Trace formulas for parameter-dependent pseudodifferential operators

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Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

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Trace formulas for parameter-dependent pseudodifferential operators Gerd G r u b b a aCopenhagen Univ. Math. Dept. Universitetsparken 5, DK-2100 Copenhagen, Denmark. E-mail g r u b b m a t h , ku. dk Trace expansions for operator families such as the resolvent, the heat operator and the complex powers are established for elliptic problems containing pseudodifferential elements. We consider operators on closed manifolds, as well as operators on compact manifolds with boundary, where suitable boundary conditions must be added. It is found in general that one can obtain expansions, e.g. of the heat operator trace, in powers t ~ and powerlogarithmic terms t ~ log t, and the stability of the coefficients under perturbations is discussed. A survey is given of the methods relying on pseudodifferential calculus that lead to these results.

1. I n t r o d u c t i o n

_

1.1. T r a c e e x p a n s i o n s a n d g e o m e t r i c invariants One of the ways to determine invariants of Riemannian manifolds is by showing trace expansions associated with elliptic operators on the manifold. Indeed, it has been known since the work of Minakshisundaram and Pleijel IMP49] t h a t for a compact n-dimensional manifold X with b o u n d a r y OX = X', the zeta-function ~(AB, s) = Tr [(AB) -s] of the Laplace-Beltrami operator A = --A on X with a b o u n d a r y condition Bu -- 0 at OX (such as the Dirichlet condition), defined for large Re s, extends meromorphically to all of C with simple poles at (at n--~ most) the points s ---- 2 , J E N (we write N = {0, 1 , 2 , . . . } ) . The residues at these poles characterize geometric properties of the manifold, cf. e.g. McKean and Singer [MS67], and, more recently, e.g. Branson and Gilkey [BG90], Gilkey [G95], Avramidi [A90], for more general cases. Instead of studying the powers A s s , one m a y equivalently s t u d y the resolvent and its derivatives O~(As - A I ) - I , or the heat operator e - t A B , in their dependence on A (for A ~ co in sectors of C), resp. t (for t ~ 0q-). In fact, there are transition formulas relating A~ ~ and e - t A B to each other and to the resolvent (cf. e.g. G r u b b and Seeley [GS96] or G r u b b [G97] for details),

A ~ _ (s-- 1)---(s--r) 1 ~i f A"-~O~(AB Jc

AI) -1 dA

1

r(~)

~0°°

tS-Ye -tABdt,

(1)

e--tAB = t--r !2~rfC' e-tAi)~ (AB _ )~/)-1 d)~

-~.~1[

dR e

t-'r(s)A~'ds,

(2)

s=c

which imply t h a t there is an equivalence between a statement on the poles of F(s)Tr AB ~, an asymptotic expansion of 0~Tr (As - AI) -1 in decreasing half-powers powers of A, and an asymptotic expansion of W r e - t A B in increasing halfpowers of t; the coefficients in these expansions are closely related to the residues. Seeley [$69], [$69'], and Greiner [G71] showed such expansions for general elliptic differential operator systems with differential b o u n d a r y condition. In particular, they lead to index formulas, in view of the general observation indexAB

=

Tre -tAB°AB

-

Tre -tABAB*

.

(3)

New invariants came into the picture when Seeley in [$67] initiated the s t u d y of similar questions for pseudodifferential operators (~bdo's) A on closed manifolds. As pointed out by D u i s t e r m a a t and Gnillemin [DG75], this leads to logarithmic t e r m s in the expansions of the resolvent and heat operator traces, corresponding to double poles of r(s)TrA-'. Related questions for manifolds with b o u n d a r y have only been taken up fairly recently. Finite

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72

G. Grubb/Nuclear PhysicsB (Proc. Suppl.) 104 (2002) 71~8

expansions are known for general pseudodifferential problems from Grubb [G96] (and its first edition from 1986), and [G92] gives a finite expansion for the Atiyah-Patodi-Singer problem stopping at the index term. Full expansions for APSproblems were established by Grubb and Seeley in [GS95], [GS96], and full expansions for rather general pseudodifferential boundary problems for differential operators have been established in [G99]. For other types of manifolds, we should also mention the work of Briining and Seeley [BS91] and of Gil [G98] concerning operators on manifolds with conical singularities, and the work of Loya [L98] treating resolvents on manifolds with singularities such as corners and edges. Underlying these studies is always a suitable calculus of Cdo's depending on a parameter.

