Can. J . Math., Vol. X X X I I , No. 6, 1980, pp. 1423-1437

MINIMAL OPERATORS FOR SCHRODINGER-TYPE DIFFERENTIAL EXPRESSIONS WITH

DISCONTINUOUS PRINCIPAL COEFFICIENTS

M . FAIERMAN A N D I . KNOWLES

1. Introduction. The objective of this paper is to extend the recent results [7, 8, 93 concerning the self-adjointness of Schrodinger-type operators with singular potentials to a more general setting. We shall be concerned here with formally symmetric elliptic differential zxpressions of the form ,$ ti ,.. Tq2: ",;I{

e c # - "8 .s{al*r 81if,> k" ,:$*?a m

(1.1) 7 = - (a, - ibj(x))afl(x) (a, - ib,(x)) + q(x) j . k = l

where x = (xl, . . ., x,) E Rm (and m 2 I ) , i = (-1)lt2, a, = a/ax,, and the coefficients a,,, b, and q are real-valued and measurable on Rm.

The basic problem that we consider is that of deciding whether or not the formal o p e r a t o r 7 defined by (1.1) determines a unique self-adjoint operator in the space L2(Rm) of (equivalence classes of) square integrable complex-valued functions on Rm. I t is well known that when the co- efficients a,,, b, and q are sufficiently smooth (see, for example, [8] conditions (S1)-(S4)), this problem reduces to deciding whether or not the restriction of 7 to Co"(Rm) is essentially selfadjoint in L2(Rm). However, when these smoothness conditions are not satisfied a riori the ,, problem becomes more difficult. w$&;;pni $.

An indication of what may be possible in the general case is given by the known theory for the case m = 1 in [ll, 5 171. Here, we may set b , = 0 without loss of generality, because the resulting operator is known to be unitarily equivalent (via a gauge transformation) to theroriginal operator; we also assume that

(1.2) q E .LIOC~(R),

Let T denote the restriction of 7 to the set

g ( T ) = {u :u E L2(R) r\ AC,,, (R), F u E L2(R))

where AC Ioc(R) denotes the set of locally absolutely continuous functions on R. Let To denote the restriction of T to the set of functions of compact

Received March 15, 1979 and in revised form August 13, 1979.

1423

1424 M. FAIERMAN AND I. KNOWLES

support in Q(T). Then it is known ([ l l , p. 681) that

(1.4) To* = T.

Thus, given (1.2) and (1.3), the problem reduces once again to d whether or not a certain minimal operator (T4 in this case) is essen self-adjoint (equivalently, the maximal operator T is symmetric).

For m > 1, it was shown in [9] that if ajk E C1+(Rm), for som b, E C1(Rm), and q = ql + 9.2 where qi E Llocl(Rm), i locally bounded below and qz is small in a certain sense, then define analogues of the operators T and TO above that satisfy t6 relation (1.4). I t was also shown in [9] that, as a consequence the standard self-adjointness criteria automatically hold in setting. Our main objective here is to extend this theory to case in which the principal coefficients of 9- may have discontinuities particular, we assume that the coefficients, ajk, bj, and g' satisfy following conditions:

(Cl) The matrix (ajk) is symmetric, and locally uniformly el the sense that for any compact set K C Rm there exists a positive X(K) such that

5 ajk(x)cj& 2 ~(K)(I: + . . f d,k-1

for all x E K and all vectors E: = (51, . . ., 5,) E Rm. (C2) For all 1 $ j , k S m, ajk E LIocm(Rm); furthermore, if

the open ball with centre 0 and radius r, there exist seque numbers (r,,) and {en] with r, -, oo as n -+ co , such that each a Lipschitz condition in each of the annular regions B,,+,,

