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

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5.4. Perturbation Results for the Schrodinger and Dirac Equations

297

We apply inequality (5.4.16) for m = 3 to the function q(x) in

(5.4.15) and to each coordinate of u E D(d) = H'(R3)4. Adding up we

obtain

f q(x)2Iu(x)I2dx=d2 fIx-x°I-2Iu(x)I2dx

4d2 fIgrad u(x)I2dx

5 4c-

(5.4.17)

The last inequality requires some explanation. For the operator (5.4.14), we

have

III' uII2 = IIcEa;Dju + iA$uII2 = II(cEQjaj - X$)

uII2 = II(cEajaj)

uII2

+ I A

12I

I u 112 We make use of (1.6.38) plus the explicit computation of the

constant 0 in (1.6.39): in this way, (5.4.17) follows.

We apply Corollary 5.3.5 to the self-adjoint operator ie and the

symmetric operator il: if

IdI 5 c/2,

(5.4.18)

the operator i (( + ) is self-adjoint, while if the inequality (5.4.18) is

strict, the operator i(d + 2) (with domain D(d)) is self-adjoint. Since

(5.4.18) imposes a definite limitation on the size of IdI, it is interesting to

translate it directly for the potential U in the equation (1.6.27). Writing U =

e I x - x° I -', the inequality turns out to be I e I < ch /2.

Obviously, this perturbation analysis can be applied to the general

symmetric hyperbolic operator d in (1.6.22) if condition (1.6.36) is satisfied.

5.4.4 Remark. When the potential function U in the Schrodinger equa-

tion (1.4.1) is "too singular" it may be the case that i(e + ) fails to be

essentially self-adjoint. It is then important to construct self adjoint exten-

sions i (d + )

of i (C9', + D.) (so that the Schrodinger equation can be

"embedded" in the equation

u'(t)

+ L)su(t),

(5.4.19)

which possesses a unitary propagator). As pointed out in Section 6, these

extensions will exist if and only if both defficiency indices of CT + 2

coincide. However, mere existence of (? + 2)S is of little physical interest;

at the very least, the domain of (d + 2)s should be clearly identified. We

present here results generalizing Theorem 5.4.1 that combine perturbation

theory with the methods used in Section 4.6 to deal with parabolic equations

with rough coefficients. In the nomenclature of Section 4.6 we consider the

differential operator

A° _ A + q(x)

(5.4.20)

298

Perturbation and Approximation of Abstract Differential Equations

in U = R', A the m-dimensional Laplacian and q real, locally integrable,

and bounded above in R m. The operator A0 is not obviously defined for any

nonzero function (qu may fail to be square integrable even for u E 6D).

However, an "extension" can be defined as follows. Let A >_ ess. sup q + I

and ¶ c L2 (l m) the completion of 6 (R m) with respect to the scalar

product

(q), p)X =fR {(grad T,grad p)+(A - q(x))-fi) dx.

(5.4.21)

Then an element u E 9C belongs to D(A) if and only if w - (u, w)x is

continuous in JC in the norm of L2, and Au is the unique element of L2 that

satisfies

A(u,w)-(u,w)x=(Au,w) (wE'JC)

(5.4.22)

(note that in the present situation we have Bju, w) = (u, w),,). The fact that

A is self-adjoint follows from (4.6.30) and Remark 4.6.8 (note that the

formal adjoint A'0 coincides with A0). We can then set (( + )S = iA and

obtain a solution of our problem. The characterization of D(A) = D(& + 2),

however, is too vague for comfort and we try to obtain a better one. Note

first that the uniform ellipticity assumption (4.6.2) is satisfied so that

D(A) c %C C Hl(Rm). On the other hand, if u E D(A), we obtain integrat-

ing by parts that

f uEgpdx= f (Au-qu)pdx

for q ) E 6D, so that Au + qu = Au E L2(R'). (Note that

I qI u = I qI u

jqI is

locally integrable by the Schwartz inequality so that A u E L l,,, (R m ).) Con-

versely, if u E %C is such that Au + qu E L2(R ') and p E 6D, we have, after

another integration by parts,

(u,(P)x=- fu&r dx+ f(A-q)ugpdx

f (Du+qu-Au)pdx.

