Product Formula for Resolvents of Normal Operators and the Modified Feynman Integral
Author(s): António de Bivar-Weinholtz and Michel L. Lapidus
Source: Proceedings of the American Mathematical Society, Vol. 110, No. 2 (Oct., 1990), pp.
449-460
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2048088
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PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 1 10, Number 2, October 1990

PRODUCT FORMULA FOR RESOLVENTS OF NORMAL OPERATORSAND THE MODIFIED FEYNMAN INTEGRALANTONIO DE BIVAR-WEINHOLTZ
AND MICHELL. LAPIDUS(Communicatedby Palle E. T. Jorgensen)
ABSTRACT. We extend the theory of the "modifiedFeynmanintegal"developedby the second authorby extending his productformula for the imaginaryresolvents of selfadjoint (unbounded)operatorsto those of normal operators. This
enables us to establish the convergenceof the "modifiedFeynman integral"for
Hamiltonians with highly singular complex (instead of real) potentials. Such
Hamiltonians arise naturallyin the study of the Schrodingerequation associated with dissipative quantum mechanicalsystems.
By slightly altering the proof of our results, we also give a very general(operator-theoretic)interpretationof Nelson's "Feynman integral by analytic
continuation in the mass parameter"that is valid for singular potentials withan arbitrarysign.
An interestingaspect of our "productformula for the imaginaryresolventsof normaloperators"is that it extends, and in some sense unifies,the above two
approachesto the Feynman integral.

0. INTRODUCTION

In [LI, L2], a Trotter-like product formula was shown to converge to theunitary group generated by the generalized sum of nonnegative selfadjoint operators,using the imaginaryresolvents of the interveningoperators;in [L5], thisresult was generalizedto the case where one of the operatorshas a negative part,
relatively form bounded with respect to the other operator (with relative boundstrictly less than one). This applies to a "modified Feynman integral"[L2-L7]
when one specializes the intervening operators to be the usual L2 Laplacian-A and the operator of multiplication by a real L' function q, with someReceived by the editors September9, 1989; this paper was presentedby the first authorto the
Encontroda SociedadePortuguesade Matema'tica,Oporto, Portugal,April 1989 and by the second
authorto the AnnualMeetingof the AmericanMathematicalSociety, Louisville, KY, January1990.
1980 MathematicsSubject Classification(1985 Revision). Primary 47D05, 81C 12; Secondary
28C20, 35J10, 47B 15, 47B44, 47D45, 81C35.
Key wordsand phrases. Semigroupsof operators,product formula for normal operators,modified Feynman integral,Schrodingerequation, Hamiltonianswith singularcomplex potentials, dissipative quantum mechanical systems, magnetic vector potentials, analytic continuation in mass,
Feynmanpath integrals,potentials with arbitrarysigned singularity.
The work of the first authorwas partiallysupportedby the J.N.I.C.T. under contract 87568.
The work of the second authorwas partiallysupportedby the NSF undergrants DMS-8703138
and DMS-8904389.
? 1990 Amencan Mathematical Society
0002-9939/90

449$1.00 + $.25 per page450

A. DE BIVAR-WEINHOLTZ AND M. L. LAPIDUS

supplementary condition on its negative part (cf. [L5, ?4]); in this manner,
the second author has established in [L5] the existence of the "modified Feynman integral" (which he had introduced in [L2]) in the most general case forwhich the Schrodingerequation (with a real potential) can be solved without
ambiguity.
In the present paper, we first show how the abstractformula can be extended
to normal operators with sign conditions on their real and imaginaryparts andrelative form-boundednessof the negative real part of one of the operatorswith
respectto the real part of the other. In the second and third sections, we consider
the application to Schr6dinger operators (hence, in particular, to a modified
Feynman integral) in two distinct situations not contained in previously known
results:(1) The case of the Schrodinger equation with a highly singular complex
potential (a magneticvector potential is also allowed), which correspondsto thepseudo-Hamiltonianstreated, for instance, in [E]. Formally, it is the operator
1 = -(V - ia)2 + V;
2+
q - q - iq, and q, q ,q are
real nonnegative functions in L o(?), q is relatively form bounded withrespect to -A, with relative bound less than 1, and Q is a nonempty arbitrary
open subset of RN. The new point is that one can consider a highly singularimaginary part in the potential V; in general, however, due to the term - iq'e
is no longer a group but only a semigroup. Physically, this corresponds
to the fact that a quantum mechanical system with (nonreal) complex potential
is dissipative (or "open," in the sense of [E]), rather than conservative.
Just as in [L2-L5], one can then establish the existence of the "modified
Feynman integral" for Schrodinger operators with a highly singular complex
potential.
(2) The case of the semigroup e -t
generated by an operator ,V' corresponding to the formal expression.1