(C3) b, E LloI(Rm) for 1 5 j 5 m. (C4) q E Llool(Rm). With reference to (C2), the regions in which we require the prin

coefficients to be Lipschitz have been chosen to be annuli, for venience; we could just as easily require that the same conditions in more general "bands", such as those defined in [3, 5 61, provided o that each band has non-zero width everywhere, and that every comp set K in Rm can be surrounded by a t least one band. I vital for the validity of the proof given below that the ajk condition (C2). Indeed, in Lemma 3 of the sequel we shall precisely the condition (C2) that enables us to relate the operators and T (defined in Section 2) with certain truncated op T(") (which are shown to possess certain properties), and this relationship that we are able to arrive a t our main result concerning TO and T. Finally, we might mention that the physical motivation for (C2) is that the coefficients ajk are allowed to have a t least bounded

I MINIMAL OPERATORS 1425

discontinuities, a situation encountered for example in diffraction problems (c.f. [lo, p. 2051).

The treatment follows, in essence, that given in [9]. A rather surprising fact that emerges is that despite the very weak conditions on the principal coefficients, the proof of the central Lemma (Lemma 2, corresponding to Lemma 1 of [7] and Lemma 2 of [9]) is essentially a direct consequence of the (local) ellipticity assumptions together with the corresponding

! result for the simple case when the principal part of the operator is just 1 the negative Laplacian.

2. Analogues of the maximal and minimal operators. In the sequel we shall set 3'f = L2 (R*), and denote by H1 the Sobolev space on Rm of order 1 (that is, H1v2 (R'")). L and Hlocl shall have their usual meanings. Observe also that, under conditions (C2) and (C3), F u makes sense as a distribution provided that u E HlOc1 and qu E Llocl.

We now define analogues of the operators T and Tmtn of [9]. Let T denote the restriction o f y to the set

Notice that, as both q and qu2 are in L1,,1, it follows that qu E LlOc1, and hence that 9-u is automatically defined in a distributional sense, as outlined above. Also, when b,, ajk E C1(Rm), 1 6 j, k $ m, the operator T is a restriction of its analogue in [9, equation (1.5)], by [7, Lemma 31. However, this difference is more apparent than real, because once we establish the analogue of (1.4) for the present situation, i t follows immediately that the two operators concerned must coincide. If, in addition, q E LlOc2 then a similar argument shows that T is just the maximal operator in the usual sense.

Let Q (To) denote the set of all functions of compact support in 9 (T) , and denote by To the restriction of T to TO). We define the minimal (closed) operator, Ti,,,,, corresponding to T by

whenever To is closable in %.

3. The main result. Following [6] , we define the norm

where 1 1 . 1 1 without the subscript denotes the usual norm in 3' f , and D5 = a, - i b , ( x ) . The completion of Com(Rm) with respect to the norm (3.1) is denoted by HB1(Rm) (or Ha1).

As in [9] we seek an appropriate analogue of (1.4) for the present

1426 M . FAIERMAN AND I . KNOWLES

situation. To facilitate this, we assume that the potential q satisfie! either of the following additional conditions of a general nature;

(C5) q can be expressed as q = gl + q2 where gl E LIocl and is lo call^ bounded below, and 92 satisfies either

and

(3.3) s lqdx-~) IQ(y)d~- -+O ass--0 Iv l<s

uniformly for x E Rm where

Q(Y) = IYI"" i f r n > 2 = -1oglyl i fm = 2 = 1 i f m = l

or, if m 2 5,

(3.4) 92 E LmI2(Rm).

(C5)' g can be expressed as q = ql + q2 where ql E L1ocm, and q satisfies

for any u E Hpl and E > 0, where 7 depends only on e and q2.

Remark. Condition (C5)' represents a variation on the standard con ditions (C5) (see [6]) in that while the assumptions on ql are stronger the assumptions on 92 are somewhat weaker. it her (3.3) or (3.4) ari sufficient for (3.5) (see [6] for a discussion of these conditions).