Hence (u, w)a is continuous in the norm of L2 for w E 6D (a fortiori, for

w E =-'K) and u E D(A) with Au = Au + qu. We have then shown that

D((9 +2)s)=D(A)=(uE%; Au +quEL2). (5.4.23)

This result is not entirely satisfactory, however, since functions in iC are not

easy to characterize. A better identification can be achieved at the cost of

some restrictions on q. Surprisingly enough, these bear on the presence of

singularities of q at finite points rather than on its behavior at infinity.

5.4. Perturbation Results for the Schrodinger and Dirac Equations 299

5.4.5 Theorem.

Let q be real, locally integrable, and bounded above

in R

m. Let

mp,

(q' x) =f

IY-xI2-m-PIq(Y)I dy

(5.4.24)

lY-xI<R

and assume there exist p, R > 0 such that mp, R

is locally bounded.9 Then

D(A) consists of all u E H 1(R m) such that I q-1 u and i u + qu belong to

L2(R M).

For the proof we need a simple inequality.

5.4.6 Lemma.

Let p be locally integrable and a.e. nonnegative,

continuously differentiable in R m with compact support. Then, if R > 0,

0<p<2,

fR P(x)Ip(x)I2dx

< 2pw

fRm(RPIgradip(x)12+Rp-2I

(x)12)mp,R(P;x)dx,

(5.4.25)

where m

p, R

is defined as in (5.4.24) and wm denotes the hyperarea of the unit

sphere in R m.

Proof

Let q(r) be continuously differentiable in r > 0. Obviously

there exists s, 0 < s < R, such that the value of I n I at s does not surpass its

average over 0 < r < R. Expressing q (s) - ,q(0) as the integral of rl'(r) in

0 < r < s, we obtain

I71(0)I< f R(Iq'(r)I+R 'In(r)I)dr.

Apply this inequality to q(r) = (x - r(y - x)), afterwards integrating in

l y - x I = 1. An interchange of the order of integration combined with the

inequality I E(Yk

- xk) D

I< I Y- x I

I grad P I

easily yields

Ib(x)I< R f

wm IY-xI-<R

We estimate now by means of the Schwarz inequality writing ly - x I

- m =

IY-xI(P-m)/2Iy-xI(2-m-p)/2 and noting that (a+b)2 <2(a2+b2). We

obtain

IiP(x)I2 < 2RP

- x12-m-P(Igrad 4'(y)I2 +

R-21'(Y)I2) dy.

wmP

fly

- x I < R

9An application of the Fubini theorem shows that M,,

exists a.e. and is locally

integrable if p < 2. We only use the fact that q is locally integrable.

300

Perturbation and Approximation of Abstract Differential Equations

Finally, we multiply by p (x) and integrate in Rm; after an obvious

interchange of the order of integration, (5.4.25) results.

Proof of Theorem 5.4.5. In view of (5.4.23), to show that any u

satisfying the assumptions belongs to D(A) it is enough to prove that if A is

as in (5.4.21), then any u satisfying

u E H' (R"), (A - q )1 /2 u E L2 (R m ),

(5.4.26)

belongs to %, that is, there exists a sequence (Tn) in 6D such that

IIgn-uiiH.-0, II(A-q)1/2(pn-u)IIL2-'0,

(5.4.27)

as n - co. To this end, let (Xn) be a sequence in 6D such that X,,(x) =1 for

I x I< n, X,, (x) = 0 for

I x I> n + 1 and having uniformly bounded first

partials. It is a simple matter to show that

Ilxnu-UIIHI-+0,

II(A-9)1/2(Xnu-u)IIL2- 0,

hence we may assume that the function u itself has compact support, say,

contained in the ball I x I < a. Let (/3n) be a mollifying sequence in 6D (see

Section 8), Jn the operator of convolution by /3n. It is immediate that

Dk(J,,u) = Jn(Dku),

hence it follows from Theorem 8.2 that q9n = Jnu = fin * u converges to u in

H', while each %, belongs to 6D with support contained in

I x I < a + 1. We

apply inequality (5.4.25) with p = A - q and _ (pn - (pn,: since 1 vanishes

for IxI>a+l,

f (A-q(x))Iq)n(x)-gPm(x)I2dx<CIIgn-9miiH"

(5.4.28)

where we have used the fact that m

p, R

(q; x) is bounded in

I x I < a + 1.