with inverse

:-lg](x) = f (where ~ = (2~r)-nd~), turns a differential operator Dxj = -iO/Oxj into the multiplication by ~j; more generally with multi-index notation, D~ = D ~ .-. D~: is turned into multiplication by ~ . So a differential operator on R n may be written

AS= ~ a~D'~f = f e'~'~ ~

a,~'~:(~)¢l~. l,~l<_m

l~l<_m

This motivates the more general definition of a pseudodifferential operator (¢do) P = OP(p(x, ~)) on R n, by the formula:

P f -= OP(p(x, ~))f(x) = / e~'~p(x, ~)f(~)fl~. (5)

Plan of the paper: This is a survey paper, presenting published as well as new results. The problems in dealing with parameter-dependent Cdo's are recalled in Section 1.2. Chapter 2 is concerned with the boundaryless case. In Section 2.1, we go through a calculus that generalizes that of [GS95] in a way that allows the treatment of complex powers of resolvents [GH01]. Section 2.2 gives the application to full trace expansions with powers and logarithmic terms. Chapter 3 is concerned with manifolds with boundary. First, we show in Section 3.1 how the results for the boundaryless case can be used in the treatment of the Atiyah-Patodi-Singer problem, including a new result on the stability of the first half-integer logterm [G01"]. Next, we explain in Section 3.2 a systematic calculus for manifolds with boundary [G01q that englobes and extends the APS methods. Finally in Section 3.3 we deal briefly with a case that is not covered by [G01'] and is needed for the study of the noncommutative residue in the Boutet de Monvel calculus [GS01]. In each of these various theories, it is the applications to full trace expansions that are in focus. 1.2. P a r a m e t e r - d e p e n d e n t calculi Recall t h a t the Fourier transform

[~-f](~) = ](~) = f e-'X'~f(x) dx,

(4)

When p is independent of x, this simply means that

OP(p(~))f(x) =

-1

(:/)

(6)

The formula (5) is a generalization:

op(p(

,

=

[p(y,

I =x.

The function p(x, ~), called the symbol of the operator, need not be a polynomial in ~. Instead it may be taken in the standard symbol space Sm(R~ x R'~), consisting of the C °o functions p(x,~) such that

O~O~p = O((~) m - M ) for all a , 3 • N'~;

(7)

here we use the notation <~)

=

(1

+

I~12)½.

For the differential operator A, the symbol a(x,~) = y~q~l
G. Grubb /Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

Hhrmander [H85] for various set-ups with local or global estimates in x. There is an important concept of asymptotic series used here: When p • S m° and there are symbols pj • S m ¢ for a decreasing sequence rnj --* - c ~ (where j --* co in N), we say t h a t

jEN

~

pj • SmJ for any J e N.

OP(p(~))OP(p'(~)) = ~ - l p ~ - l p ,

f

For general symbols, this holds in the first approximation: OP(p(x, ~))OP(p'(x, ~)) = OP((p

o

O P ( a ) O P ( a (-1)) -- 1 + R1, OP(a(-1))OP(a) = 1 + R2, with ~bdo's R1 and R2 of order - o o . In the study of resolvents one needs to carry out such parametrix constructions for symbols a - AI containing the extra paxameter A (and, notably, one needs a good control of how A interacts with the other variables). Here it is remarkably more difficult to handle pseudodifferential than differential operators. Let for example a(x, ~) be homogeneous in ~ of order m > 0 for I~1 > 1, C °O and strongly elliptic: Re a(x, ~) > c]~lm for I~1 -> cl, with c > 0. Let A -- OP(a), then the resolvent Q~ = ( A - A I ) -1 is defined for - X E It+ with I)q sufficiently large. It is a Cdo with symbol

q-m = (a(x,~) - A ) -1, q' of order < - m -

-m

~! ~p 0~ ~p, (-i)l~10~

I,~l_>l

(-i),'"' O~pO~p' in S m+'¢.

1.