The main result of this paper is the following

THEOREM. Under conditions (C1)-(C4), and (C5) or (C5)', To is (

densely dejined symmetric linear operator in & and

To* = T,,,,* = T.

We begin the proof by establishing a number of lemmas. I

I LEMMA 1. Let

where ajk, 1 5 j, k 6 m, and b,, 1 6 j S m, are real-valued functions 21

LIOcm. Let the matrix (ajk(x)) be positive dejinite for each x E Rm. Then, z

MINIMAL OPERATORS 1427

u E HloC1 and 22'u E L,,,'

(3.7) 9 0 ( u l 2 Re[(sgn d)u]

where 90 denotes the operator 049 with bk = 0, k = 1, . . . , m, and sgn (x) = ii (x)/lu (x) 1 if u (x) Z 0 and is zero otherwise.

This is the analogue in the present situation of the well-known Kato distributional inequality (see [6, Lemma A]). As it happens, we only require this lemma in the case alk = being the Kronecker delta. However, the'full result has some independent interest, and its proof is no more difficult. We delay the proof until Section 4.

The next lemma is the analogue of Lemma 1 of [9 ] ; as usual most of the technical difficulties of the paper are encountered here. We define formal operatorsY(") as follows: 9") denotes a formal operator that is identical with Y for 1x1 < rn + en, but has coefficients ajktn) = afk, and q ~ ( ~ ) constant for 1x1 2 r, + en. As usual, b, and p2 remain unaltered in this construction. We will also need the quadratic forms a(n)[.], cl(")[-], and c2[.] defined by

where D, is defined above,

LEMMA 2. Let Ten) denote the operator corresponding t o r ( " ) via (2.1). Let h(") = a(") + clcn) + c2, where a("), el("), and c2 are dejined by (3.8)-(3.10). Then Ten) is a self-adjoint operator with

where -p* is any lower bound for plcn) on Rm and g((ql(") + p*] 'I2) denotes the domain of the self-adjoint operation of multiplication by (glen) + q*)'I2 in y. Furthermore

(3.11) h(")(u,v) = (T(")u,v)

for any u E 9(Tcn)) and any v E 9 (hen)), and

(3.12) h(")[ul 2 k l l ~ l l ~ b 1 ~ - (P* + M ) I I u ~ ~ ~

for some positive constants k and M, and all u E 9 (hen)).

Proof. We assume firstly that conditions (C1)-(C5) hold. The modifica- tions that need be made when (C5)' replaces (C5) will be noted a t the end.

1428 M . FAIERMAN AND I . KNOWLES

We begin by observing that (3.12) follows easily from (Cl) and Proposition 11, a consequence of (3.3) or (3.4). The constants k and depend only on n and q2. Clearly then, h(") is densely defined, se bounded and symmetric. Also, using [6, Proposition 11, it is not hard see that h(") is closed. Hence, associated with the form h(") is a uniq self-adjoint operator S(") with domain 9 (S("))C 9 (h(")), having t same lower bound as h'"), and satisfying

for all u E 9 (Sen)) and v E 9 (h(")). Let A(") denote the formal operator

and let 1'") denote the form defined by

m

L ( ~ ) [ u , v ~ = j=l zJRm I D ~ u ( x ) ~ ~ ~ x + J ql (n)(~) l

with domain 9(1(")) = Hbl n 9 ) 'I2). Finally, denot and J respectively the Hilbert spaces 9(hcn)) and g( l (") ) with

( (hen) + c~)[u]) 'I2 2 llull and ( ( I (") + c ~ ) [ u ] ) ~ / ~ 2 I I u I I , where c is some suitably large constant. Since there exist constants and hz (depending only on n) such that

m 2 2

A1141 I C ajk("'4Zk I A2141 f , k = l

for all complex m-vectors f ' , it is clear from [6, Proposition 11 that two norms defined above are equivalent norms on q*)l12). We now note that Com(Rm) is a form core for I this is essentially just Lemma 4.6b, p. 349 of [5]. I t is a simple ma using Lemma 1, to adapt this proof for magnetic potentials satisf (C3). Finally, as the spaces K and J are identical, Corn (Rm) is also a for h(").