where a E L10 (Q; Ri), V = q - iq-0-(V-iad)=-.2-p+i(q-iq),p a nonnegative real function Withsome adequate relative form-boundedness
and local Lp conditions, a, q, q' with the same hypothesesas in (1), but without any restriction on q other than L1l0(Q) (or, alternatively, q E L 0+(C)E > 0, depending on the restrictions on p ). In this case, the proof has to beslightly modified, as the abstract result does not apply directly. The semigroupcorresponds to a "Schr6dingerequation with imaginary mass" and potential
q + i(p - q'). The application to the Feynman integral for a real mass can beobtained (at least when p _ 0) by analytic continuation in the "mass parameter," a device introduced by E. Nelson (cf. [N]).
The theorem in (2) partiallygeneralizesTheorem 4.1 of [B2] (detailed in [B3];see also [N, K, and BP] for related results); we note that the product formula

MODIFIED FEYNMAN INTEGRAL WITH SINGULAR COMPLEX POTENTIAL

451

considered here holds for a scalar potential with the hypotheses of [BK] in theirfull generality, and a vector potential with optimal restrictions, which was not
the case in [B2, B3], where, nevertheless, other formulae were considered.
1. THE ABSTRACT RESULTLet T be a (not necessarilybounded) normal operatorin a (complex) Hilbert
space H with inner product (-, ) and norm 11 11 (i.e., T is a densely definedclosed operator on H such that T*T = TT*) . Then one can write
T= T1- iT2,
with T1, T2 selfadjoint operators on H, where T1, T2 are respectively thereal and minus the imaginaryparts of T, which can be defined by using theoperational calculus given by the spectral theorem for normal operators, withthe functions 9(z) and -3(z) defined on the spectrum v(T) c C. (For all
the elementaryfacts about unbounded normal operators, see, e.g., [R, pp. 348355].) Since D(T) = D(T*), it easily follows that

T* = T1+ iT2.
If one supposes T1, T2 to be nonnegative, then iT is seen to be m-accretive(i.e., iT generates a contraction semigroup on H); this follows from the obvious fact that iT and (iT)* are both accretive (see, e.g., [P, Corollary4.4, p.
15]). We shall denote byQ(T) =D(JTJ1,2) =D

T,l n DT2)

T (j = 1, 2) can also be defined by
the form domain of T; T 112 andmeans of the spectral theorem.
If now A = A1 - iA2, B=B1 - iB2 are normal operators on H with A1,
B1 nonnegative (j = 1, 2) andQ = Q(A) n Q(B) dense in H,
one can define the form sum A-+B as the operator in H associated with thesesquilinearform
s:QxQcHxH
(u,v)

K

+

Alu,

-C

Aiv)-iB1u, Biv) - i

A2u,

B2u,

A2v)

B2v);

i.e., u E D(A+-B) iff v I-, s(u, v) is continuous on Q for the H-topology,(A+-B)u being the unique vector of H, given by the Riesz representationtheorem, such thatlv E Q.
((A-+B)u, v) = s(u, v),i(A+-B) is obviously accretive, and using the Lax-Milgram lemma, one caneasily show that it is in fact m-accretive. The same conclusions still hold if

452

A. DE BIVAR-WEINHOLTZ AND M. L. LAPIDUS

one replaces the hypothesis " B1 nonnegative" by a weaker assumption on its
negative part B_ . (Note that B1 = B+ - B_, where B+, B are respectively
the positive and negative parts of B, given by the spectral theorem.) Namely,
we shall assume, as in [L5], that B_ is relatively form bounded with respect toA1, with relative bound < 1, i.e.,(l.i)Q(B_) D Q(Al),and there exist positive constants y < 1 and J such that:2(l.ii)

llAu|Bu

2

?Y

A1u

Vu

+(51U12,D(A1).