Here A, or rather,/~ = ( - X ) ~ , can be considered as one more "cotangent variable" in addition to ~1, ~2,-.., ~n, and q-m is homogeneous of degree

p')(x, ~)),

where

aEN ~

(where r °k = r o r o ... o r with k factors), then we find that a o a (-1) ..~ 1. The symbol a (-1) is called a parametrix symbol; it defines an operator such that

q(x, ~, A) = q-re(x, ~, X) + q'(x, ~, A),

= OP(p(()p'(~)).

+ ~

°k

k=0

(8)

This does not at all mean that the series y~q pj converges to p, only that for any J, the sum up to j = J - 1 differs from p by a symbol that has the next order of magnitude O((~)mJ ) (along with the associated estimates of derivatives). This is similar to the way Taylor's formula shows how a Coo function is close to a polynomial of a given order, in the way t h a t the error term is of the next order of magnitude. For any given sequence pj • Sine with mj "N - 0 % one can construct a p such that (8) holds, by a generalization of the Borel construction of a Coo function with given Taylor coefficients. We call the symbols in S m ( R n x R ~) classical (or one-step polyhomogeneous), when they have expansions in series of terms p m - j ( x , ~), j • N , that are homogeneous in ~ of degree m - j for I~1 >1, such that p ~ }-~jeNPrn_j in S m ( R '~ x Rn). A composition of two operators is very simple in the case of x-independent symbols:

= ~

a o a - 1 _- 1 - r for some r E S - 1,

a (-1) ~ a - l o ~ r

O<_j
pop'~pp'

The formula, sometimes called the Leibniz product, is used in the construction of approximate inverses (paxametrices) for elliptic operators. If for example a C S m with a -1 E S -'~, then '

so if we take

p ~ y ~ pj in S m° if

p-

73

(9)

in

(¢,

More precisely, if A is a differential operator, then m is integer and a(x, ~) - A = a(x, ~) + #'~ is polynomial in (~, #), hence homogeneous and Coo in (~, it) • ~++1, so q-re(X, ~, A) is homogeneous of degree - r n and C °~ outside (~, #) = (0, 0). But

74

G. Grubb/Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

if A is truly pseudodifferential, the strictly homogeneous version of the symbol a(x, ~) will in general have a lack of smoothness at ~ = 0 (its ~-derivatives are bounded up to order < m only), so a(x, ~) + #m will have this lack of smoothness on the whole halfline {(0, #) lit _> 0}, and so will q-m. When we modify the strictly homogeneous symbol in a bounded neighborhood of ~ = 0 to be C °°, the ensuing modification of a(x, ~) + #m takes place in an unbounded set. This is also reflected in the estimates of q_re. Here one has (for I~l >- 1):

D~q-m = O(((~,#))-m-I<~l), for I~1 -< m, D~q-m = O(((~,#))-2"~(~)m-I<~l),

for

lal 2

m,

polyhomogeneous pseudodifferential operators

2. W e a k l y

2.1. P o l y h o m o g e n e o u s s y m b o l classes We here sketch a generalization of the symbol classes introduced in [GS95], now allowing quasihomogeneous symbols. It was inspired from reading Loya [L98], but is adapted to treat also e.g. complex powers of resolvents. A detailed presentation is given in Grubb and Hansen [GH01]. Consider symbols p(x, ~, A) that are C °O functions o f x and ~ E R ~, A E F (a sector of C\{O}). Let a E R + . We shall say that: p is s t r o n g l y a - h o m o g e n e o u s of degree m, when

p(x, t~, t~A) = trap(x, ~, A) for

where the estimate in the first line extends to all a if and only if a is polynomial in ~. In the polynomial case one can apply the usual symbolic calculus, just in one more variable, getting simple and straightforward results, whereas in the general case the fact that only the estimates up to order < m are standard (the so-called regularity number is only m), gives severe trouble. We give in Chapter 2 below a definition of parameterdependent symbol classes (more restrictive than in [G96] but more general than in [GS95] and [L98]) that handles the problems with resolvents of Cdo's in an efficient way, leading to full trace expansions. For boundary value problems there are similar phenomena. In the differential operator case, the resolvent parameter enters as another cotangent variable, on a par with the others, whereas for a pseudodifferential boundary operator (¢dbo), a resolvent parameter, when considered as a cotangent variable, gives symbolic estimates where only the first finitely many are "good". Again one can assign a regularity number to the operator and keep track of it in the calculus, this was an important point in [G96], leading to trace expansions with a finite number of power terms (related to the regularity number). In Chapter 3, we present several cases of more delicate (and restrictive) calculi which allow complete trace expansions with powers and logarithms.