We are now going to show that

(3.14) Sen) C T(").

To this end let u E 9(Scn)). Then, observing tha see from (3.13) that for every 4 E Com(R")

(S%, 4) = h(") (u, 4) = (T(")u, 4)

M I N I M A L OPERATORS 1429

~vhere the expression in angular brackets denotes the value of the dis- tribution T(")u a t 4; (3.14) now follows. T o show that equality holds in (3.14) it suffices to show that 9 ( T c n ) ) C 9 ( S c n ) ) . Accordingly, let

zb E 9 ( T c n ) ) . We shall now proceed in several steps.

Step 1. Let us assume first that u has compact support. Then u E 9 (hen)), and

for all 4 E Corn (Rm), which is a core for h'"). Thus by [S, Theorem 2.1 (iii), p. 3221 u E 9 (S'"').

Step 2. Let { E Corn(Rm) satisfy 0 5 { 5 1 , { ( x ) = 1 for 1x1 5 1, and { ( x ) = 0 for 1x1 2 2. For R > max(rn + en, 1 ) put r R ( x ) = { ( x / R ) . Observe that there exists a positive number A , independent of x and R such that la j{R(x) l 5 AR-I, and la,a,{,(x)( 5 AR-2 for 1 5 j , k 5 m. From the distributional identity

where A = x';"=l a j Z , it is easy to see that l R u E Q ( T ( " ) ) , and since rRu has compact support it follows from Step 1 that rRu E g ( S ( " ) ) . Hence from (3.11) and (3.12)

Fixing our attention upon (3.16), we observe that

and hence

From these results and (3.18) we conclude that

1430 M. FAIERMAN AND I. KNOWLES

where k, A , and C1 do not depend on R. Consequently we have

2 (3.22) IltRullaal S C2,

where C2 does not depend on R. Returning now to (3 (3.16) and the estimates (3.19)-(3.21) again,

by (3.22), for all R large enough, where Ca and C4 a: of R. Since tRu + u in %, it therefore follows from p. 3151 that u E 9(h(")).

Step 3. Observe now that by (3.15) and the I

u E 9 (Sen)), as required.

Our final task is to discuss the consequences of repla Here we note first that $9 (h(")) = Hbl, provided that perturbation of q l ; this is certainly assured by (3.5). I Com(Rm) is a form core for hen). Thus no other restrictic

The next result, though simple in form, is crucial enables us to relate the properties of the truncated those of T. For n = 1,2, . . . , let 4- E Co*(Rm) 0 6 4- I 1 and

By analogy with the operator TO, let TO'^)) denote support functions u E 9 (T(")) ; for u E 9 (To(")) set

LEMMA 3. Let u E 9 ( T ) u 9(Tc")) , and let 4" be d{

+nu E 9 (To) A 9 (To(")) an& ~ ~ ( " 1

Proof. We note first the following (distributional)

- 5

Let u E 9 ( T ) (the proof is similar if u E &u E Ha1 A H1. Also, it follows easily from (C2) i

and 3.15 of [I] that

The assertion now follows immediately from (3.24).

MINIMAL OPERATORS

We now have

LEMMA 4. TO and TO'") are densely defined symmetric linear operators in 8.

Proof. To see that TO is densely defined, let f E 2, and choose e > 0. Choose so that Il&f - fll < e/2. Since Ten) is densely defined, there is a g E 9(Tcn)) with llg - f l l < e/2. Then

II4ng - fII 5 II4ng - 4nfII + II4nf - f I I < e,

and E TO), by Lemma 3. Similarly we can show that To(") is densely defined; and i t is clear that To@) is symmetric. To prove that To is symmetric, let u, v E TO) be chosen arbitrarily, and choose n so that

supp v W supp u C B,,.