Letting m - oc, we obtain the second relation (5.4.27), thus u E X as

claimed. The opposite implication follows from (5.4.23) and the fact that

u E 9C c H'; that

I qI u E L2 is a consequence of the definition of the norm

of `3C.

Note that the assumption on m

p, R

is independent of R. Observe also

that an operator A with domain described as in Remark 4.6.8 could be

defined without any special assumption on q by simply replacing iC by the

space of all u E H' with /u E L2 endowed with the scalar product (5.4.21).

Thus, the importance of the result lies in the approximation property

(5.4.27) and the subsequent identification of D(A).

The result just proved, although partly more general than Theorem 5.4.1,

does not include, say, potentials having quadratic singularities c l x l- 2 in 68'.

Attempts to apply Corollary 5.3.5 fail since the domain of A (even after Theorem

5.4.5) is not easy to work with; for instance, the Fourier transforms of the elements

of D(A) allow no easy characterization. On the other hand, inequalities of the type

(Pu, u) < al (Au, u) I + b(u, u) are easier to establish (as we shall see in the next

Theorem 5.4.9) and it is then of interest to have a perturbation theory based on

5.4. Perturbation Results for the Schrodinger and Dirac Equations

301

these. We limit ourselves to a simple result involving perturbations of symmetric

operators.

5.4.7

Theorem.

Let 921 be a densely defined symmetric operator in a Hilbert

space H such that

(9fu,u)>0 (u(=-D(1)), (5.4.29)

and let 13 be another symmetric operator such that D( 3) D(9f) and

(93u,u)<a(S2fu,u)+b(u,u) (uED(9f))

(5.4.30)

with 0 S a 51, b > 0. Then there exists a self-adjoint extension (W - tI3 ),, of W -

with

((9f - l3)Su, u) > - b(u, u)

(u E D(%- t3)S). (5.4.31)

Proof

Define

(u,v),$)u,v)+(1+b)(u,v)

for u, v E D(s9f). Since

((% -$)u,u)>(91u,u)-( tu,u)>(I-a)(%fu,u)-b(u,u),

(5.4.32)

it follows that

has all the properties pertaining to a scalar product and we can

complete D(1t) with respect to the norm 11.11, = (., .)l/2 Since

Hall, > hull,

(5.4.33)

the completion H, D D(9f) is a subspace of H. We define an operator 93 in H as

follows: the domain of F8 consists of all u E H, that make the linear functional

w - (u, w), continuous in the norm of H; since H, is dense in H, we may extend the

functional to H and obtain a unique element v E H satisfying

(u,w),=(v,w).

(5.4.34)

We define T u = v. Let u, v E D(93); then

(0u, v) = (u, v), = (v, u), _ (8v, u) = (u, 8v)

so that 8 is symmetric. We prove next that 93

E (H). Since 1113 u I I l l a l l > (8 u, u) _

(u, u), > Ilullz, we have

hull for u E D(8), thus 0 is one-to-one with

bounded inverse T - '; it only remains to be shown that

D (T) = E. To see this we

take v E E and consider the functional w -> (v, w). Obviously, this functional is

continuous in H, so that there exists u satisfying (5.4.34); by definition of

u E D(F8) and t

u = v showing that 0 is onto as claimed. It is obvious that 0 -' is

also symmetric, hence self-adjoint; that 93 is self-adjoint follows easily from the

formula

o*)-'

_ (-')*,

(5.4.35)

whose verification is

elementary.

is also obvious

that (W - $), = 0 -

(1 + b) I is an extension of If - l3 and that (5.4.31) holds. This concludes the proof.