In this case, in the definition of the sesquilinear form s, one has of course
to replace the term ( B1u, /Bv)by ( B\ u, IBv) - ( B_ u,
B_v) .
We can now state the following abstract result, which extends from selfadjoint
to normal operators the "product formula for imaginary resolvents" obtainedin [L5, Theorem 1, p. 263].
Theorem 1. Under the above hypotheses,([I + i(t/n)A]'[I + i(t/n)B]-)nune-it(A+B)ufor all u E H, uniformlyin t on boundedsubsets of [0, +oo).
Proof. The proof parallels that of Theorem 1 in [L5], and we shall only indicate
the changes to be made in the argument. Fix A > 0, v E H; for t > 0, set
Ut = [I + t(A + iA)]1

Vt = [I + itB]

and
Wt,

=

t[Uv1

-/],

wt =W

t,v.

Let us show that wt[A+ iC] 1v, as t - )0+, where C = A?B.
Wt E D(A) c Q(A) c Q(B_), and, by definition,v = Awt+ A2wt + iAlwt + t(2)

One has(I - Vt)Wt.

By using the spectral theorem for normal operators, one obtainstwhere RB I B'
given by

(I-Vt)= RB + IB/?i(IB,+-IB-)

IB+, and IB- are nonnegativebounded selfadjoint operators

RB =tIBl [(I +tB2)2 + t2B1],

IB =B2[(I + tB2) + t2B1]

and
IB =B ,[(I+tB2)2 + t2B,] 1.
Taking the inner product of (2) against wt, one then has easily

llwtII< A- Ii HI;

MODIFIED FEYNMAN INTEGRAL WITH SINGULAR COMPLEX POTENTIAL

453

hence, also A1A-2wt11I H RBWtJJ,and
IB:Wtll are bounded independently
of t. This, along with the observation (made in [L5],just before Equation (12),p. 266) that
11X-1<,

1B

V8 E Q(B_),(a consequence of the spectral theorem), allows us to apply hypothesis (1) todeduce, as in [L5, p. 266], that 11B?wtll and A1/4wtllare also bounded
independently of t. We can then pass to the limit, weakly in H, along asequence tn ? 0; wt tends to a certain w, and to identify the limits of theother bounded sequences, one can use the spectral and Lebesguetheorems. We
first obtain, in particular,that w E Q(A) n Q(B) = Q. Note also that by them-accretivenessof iB, one has
5(B) c {z E C: 3(z) < 0};so that, for z = x + iy E C(B), we obtain_ ty)2t,Z,2[(ltlZ= lZ_+ t2x2]-l-2(f1 -ty)2
<
*K

(3)

222

?tx2

(f1 -ty)2?+t2x2

lzl = lzr

Hence, in fact,
11 R

,t

0, 1

V? E Q(B)from which one easily deduces now that, along tn n
RBwt

0,

weakly.

Now, let y E Q = Q(A) n Q(B); the inner product of (2) with y gives, upon
passing to the limit in the second member,(V,y)=

(wt

y)?K

A2wt
4

+ i

A(4wt

Aly

A2Y)+K
+

B+Wt

RBWt,

RBY)?K

SY Bt

B- (A

IBWt

Wt

IB;Y)

Sy)

=A(w, y) + is(w, y);so that, by definition of C = A+-B, w e D(C), and(A + ic)w = v,i.e., wt tends weakly to w= ( + iC)1 Iv .
The remainder of the proof can be carried out just as in [L5, pp. 267-268],
and we omit it. oRemark 1. (a) A corollaryanalogous to [L5, Corollary 1, p. 269]-in which theform domain Q(C) is not necessarily densely defined in H-is also available.

454

A. DE BIVAR-WEINHOLTZ
AND M. L. LAPIDUS(b) Even in the case of nonnegative selfadjoint operators [LI, L2], let alone
in the case studied in [L5] or the present situation of normal operators,it is notknown whetherone can replacethe "imaginaryresolvents"by the corresponding
semigroups in the statement of Theorem 1. See [L2-L7], especially [L2], for a
discussion of this very difficult and intriguing problem.
2. APPLICATION TO A MODIFIED FEYNMANINTEGRAL

2N

We specialize now to H = L (Q), Q a nonempty open subset of R
N > 1. Let A = -A be the nonnegative Laplacian in H, and B the maximal
multiplication operator by a function V = q - iq', where q+, q,q'
are real nonnegative functions in LIl(Q), q relatively form bounded withrespect to -A, with relative bound less than 1. In this case, the form sum
A+-iB extends, hence coincides with, the natural realization F of -A + V(x),with domain
D(Zf)

=

{uE Ho ():Vu E Ljl(c(Q),-Au + Vu E H}.