I~1 +

(10)

-> 1, t >_ 1, (~, A) E R '~ x (F U {0}).

p is w e a k l y a - h o m o g e n e o u s when

of degree m,

p(x,t~,taA) = tmp(x,~, A) for

(11)

> 1, t _> 1, (~, A) E R n x r .

E x a m p l e 2.1. Let a(x,~) be C °O on R n x R n and homogeneous in ~ of degree r > 0 for ]~] > 1, taking values in a closed sector C \ F. Then for A E F, a(x,~) - A and (a(x,~) - )k) - 1 extend to: s t r o n g l y r-homogeneous symbols of degree r, resp. - r , if r is integer and a is p o l y n o m i a l in (it is the symbol of a differential operator); w e a k l y r-homogeneous symbols of degree r, resp. - r , if a is n o t polynomial in ~ (it is the symbol of a genuine Cdo). For example, when r = n -- 2, then a(x, ~) = ~2 + ~22 enters in the first case, and a(x,.,) =

+

+

(for

>

1)

enters in the second case, with F = C \ R+. For the weakly a-homogeneous symbols, we need extra conditions on the behavior when A is large in comparison with ~. Therefore we introduce the following symbol spaces: D e f i n i t i o n 2.2. Let m E R, a E R + .

75

G. Grubb/Nuclear Physics B (Proc. SuppL) 104 (2002) 71-88

S# (R × R , F) consists of the C ~ functions p(x,~, A) that are holomorphie in A ~ F ° for I(~, A)I >- ~ (for some s > 0) and satisfy, with I=Z,

s~,~(r), s~"',~'(r) c s~'+m"8+6'(r),

with uniform estimates for [z[ _< 1,

(12)

P E OP(S~'a(F)), P ' E

Moreover, we set, for any 5 ~ C, rn5 s~, (R n x a n ,r) = A~S~",°(R"xR",F).

(is)

and since one can refer to the standard rules for S m symbol classes, which must here hold uniformly in z as in (12). One finds for example that

O~p(.,.,~) is in S m + ~ ( R ~ x R ~ ) for 71 ~ F,

!Z in closed subseetors o f F .

The operators have good composition rules, since clearly

op(s~",a'(r))

P P ' E O P ( S ~ +m''~+z' (F))

(19)

(13)

The indication ( R n x R n, F) is often abbreviated

to (r)

and the resulting symbol is described by the usual formula:

(p o p')(x, ~, A)

We have: T h e o r e m 2.3. When a is as in Example 2.1, then for any N E N ,

(a(~,~) (~(~, ¢)

-

-

A)-N ~ s z ~ , ° ( r ) n s°,-~(r). A)N e sFN,°(r) + s°,N(r).

(14)

Moreover, ff 5 E (2 with Re ~ < O, then

(~(~,~) - A)-N+~ e sZ~,~(r) n s°,-~+~(r); (a(x, ~)

-

A)N+~ E s:N'a(F) + S°'N+~(F).

(15)

,,~ ~

(-i2,'"'O~p(x,~,A)O~p'(x,~,A)

(20)

in S~m+m''~+~' (F). This is an expansion in terms with decreasing m-exponents m + m' - j, j -~ co (j = la]). Here we use asymptotic series as follows: When pj E Sam~'~(F) for a decreasing sequence mj -~ - c o (for j --* co in N), and p E S~°'~(F), we say that p ~ ~-~dpj in S~°,a(F) if p - ~-'~pj E S 2 J " ( F ) for any J E N.

(21)

j
One cannot in general obtain (14) for noninteger N. The A-independent symbols fit into the calculus as follows:

For any given sequence pj E S~J'~(F) with mj - c o , one can construct a p such that (21) holds. For the present special symbols there is another type of expansion that is of great interest:

T h e o r e m 2.4. When a s y m b o l p ( x , ~) of orderm is considered as depending on one more variable A, it lies in S~'°:

T h e o r e m 2.5. When p E S m a ' ~(R ,, x R ,, ,F), then the limits

m)O Sm(R"xR") C S~ (Rn xR n ,r)

p(a,k)(X,~) = lim l o~(zap(x,~, _~)

(16)

z-"+O

for any F c C, any ~ E R+.

exist and belong to Sm+~k(R ~ x R'~), and p has an expansion in terms A~-kp(s,k)(x,~ ) such that for any N ,

We set

sY'~(r) = s;~,~(r); p(=, ~, A) -

mER

~

A6-~p(6,~)(=,~)

O
U

rnER

sT'~(r)

=

s~°'a(r) •

(17) E S~'+~N'~-N(R" × R ~, r).