Note that we have obtained a reasonably precise characterization of

D ((W - $j; another look at the proof of Theorem 5.4.7 shows that

D((%- $)s) = H, n D((%- $)*)

(5.4.36)

while (If - $)S is the restriction of (If - 3)* to H,. It is also interesting to note that

302 Perturbation and Approximation of Abstract Differential Equations

if (5.4.30) holds with a < I and if

($u,u)>-a'(521u,u)-b'(u,u) (uED(921))

(5.4.37)

for some a', b'> 0 (without restrictions on their size), then H, is independent of q3

in the following sense. Define

(5.4.38)

Then it follows from (5.4.32) that

(u,u),> (1-a)(u,u)o,

while on the other hand,

(u,u),<(1+a')(%Cu,u)+(l+b+b')(u,u)<C(u,u)o,

hence we may use ( , -)0 (which does not depend on $) in the definition of H,. We

note also that the argument in the proof of Theorem 5.4.7 can be adapted to show

that any symmetric operator satisfying (5.4.29) has a self-adjoint extension with

domain H, n D(%*). Finally, we point out that Theorem 5.4.7 is a sort of abstract

version of the reasoning used in Section 4.6 to construct the operators A (ft) there;

an even more similar method would be to define H, using ( , -)0 and apply Lemma

4.6.1 to the functional B(u, v) = (u, v),.

In many applications (such as in Theorem 5.4.9 below) a generalization of

Theorem 5.4.7 must be used. In it, the role of (' u, v) is taken over by Tl (u, v),

where Ij is a symmetric sesquilinear form with domain D(C)), that is, a map from

D (b) X D (b) into C (D (b) a subspace of E) which is linear in v for u fixed and

satisfies Ij(v, u) =t (u, v). The result is:

5.4.8

Theorem .

Let 521 be a densely defined symmetric operator such that

(5.4.29) holds, and let b be a symmetric sesquilinear form with domain D(t) 2 D(91)

such that

Ij(u,u)<a(51tu,u)+b(u,u) (5.4.39)

with 0 < a < 1, b > 0. Then there exists a self-adjoint operator Cs with domain D(Cs)

D(91) such that

(Csu,v)=($Cu,v)-I)(u,v) (u,vED(91)) (5.4.40)

and

((Y u, u) > - b(u, u). (5.4.41)

If a < 1 in (5.4.39) and if there exists a', b'>- 0 with

Tj(u, u) > - a'(S?Xu, u)- b'(u, u),

(5.4.42)

then (Y can be described as follows: the domain of (Y consists of all u E H, (H, the

completion of D(81) with respect to (5.4.38)) that make the linear functional w - (u, w),

continuous in the norm of H, where (u,w), is (the extension to H,) of

(u,w)c =(9tu,w)-I)(u,w)

and

details.

(u, w), =(1. u, w) (uED((Y),wEH,).

The proof is essentially the same as that of Theorem 5.4.8 and we omit the

5.4. Perturbation Results for the Schrodinger and Dirac Equations

303

Theorem 5.4.8 will be applied to A, the operator defined in (5.4.22) and

lj(u, v) = C),(u, v)+ t 2(u, v), where 1ji(u, v) "is" (31u, v), $, the operator of

multiplication by q1( x ), j = 1,2. However, the highly singular behavior of the qi (see

(i) and (ii) below) precludes the direct definition of the operators 13,. We make the

following assumptions:

(i) q, is locally integrable with mP, R (q, ; x) bounded in R' for some p > 0.

(ii) q2 is a finite sum of terms of the form ci I x - xi I -2.

The sesquilinear form ljj has domain 'D and is defined by

lji(u,v)= fgi(x)u(x)v(x)dx (j=1,2).

(5.4.43)

Since lu(x)v(x)I <Z(lu(x)I2+Iv(x)I2), we only have to prove that 1)i(u,v)

exists for u = v. For j =1, this is an obvious consequence of inequality (5.4.25): in

fact, under hypothesis (i),

1j,(u, u) <Cp-1RI llgrad u1I2 +C'(R)IIu1I2, (5.4.44)

where C does not depend on p or R. Note that by simply interchanging q, by - q,

an estimate of the same type can be obtained for - tj,(u, u), thus (5.4.42) holds for

17i.

The second sesquilinear form is estimated in the same way, but we use

(5.4.16):

Cj2(u,u)< E ci flx-xil-2lu(x)I2dx

ci>o

< S 4 2

Y- ci IIIgrad u1I2.