Then, noting that TOU = TO(+&) and Tov = To(4,v) on B,,,

(TO%, 0)' = (T0(4nu), 4nv)

= (TO'") ( 4 , ~ ) ~ 4"v) by Lemma 3,

= (&u, To(") (4"~)) since TO(") is symmetric,

= (u, Tov).

Thus To is closable, and T,,* is a well-defined operator in%.

LEMMA 5. For all u E 9 (T) and v E 9 (TO) we have

(3.25) (Tu, v) = (u, TOO).

If we choose n so that supp v C B , and make appropriate use of the function 4n, then the proof of this lemma is identical with that of Lemma 4 of [9].

LEMMA 6. The operator To(") is essentially self-adjoint in %. i.e.,

Proof. Since T(") is self-adjoint in 8, i t is sufficient to show that Te) C ~7). Let f denote the function in Co"(Rm) defined earlier. For s > r, + andefine f,(x) = f(x/s). Then, as s -+ oo, f , -+ 1, a,{, -+ 0, and a5akp, -+ 0 boundedly on Rm. Note also that

The proof now follows that of [9, Lemma 51.

1 1432 M. FAIERMAN AND I. KNOWLES

LEMMA 7. If u E TO*), then for each n h 1,

4,u E 9 (To*) n 9 (To@)*) and To* ( 4 , ~ ) = To(")* ( 4 , ~ )

Proof. Let u E TO*) and set To*u = f . Then

(3.26) (u , TOY) = ( f , V )

for all v E TO). Since 4,v € TO) for any v € 9 ( T o ) by Lem~ it follows that

(3.27) ( ~ 1 To(4nv)) = (f14nv)

for all v E 9 ( T o ) . On the other hand, if v E 9(Tocn)), it also follows Lemma 3 that

and that

TO (4nv) = TO(") ( 4 " ~ ) . Thus

by (3.26), for all v E TO(")). Now, let K denote the Hilbert associated with the form h(") as defined earlier, and with norm

where c2 2 q* + 1 + M. Then denoting the anti-dual of K by H have,

9(To(") ) C Q(Tcn)) C K C Hhl C Hb-' C K*

where the inclusions are continuous, and we may regard 9 ( T o ( " ) ) dense subset of K*. I t is well-known, and easily confirmed (see [7, Le 21) that T(") can be extended to a continuous map H(") on K to K*, 1

H(") is actually a restriction o fF(" ) . I t is also known that c2 + Hen) K onto K* bicontinuously.

From (3,28) we have

(3.29) (&u, To(")v) = ( f *, v)K*

for all v E 9 (TO(")), where (. , . )K+ denotes the inner product in K* f* is given by

m

(3.30) f* = 4nf lf- Dk(ajk@) ' a,& - 5 ajkDj(akhi u) . j .k-1 j.k=l

Clearly f* E H6-' C K* by an argument similar to [6, p. 1431. that, by the same reasoning we also have from (3.27) that

MINIMAL OPERATORS

for all v E TO). Returning now to (3.29) we see that

E 9 (To(")') and TO(")' ( 4 % ~ ) = f*,

where TO'")' denotes the adjoint of To(") regarded as an operator in K*. Set

(To'") + c2)' (&u) = g E K*.

By the preceding remarks there exists a function z E K with

(H(") + c2) ( z ) = g.

Consequently,

(by continuity of H(") on K )

= ( z , (To(") + c2)v) F

1 for all v E 9 (TO(")). Hence I

(&u - z , (TO(") + c2)v) = 0 for all v E TO'")).

Now, since To'") is essentially self-adjoint, i t follows ([12, Theorem X.261) that the range of To(") + c2 is dense in 2 f ; hence we have that 4 % ~ = z E Hbl, and therefore that u E H,,,l. Thus, from (3.29) and (3.30),

4nu E 9 (To(")*) and To(")* ( 4 " ~ ) = f* E 8.