(22)

76

G. Grubb/NuclearPhysicsB (Proc. Suppl.) 104 (2002) 71-88

In the proof, one reduces to the case (i = 0 by multiplication by A-~. Then the expansion is essentially a Taylor expansion in z = ~ at z = 0; it exists because of the uniform estimates for z --~ 0 in the sector. Note that in (22), the order of P(5,k) increases with increasing k, whereas the power of A decreases. A very simple example is

(1~[2 }_/~)--1___ )--1(1- [~2-[-[I;4--...). C o r o l l a r y 2.6. When p • S~-°°'~(F), the kernel K(x, y, A) of OP(p) has an expansion

The type of parameter-dependence mentioned in Remark 2.8 was used by Agmon and by Agranovi6 and Vishik in resolvent studies for differential operators; for Cdo's this is the kind of parameter-dependence studied e.g. in Shubin [Sh78] and other works. It is a mild generalization that does not cover resolvents ( P - A I ) -1 and parabolic operators such as 0 / ~ + P when P is truly pseudodifferential (as treated e.g. in [G96]). The operators behave in a standard way under coordinate transformations, so they can also be defined to act on sections of smooth vector bundles over smooth manifolds.

(23)

2.2. Applications to kernel a n d t r a c e expansions Both the expansion in Theorem 2.5 and the expansion in Definition 2.7 enter in the proof of:

Definition 2.2 contains no homogeneity requirements, but we now define a polyhomogeneous subspace:

T h e o r e m 2.9. Let p E sm°'~(F) be weakly apolyhomogeneous as in Definition 2.7, and assume furthermore that the pj with mj >_ - n are integrable with respect to ~ for each A. Then OP (p) has a continuous kernel Kp(x, y, A) with an expansion on the diagonal

K(x, y, A) ~ ~

A~-kKk(x, y),

kEN with Kk E C ~°.

Definition 2.7. A symbol p e S~°'~(F) is called (weakly) a - p o l y h o m o g e n e o u s of degree mo + a6, when p ~ ~ j e N PJ, with pj 6 S~J'~(F), mj x~ _co for j --~ co, j E N, such that the pj are weakly a-homogeneous of degrees mj + a6 (cf. (11)). If the pj axe in addition strongly ahomogeneous (el. (10)), and

oo Kp(x, x, A) ~ ~

j=O oo

+

k=0

[4 (x)log +

= 0(((5 N for all indices a, j3, J, then p is called s t r o n g l y a-polyhomogeneous. The a-polyhomogeneity is called classical (or one-step) when the sequence rnj is of the form mj = m 0 - j, j C N. R e m a r k 2.8. One has in particular that classical Cdo symbols in n + 1 cotangent variables p( x, (~, ~n+l ) ) give strongly 1-polyhomogeneous symbols when ~+1 is replaced by A.

(25)

for [A[ --* co, uniformly for A in closed subsectors o f f ([A[ > 1). The coefficients c~(x) and ~ ( x ) (with k =

j
"~+'~ cj(x)A ~+

are d e t e r m i n e d

by pj(x,e,

for [el -> 1 (in particular, they axe "local"), while the %I ! (x) in genera/depend on the full symbol (are "global"). The proof, given in [GH01], is a straightforward extension of the proof of Theorem 2.1 in [GS95]. A brief explanation in local coordinates: One uses the general principle that "remainders contribute to c~ terms," by Corollary 2.6. The pj contribute with

