the cj and the cjk are real. For terms of the form ccI xiI -', we note that

f Ixj1-2iu(x)IZdx= f Ixjl -211

ju(x')I2dxj.

(5.4.12)

We divide the domain of integration in I x'

1 and

I x' I < 1; each of the

integrals is estimated as follows:

Ix'1-21Ii u(xj)l2dx'<f

Ilju(xJ)I2dxj

IxJI>1

Ix1I>1

< fu3nlu(x)I2 dx = IIU112,

Ix'I-21IiU(xj)l2dxj<(a11C9''U112+bIIUl12) f

Ixjl-2dxj

Ix'j 51

Ix)I<l

< (a'Il6TuII+

b'IIU11)2,

where a' can be taken as small as desired (we have used here the inequality

a2 +a2 < (a+/3)2).

Consider next any of the other terms, say c 121 x' - X21 -'. The change

of variables y' = x' - x2, y2 = x2,..., y" = x" yields

fIx'-x21-21u(x)12dx= fIx'I-21v(x)I2dx, (5.4.13)

296

Perturbation and Approximation of Abstract Differential Equations

where v(x'..... x") = u(x' + x2, x2,...,x") is easily seen to belong to D(d)

if u does, with I I & v I I < C I 16 u 11. Collecting our observations, we see that 2, d

satisfy (5.3.1) with a arbitrarily small, so that i((? + 2) with domain D(d) is

self-adjoint.

5.4.3

Example. Perturbation theory for the Dirac equation. As a conse-

quence of the results in Example 1.6.6, we know that the operator

du = cEajD'u + iX/3u (5.4.14)

in L2(R3)4 (A = mc2/h) whose domain consists of all u = (u1, u2, u3, u4) in

H' (R 3)4

, belongs to (2(1,0); equivalently, id is self-adjoint. We consider

now the case U* 0 in (1.6.34); as for the Schrodinger equation, we write

q = - h -'U, but we limit ourselves to examine here the Coulomb potential

q(x) =

x°I

(5.4.15)

where x° is a point in three-dimensional Euclidean space. A key role in the

following argument is played by the inequality

Ix-x°I-2Iu(x)I2dx<

4 2 f lgradu(x)I2dx

(5.4.16)

(m -2)

valid for any function u E H' (III m ), m >, 3 and any x ° E R R. We prove it

first for a smooth function assuming (as we may) that x° = 0. Let q = qp(p)

be a function in 6 (R' ). Then

(pm-2l

l2),=

(m-2)pm-3Iq,l2+2pm-2Re\m'v;)

Integrating in 0 < p < oo,

fpm-3Iw12dp=_ (m2 2)

fpm-2RedP

2 1/2 1/2

m-2

(.l/Pm-3I,T I2dp)

(fPm_1ipl2dP)

by the Schwarz inequality, so that

fpm-31q)12dp <

(m-2)

2 zfpm-1I12dpWe

take now u E 6D (R m) and apply this inequality to q2 (p) = u (px ), IxI =1;

observing that Iro'(P)I = I ExkDku(Px)l < IxI Igrad u(px)I and integrating

the resulting inequality in

I x I = 1, (5.4.16) results. To establish the inequal-

ity for an arbitrary u E H' (III m ), we approximate u in the H' norm by a

sequence (T") in 6D and apply (5.4.16) to each %. Obviously we can take

limits on the right-hand side; on the left-hand side, we select if necessary a

subsequence so that (q,") is pointwise convergent to u and apply Fatou's

theorem.