In fact, F + A is known to be m-accretive for some A > 0 (see, e.g., [BK]).
In the case Q = RN, with some supplementaryassumptions on q , one caneven replace " u E Ho (Q) " by " u E L2(Q) " in the definition of the domain (see[BK]). (The hypothesis q' > 0 is not requiredfor the results of [BK] to hold anda somewhat strongerhypothesis is assumed on q , but then it is the closure of
the operator F + A that is known to be m-accretive. In the case when q' hasa sign, however, even with the stated hypothesis on q , the methods of [BK]
easily adapt to show that F + A itself is m-accretivewith D(Z) c D(A -B) .)We can then apply Theorem 1 to A and B, which gives us a representationof the solutions of the Schrodingerequation
jOT =

T

(t>0)

corresponding to the pseudo-Hamiltonian F. (See [E] for a description of
the properties and physical relevance of such operatorsto the study of "open",i.e., nonisolated, quantum systems.) When Q = RN, this gives us an explicit
representationof e'ie as a limit of iterated integraloperatorsthat generalizes
the modifiedFeynman path integral introduced by the second author [L2-L7]
to the case of a complex potential with a highly singular (merely L' ) negative
imaginary part. (See [L2, ?6.B], as well as [L5, ?4, in particular Theorem 2,
p. 270], and [L6, ?4c] or [L7, Part I] for the definition and properties of themodified Feynman integral in the case of real potentials.)
We can also replace -A by a more general elliptic operator L with a vector
potential, formally,N(4)

L = L(d)=-

Z

j,k=i

(Ok-iak)[bjk(01-iaj)],MODIFIED FEYNMAN INTEGRAL WITH SINGULAR COMPLEX POTENTIAL

455

where the blk form a symmetric uniformly elliptic matrix of real-boundedmeasurable functions on Q and a is an N-dimensional vector of L2 realfunctions on Q. L is the nonnegative selfadjoint operator associated with theclosure of the minimal energy form correspondingto (4) (i.e., the closure h of
the form defined on C(?(Q)). The form domain of L is
Q(h) = Q(L) = {u E L2()u

Ho (0), Vu - ia'u ELL(Q)},and L is the maximal restriction, as an operator on H, of the operator
L: Q(h)Q(h)' c 9/'(Q)-Nu I- Lu = h(u,)

=-

E J

bjkI(

;- iaj)u][(Ok-iak)-] dx.

j,k=lSee [B1-B3] for the definition of such operators and for the properties thatenable us to apply Theorem 1 in this case.
Remark 2. It would be interesting to extend to normal operators the abstract
perturbationtheorem established in [L6, Theorem 3.1, p. 43] in order to extend to singular complex potentials the "dominated convergence theorem" forFeynman integrals obtained in [L6, Theorem 4.1, p. 52, and Corollary 4.1,
p. 57].
3. THE CASE OF IMAGINARY MASS
With the notation of the preceding section, we can also take A = -iA2 = iA,

and B as in ?2. If one wanted to apply Theorem 1 directly, hypothesis (1)would impose that q is essentially bounded, as now A1 = 0. In this case,however, we can modify the proof of Theorem 1 to obtain the analogous result
by assuming only that q E L1c(Q); i.e., the real part of the potential can be
any locally summable function on Q0. We can also add a nonnegativeimaginary
part p to the potential with some boundedness hypothesis relatively to -A,
provided that, in the product formula, p appears in the same resolvent factor
as -A. (Otherwise,the resolventassociated with the potential would, in general,be unbounded.) Finally, as above, we can replace -A by the general operator
L. Under these general hypotheses, one can no longer define " A + B " as aform sum in a suitable way, but we can consider an m-accretiverealization of" i(A +B) ", defined in [BK] for the case A = iA and in [B1, B3], for A = -iL.
We shall consider the following set of hypotheses, exactly the same as in [BK]:(5.i)(5.ii)V EL'o(QV = q + i(p - q )q, p, q real-valuedfunctions,(P=(5.iii)p E L(Q)+ LP(Q),withp, q > 0,
N

p> I
p= I

when N> 3
when N = 2
when N= 1,

A. DE BIVAR-WEINHOLTZ AND M. L. LAPIDUS

456

and
either q E L'j+ (0) or p E L (NI2)+(Qwhen N > 2, for some arbitrarilysmall E> 0.