(5.4.45)

1 (m-2) c;>o

It is again obvious that a similar inequality (with a different constant) can be

established for - lj 2 (u, U)-

5.4.9

Theorem. Let m > 3,

q=qo+q1+q2,

where the qi are real. Let qo be locally integrable and bounded above, and let q, (resp.

q2) satisfy assumption (i) (resp. (ii) with

E ci <(m -2)2/4).

(5.4.46)

c;>o

Then the operator A described below is self-adjoint: D(A) consists of all u E `3Co (the

completion of '5D in any of the norms

Ilullx= f (Igrad u(x)12+(A-go(x))Iu(x)I2)dx

(5.4.47)

with A > ess sup qo + 1) such that Au + qu E L2(R "') and

Au = Au + qu. (5.4.48)

If mP, (q ; x) is locally bounded forsome p > 0, then D(A) consists of all u E Hl (Rm)

with I qO I u and t u + qu in L2 (R-).

304

Perturbation and Approximation of Abstract Differential Equations

Proof

Let A° be the operator defined in (5.4.22) with q = q0,

S2t the

restriction of Al - A° to 6D. We have

(2tu,u)> (u,u)a>Ilgradu112.

(5.4.49)

Combining this with (5.4.44) for R sufficiently small, we obtain (5.4.39) for tt, with

arbitrary a > 0 (note that if assumption (i) is satisfied for R > 0, it is satisfied as well

with any R' < R). On the other hand, (5.4.39) for tt 2 follows from (5.4.45) and

(5.4.46). We have already noted that (5.4.42) holds for tt, and 1) 2. Accordingly, we

can apply Theorem 5.4.9 to the operator 221 and to the sesquilinear form Ct = 01 + Ct 2.

Let G be the self-adjoint operator in the conclusion. Since the norms derived from

(5.4.38) and (5.4.47) are equivalent in D(A), H, and 9C0 coincide. Let u E' . Then

we have

( u,

(g)EGl). (5.4.50)

Since u belongs to Gt, q0 u, q, u, and q2u belong to L'(Rm), thus it follows that

Csu=Au - Au - qu

(5.4.51)

since (

is arbitrary. On the other hand, let u be an element of D((s),

a sequence

in 6D with u -> u in L2. In view of (5.4.51), if 9) E GD, we have ((E u, q2) = (u, CS (p)

lim(u,,, (Eq)) =

q)) = lim(Au - Au - qua, (p) = (Au - Au - qu, q2),

the

last limit relation valid because Au - 0u - qu - Au - 0u - qu in the sense of

distributions. Accordingly, (5.4.51) holds for every u E D((Y). Conversely, if u E `3C0

is such that 0 u + qu E L2, then we show using the definition that u E D ((Y) with C u

given by (5.4.51). This ends the proof of Theorem 5.4.9; the additional statement

concerning mp, R(q0; x) locally bounded follows from Theorem 5.4.5.

The translation of (5.4.46) in physical units for m = 3 and the Schrodinger

operator d = i (h /2 m) A u - i (1 /h) U is this: if U2(x) = Eejl x - xj l

-2 is the

por-

tion of the potential corresponding to q2, we must have

e;<0

5.4.10 Remark. The applications of Corollary 5.3.4 presented so far

have been restricted to the self-adjoint case. For an application to the

Schrodinger equation where the resulting semigroup is not unitary, see

Nelson [1964: 1].

5.4.11 Example. Develop analogues of the perturbation results for the

Schrodinger equation in dimension m =1,2 (see Nato [1976: 1, Ch. VI]).

5.5. SECOND ORDER DIFFERENTIAL OPERATORS

We apply the theory in Section 5.3 to the operators in Chapter 4; for the

sake of simplicity, we limit ourselves to an example involving Dirichlet

boundary conditions.

5.5. Second-Order Differential Operators

305

Let A0 be the operator defined by (4.4.1), that is,

Aou=EE ajk(x)DjDku+Y, bj(x)Diu+c(x)u. (5.5.1)

5.5.1

Theorem. Let 1 < p < oe, 2 a bounded domain of class C(2)

for 1 < p < 2 (of class C(1+2) with k > m/2 if 2 < p < oo). Assume the ajk are

continuous in S2 and satisfy thett uttniformttellipticity condition

Y. Eajk(x)SjSk>KIS12

(5.5.2)

and that the bj, c are measurable and bounded in 2. Let l3 be the Dirichlet

boundary condition. Then the operator A($) defined by

A(/3)u = Aou (5.5.3)

with domain D(A(l3)) = W2°P(S2)p (consisting of all u E W2°P(S2) that satisfy

the boundary condition /3 on IF in the sense of Lemma 4.8.1) belongs to ?.