Finally, from (3.31)

+nu E TO*) and To*(+"u) = f* = To(")*(4,,u).

We now complete the proof of the theorem stated earlier.

Proof of theorem. By Lemmas 4 and 5 i t is sufficient to show that

(3.32) To* C T.

Let u E 9 ( T o * ) and set To*u = f E 2 f . Choose n arbitrarily. By Lemma 7,

4.24 E 9 (To*) A 9 (To(")*) and

(3.33) To* ( 4 " ~ ) = To(")* ( 4 " ~ ) .

Since TO(")* = T(") by Lemma 6 , i t follows that u E H1(Brn) and ¶u2 E L1(B7,) for each n. We now show that for almost all x E B,n,

1434 M. FAIERMAN AND I. KNOWLES

Clearly, as 4, = 1 on B = Bm+(l14)rn

TO*^ - To*(+nu), v) L2(B) = 0

for all v E 9 ( T o ) with supp v C B. Thus (3.34) will be established if we can show that the set of all such functions v is dense in L2(B). Let w E L2 (B) and define w* on Rm to be equal to w in B, and zero eslewhere, Then given e > 0 there is a g E TO) with llg - w*ll < 3 e . Choose + E Com(Rm) so that

0 5 + 5 1, supp $ C B and + = 1 on BTn+(1/4)t.-6

where 6 is chosen small enough to ensure that

Il+w - wll L ~ ( B ) < 3 e .

Clearly +g E TO) (by the method of Lemma 3), supp (+g) C B, and

Finally, for almost all x E B,, we have by (3.33) and (3.34)

f (x) = TO*U (x) = To* ( 4 , ~ ) (x) = To("'* ( 4 , ~ ) (x) = T',' ( 4 , ~ ) (x)

= F("' (4nu) (x) = r (u) (x) . Since f E 2i? and n was chosen arbitrarily it follows that r u E #, u E HlOc1 and qu2 E LIOc1. Thus u E 9 ( T ) .

As in [9] we have

COROLLARY. The operator T is self-adjoint if and only if it is symmetric. I n this case TO is essentially self-adjoint and dejines a unique self-adjoint operator (TO) in %.

Thus, by the reasoning of [9], it follows that virtually all of the known criteria for essential self-adjointness (and in particular those given in [2, Theorem 2 ; 4 ; 81) are automatically valid under the considerably weaker a priori smoothness assumptions (C1)-(C4), and (C5) or (C5)'.

I I

4. Proof of lemma 1. We let 9 denote the space of test functions on Rm with the Schwartz topolbgy and 9' the corresponding space of dis- tributions. Let

j(x> = a exp ( - (1 - /x12)-1)

for 1x1 < 1 and j(x) = 0 for 1x1 2 1, where a is chosen so that

Note that j a (x) E C"(Rm), j , (x) = 0 for 1x1 2 6, and

j,(x)dx = 1.

Let

6 (4.1) u .(x) = Jsu(x) = j6(x - y)u (y)dy,

and for E > O let

U, = (1~12 + t2)lt2 and u t6 = ((u6I2 + e2)'I2.

Then an argument similar to [6, Section 5, Lemma 31 shows that

- where, for v E Hlocl, y ( v ) denotes the distribution defined by

(here 4 E $3 and (3, 4) denotes the value of the distribution S a t 4). We now complete the proof in two steps.