(5.iv)

We note that hypothesis (5.iii) can be replacedby the weakerassumption thatp E L c(Q) is infinitesimally form bounded with respect to -A, i.e., V(5> 0,
there exists A > 0 such that
(5.v)

fX
p

2

+

Vu EHo.

lul,

Since
1Vlull < (V - id)ul,

Vu E Q(L)(cf. [LS] or [B3]), this hypothesis implies, in particular,that we can define theform sum L-+(-p + A) = L-p + A, as a positive selfadjoint operator withform domain Q(L), a being the ellipticity constant of [bjk], Aa given by(5.v). (L p + ,a is the maximal restriction to an operator on H of
L - p + A e Y(Q(L), Q(L)'),see [B3, II, ?2].)
Then it is known that the operator iC such that
D(C) = {u E Q(L): Vu E L'(i)Cu = -iLu + Vu,

Lu + iVu E H},
Vu E D(C),is closable, its closure iC being such that iC + Aa is m-accretive.See [BK] forthe case L = -A and [B3] for the general case. In the adaptation of the proof
in [BK] to a general L, the following lemma plays a crucial role and it will also
be used later.
Lemma 1. Under the above hypotheses,let g E L??(Q) with a nonnegativerealpart, and let u E H; then,for A > A:
(L(a-) -p + g + A) u I < (L(O) 'p + A)-

u n1

..

Proof of Lemma 1. This lemma is closely related to Lemma 6 of [LS], whose
proof can be easily adapted to this situation (see [B3, Lemma 2.1, p. 28]). We
just note that in [LS], although g is taken nonnegative and p 0O, g is only
1
supposed L . Since we suppose that g E Lo?(Q), we do not need to startwith u in L?? and the proof does not require the use of Lemma 4 in [LS]. o
The solution of the general Schrodingerequation with imaginary mass andcomplex potential V:
jOt = (-iL(a) + V)T,(t > 0)is given by the semigroup el"tC, for which one has the following productformula.

MODIFIED FEYNMAN INTEGRAL WITH SINGULAR COMPLEX POTENTIAL

457

Theorem2. Underthe above hypotheses,([I + (t/n)(L'p[I + i(t/n)(q-iq')]-)

+A )]

e

nu

for all u E H, uniformlyin t on boundedsubsets of [0, +oo).
Proof of Theorem2. We shall use the notation of the proof of Theorem 1, with
A =-iA2=2-(LB = q-iq

P + Aj'.

We now take v E L2 (Q) n L??(Q), and we first show that:
I'Vt-[A + i(C

- iAu

Iv1

along a sequence tn -? 0, weakly in H.
n

One hasWt E D(A)= D(L p) c Q(L),but now wt is the solution of the following elliptic equation:(6)v = (L-p + AJ)wt + Awt+ Rtwt + Iq Wt + iIq,Wt,

with
Rt = t(q2 + q 2)[(,q [(I + tq')2 + t2q2]-

Iq+ tqi)2 + t2q2]-land
Iq =q[(I + tq)'2

2+ tq]2-1I

As in the proof of Theorem 1, one can easily boundllwl Wt,

w

|| \/

,

||'/ Rtwt

||,

,

and

I|

qw|

independently of t; and so we can justify that for a sequence tn -?n 0, wt
convergesweakly in H to
w E Q(A2) = Q(L),

(7)

with
q lwl 2(8)

L1(Q).