Proof

It is not difficult to see that given e > 0 we can find functions

alk infinitely differentiable in 9, having modulus of continuity uniformly

bounded with respect to a and such that

Iajk(x)-ajk(x)I <e (xES2,

j,k<m).

(5.5.4)

This condition clearly implies that the aj' are uniformly bounded in n

independently of ;moreovertt,

0 < K' < tc, we shall have

L Lajk(x)SjSk > K 112

(x E 0, E Rm)

(5.5.5)

for e sufficiently small. Let

A,($)u=EEat(x)DjDku (5.5.6)

with domain W2'P(S2)p = W2'P(0)n Wo'P(S2). Then it follows from Theo-

rem 4.8.4 and an approximation argument (see Lemma 4.8.1) that there

exists C > 0 independent of e (for e sufficiently small) such that

IIUIIw2.P(a) <C(IIAf(Q)uII LP(2)+IIuII LP(o))

(u E D(AE(/3))). (5.5.7)

On the other hand, if P,(/3) = A($)- A,(#) and C' is a bound for 1 b 1 1 ,

. , . ,

Ibml and ICI,

IIPE(l3)ulIL5(n)1< m2ellullw2.P(0)+(m+1)C'Ilullw'.P(a).

(5.5.8)

Combining this inequality with (5.5.7) and Lemma 4.8.9, we deduce that for

any a > 0 there exists b > 0 such that

II Pe(a)uII LP(a) < all Ae(/3)IILP(St)+bllull LP(O).

The result then follows from Corollary 4.8.10 and Lemma 5.3.6.

5.5.2 Example. State and prove a version of Theorem 5.5.1 for boundary

conditions of type (I).

306

Perturbation and Approximation of Abstract Differential Equations

5.6. SYMMETRIC HYPERBOLIC SYSTEMS

IN SOBOLEV SPACES

We consider again the differential operator

Lu= Ak(x)Dku+B(x)u

(5.6.1)

k=1

of Section 3.5, using the notations employed therein; in particular, do is

defined by

dou=Lu, D(O

,0)=61

and d0 is defined in the same fashion using the formal adjoint. The results

of Section 3.5 establish that, under adequate assumptions on the coefficients

of (5.6.1), the Cauchy problem for the symmetric hyperbolic equation

u'(t) = Cu(t), (5.6.2)

where 9 = (Coo)*, is properly posed in L2 (118 m )". However, the results there

leave us in the dark about regularity of the solutions of (5.6.2): if, say, the

initial value uo = u(O) of a solution

has a certain number of partial

derivatives with respect to x1,...,xm, does u(t) enjoy a similar property for

all t? We know of course the answer to this question when L has constant

coefficients (see Section 1.6); from this particular case, we can surmise that

the "right" treatment for the problem is to examine (5.6.2) in the Sobolev

spaces HS = HS(IIRr)P We do this beginning with the case s = 1, and we will

find it convenient to introduce an abstract framework to handle this and

other similar situations.

Let E, F be Banach spaces; when necessary, norms in E or F will be

subindexed to avoid confusion. We write

F- E

to indicate that: (a) F is a dense subspace of E, (b) there exists a constant C

such that

IIuIIE s CIIUIIF

(u E F). (5.6.3)

A simple example of this arrangement is that where F = D(A), (A a closed,

densely defined operator in E) and the norm of F is I I UII

F = IIuIIE + II AuII E,

the "graph norm" of D(A).

Since at least two spaces will be at play simultaneously in the

succeeding treatment, we shall write, for instance A E+(C, co; E) to

indicate that A belongs to (2+(C, w) in the space E (A, or restrictions

thereof, may be defined in different spaces). Expressions like "A E (2+(E)",

"A E E (w; E)",... and so on will be correspondingly understood.