Step 1. Here we hold E fixed and examine the behaviour of (4.2) as 6 -+ 0. Firstly, it is clear that there exists a null sequence of values of 6, (6,:~ > 1) such that up(x) = u6p(x) +u(x) and utP(x) -+ u,(x) a.e. pointwise in Rm as p -t co. Moreover, from the relations

and akuP = ( 8 , ~ ) ~ (where ( )P = ( )6p), i t follows that

8kU; -+ Re [z &U]

LlOc2 as p -+ co. Hence u, E HlOc1, utp -+ u, in HIoc' as p -+ co , and

(4.3) sku, = Re

Turning now to the left-hand side of (4.2) we see that for 4 E 9,

Clearly then, Yo(u,p) -+L?o(u,) in $3' as P + co . Consider now the first term inside the bracket on the right-hand side of (4.2); i.e., we consider

M. FAIERMAN AND I. KNOWLES

the distribution

I t is clear that J

and

7 a (4.6) 7 a $ak (up) -+ - a,&,aku in 5@'

I ue

as p W . Also, for 0 E 9 we have -

(5 jkbkuv) 9 0 ) = -(a16k3, - a j(0 ' 5)) up

= -SRm ajk bk . up' ue 7 aj+ -

- sRm ajk bk up { y - d .,%id

u: u /

Since U P --+ u and U P -, u, in HI,,^, it follows that - up 21

(4.7) ;;i aibrdkuv) - a,(a,kbr)

in 9'. In a similar fashion, one can show that as $4 w ,

-2 a (4.8) 2 a,(o ,wv) -+ - a , (a jkak~) u e

in 9'. Thus from (4.4144.81 we see that

in 9' as $ P a. One also has a similar result for the second 4 (4.2). Combining these then gives 4

Step 2. Our final task is to investigate (4.9) as e -, 0. Cle (p E 9, we have (akue, 0 ) -, (akJu), 0 ) as as -+ 0. On the other$ follows from (4.3) that ak(u,) -, Re[sgn ii. sku] in L1,,* as c 4 @ we have !

drlul = Re[sgn ii. a*u] a.e. in Rm, and i 4 aklul in L, , ,~ as e 4 O. Bii

MINIMAL OPERATORS 1 A 9 . I

A simple calculation now shows that 9 o ( u , ) -, 9 o ( ( u l ) in 9' as e -, 0. I

It is also clear that

I in $9' as e - 0. The assertion of the lemma now follows from (4.9).

1. S. Agmon, Lectures on elliptic boundary value problems, Math. Studies (Princeton, N. J., Van Nostrand, 1965).

2. A. Devinatz, Selfadjointness of second order elliptic and degenerate elliptic d$erential operators, Proceedings of the Uppsala 1977 International Conference on Dif- ferential Equations, Acta Universitatis Upsaliensis (Stockholm, Almqvist & Wiksell, 1977), 37-51.

3. M. S. P. Eastham, W. D. Evans and J. B. McLeod, The essential self-adjointness of Schriidinger-type operators, Arch. Rational Mech. Anal. 60 (1976), 185-204.

4. W. D. Evans, On the essential self-adjointness of powers of Schrodinger-type operators, , Proc. Royal Soc. Edinburgh Section A, 79 (1977), 61-77.

5. T. Kato, Perturbation theory for linear operators (Heidelberg, Springer, 1976). Schrodinger operators &th singular potentials, Israel J . Math. IS (1972),

7. - A second look at the essential self-adjointness of the Schrodinger operator, in Physical reality and mathematical descriptwn (Dordrecht, Reidel, 1974). 1 8. 1. Knowles, On essentkl self-adjointness for singular elliptic dgerentk l operators, Math. Ann. ,527 (1977), 155-172.

9. - On the existence of minimal operators for Schriidinger-type differential expres- sions, Math. Ann. 933 (1978), 221-227.

) IO. 0. A: Ladyzhenskaya and N. W. Ural'tseva. Linear and quasilinear elliptic equations (New York, Academic Press, 1968).

11. M. A. Naimark, Linear differential operators 11 (New York, Ungar, 1968). 12. M. Reed and B. Simon, Methods of modern mathematical physics, Vol. 11: Fourier

analysis, self-adjointness (New York, London, Academic Press, 1972).

University of the Witwatersrand, Johannesburg, South Africa