In particular, q'w E L I (Q). If we multiply (6) by a test function,E CO(Q) c Q(A2) n L(2

qdx)

nL2 (

2+ q2dx)

and integrate on Q , we can pass to the limit in the right-hand side of theequation obtained in this fashion, much as in the proof of Theorem 1. We
point out the differences in the argument. For the term in Rt, one has to use
R

Bt

0,
t

V? E CO (Q),458

A. DE BIVAR-WEINHOLTZ AND M. L. LAPIDUS

a consequence of Lebesgue theorem, and estimate (3), combined with the hypothesis V E L ,(Q) . For the last term,(9)

jiIqtwt(

we notice that, by the Lebesgue theorem,
Iqt(p

q (p,

either in L1 (Q), or in L1(), according to the hypotheses on q (cf. (5.iv)).
To pass to the limit in this term, it is then sufficient to obtain a bound on wt
either in L("+e)(supp (p) or in L' (supp q), independently of t. This can beachieved by using Lemma 1 with g
Rt ?Iq; + iIq
withone
Combining(6),
obtains:
wtl< V,,a.e. inwhere w is the unique Ho (Q) solution of(10)

L)%/-p/ + (A+ AJ / = I

But now, hypothesis (5.v) and an easy generalization of Theorem 2.3 of [BK]
ensure that
Lp (Q).
E n

V/

2<p<ooHence, in particular, wt is uniformly bounded in L(l+e)(Q), which allows toconclude in the case q E L +j(0 ).
In the alternativehypothesis p E Loc/2)+(Q), we can apply standardelliptic
local regularityto equation (10) to conclude that vy E L' (Q). Hence wt is
uniformly bounded in L?'(supp p); so, once again we can pass to the limit in(9), which guarantees, f E CO'(Q) being arbitrary,thatqw E ~~~~~~~~~~l1oc(f)(l l)From (5.v), (8), and ( 11), we deduce thatVw (EL'o()( 12)

We can then pass to the limit in (6) in the sense of distributions, which gives

fv v= ( L p?+ vw
(13)

=

h(w,

) -

p

) +

?A

q + iq)w

wfo+ J(A+ q' + iq)wp,

F( E CO (Q), and by definition of L, we obtain from (13), in 9'(Q):v = Lw - pw +A? w + (A + q + iq)w = Lw + iVw + (A + A)w.

MODIFIED FEYNMAN INTEGRAL WITH SINGULAR COMPLEX POTENTIAL

459

We thus have, in particular,
Lw + iVw = v - (+ ji)w E H;
(12) and (14) now imply(14)

w E D(C),[A+ i(C - iA)]jw = v.

We have concluded that along a sequence tn tending to 0+, wt tends weaklytow = [A + i(C-iA)J]-v.

Using the same argument as in the proof of Theorem 1 of [L5], we can easily
conclude that we have strong convergenceas t - 0+. The density of L2(Q) n
L?' (Q) in L2(Q), and the boundedness of the operators involved, now allow
us to reach the same conclusion for a general v E L2(Q) . The remainderof theproof can be performed exactly as in [L5]. El
Remark 3. (a) The above results in ?3 allow more general conditions on thecomplex (time independent) potential than those allowed by probabilisticmethods about the "analyticoperator-valuedFeynman integral." (See e.g. [J, ?3] andthe references therein, especially [C, N, and H].)(b) Assume that Q = IRN\F as in [N], where F is a closed set of capacity
zero. Then, as was noted in [K, Remark 2.5 and Lemma 2.6, p. 107], we
1
1
2
2
N
N) .
have L (Q) = L (R1 ) and Ho(Q) = H (JRHence, the results of ?3 can beapplied to potentials that are arbitrarilysingular on F (and, in particular,aremore general than those in [N], [H], or [J]).
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A. Bivar-Weinholtzand R. Piraux,Formulede Trotterpourl'operateur-A + q+ - q- + iq ,Ann. Fac. Sci. Toulouse Math. 5 (1983), 15-37.
[B1] A. Bivar-Weinholtz,Sur / 'operateurde Schrodingeravecpotentielsinguliermagnetique,dans
un ouvertarbitrairede R , C. R. Acad. Sci. Paris Ser. A 294 (1982), 2 13-216.
, Operateurs[B2]
de Schrodingeravec potentiel singulier magnetique,dans un ouvertarbi-2
trairede RN ,Portugal Math. 41 (1982), 1-12.
[B3], Operadoresde Schrodingercom potenciais singulares,Ph.D. dissertation, Universityof Lisbon, Portugal, 1982.
[BK] H. Brezis and T. Kato, Remarkson the Schr6dingeroperatorwith singularcomplexpotentials, J. Math. Pures Appl. 58 (1979), 137-151.
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C.M.A.F., 2 AVENUE
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Current address, Michel L. Lapidus: Department of Mathematics and Computer Science, University of California, Riverside, California 92521-0135