Kpj(x,x,A) =/rtP~(X,¢,A)¢l~

= fl~l>lAl~ PJql~-k fl~l
_

J1
l pj¢I~

G. Grubb/Nuclear Physics B (Proc. Suppl.) 104 (2002) 71~8 =

+ 12 + x3,

(26)

where 11 gives a power term (part of the cj term), /2 contributes to the c~ terms, a n d / 3 gives the rest of cj, and c~ with k = -m#+na if this is an integer > 0, plus some contributions to the c~ terms. One has of course to show that the contributions to the c~ pile up in a controlled way. When the operator acts on a compact boundaryless manifold, integration of Kp(x, x, A) (of its fiber trace if P acts in a vector bundle) in x gives a similar expansion of the trace of the operator: C o r o l l a r y 2.10. Let P be a A-dependent ~bdo acting on the sections of a smooth vector bundle over a compact manifold X of dimension n, with symbol satisfying the hypotheses of Theorem 2.9 in local coordinates. Then it is trace-class, the trace satisfying OO

m .+n

Tr P ~ E c J A ~ + ~ j =0

OO

+ E [c~ log A+c~]A~-k,(27) k=0

for IAI ~ c~, uniformly for A in dosed subsectors of F. The coet~cients are derived from those in (25) for coordinate patches by integrating the fiber traces over X .

oo

+ E ( c ~ ( x ) l o g A + c~(x))A - N - k ,

(28)

where A is a classical elliptic Cdo of positive order m acting in a vector bundle over a compact boundaryless manifold X of dimension n, with principal symbol am(x, ~) having no eigenvalues on R _ , S is a classical ¢do of order u acting in the bundle, and N is chosen so large that u - N m < - n . Since O~,(A - AI) -1 = r!(A - AI) -1-*,

(29)

this also pertains to expressions with a differentiated resolvent SO~ (A - A I ) - 1. By use of (14) one finds that for a narrow sector r around R _ , the symbol is in S,~-Nm'°(F) N S ~ - N (r) and weakly m-polyhomogeneous. Then Theorem 2.9 and its corollary apply. The kernel satisfies on the diagonal: g(x,x,S(A-

AI) - g ) "~ ~3=~--~'°°oCj(X)A~+~-~-g

(30)

k=O

for IAI --* ~ , uniformly in closed subsectors of r . Consequently, one has for the trace: ~ c ~

~+~-~

Tr S ( A - A I ) - N ,~ ~-~3 "=OcjA

'~

-N

+ E ( c ~ log A + c~)A - N - k

(31)

k=0

(by integration of the fiber traces over X). If S is a differential operator (in particular if S = I), then do(X ) = 0 and the complete coefficient of A-N is locally determined (by a generalization of the proof of this point in [GS95], Th. 2.7). In view of (29) and the general transition formulas (1), (2), we then also get trace expansions for the following operator families: T r ( S e -tA) ~ 2_ jt " . . . . j=0 oo

+ E(--~

logt + ~ ) t k,

(32)

k=0

The result applies in particular to expressions containing a resolvent power: P(A) = S ( A - AI) - N ,

77

r(s)

(sA-s) ~

j=0

+

Tr(Sn0(A))

s +

s

m

k.~o(-(s- -+-5~- -k)- ~ + ~___~__~ ~ s + k]"

(33)

For (32), it is required that the eigenvalues of the principal symbol of A have positive real part; (32) is considered for t --* 0+. In (33), the sign ",,~" indicates that the left hand side is meromorphic for s E C with pole structure as in the right hand side; here H0 (A) is the orthogonal projection onto the zero eigenspace of A. The constants 5k are related to the c}¢ by universal factors. Finally, let us mention a new type of result, made possible by the introduction of the present symbol class allowing noninteger 5:

Theorem C.

2.11. Let m c R + , u c R a n d s C Consider P(A) = S ( A - AI) -8, where A is

G. Grubb/Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

78

a classical elliptic Cdo of order m acting on the sections of a smooth vector bundle over a compact manifold X of dimension n, with principal symbol having no eigenvalues on R _ , S is a classical Cdo of order w, and s satisfies Re s > (n + y ) / m . Then for A in a sector F around R _ , the symbol of P is in S~4-s(F) (and in S ~-n~s'~ for each A) and weakly m-polyhomogeneous. The kernel satisfies on the diagonM: K ( x , x , S ( A - M ) -~) ~

oCjtX)a

oo

+ Z(C~k(X)log A + cg(x))A - s - k ,

(34)

k=O

where E1 and E2 are N-dimensional Hermitian vector bundles over X. The restrictions of the Ei to the boundary X ' are denoted E I. We assume that a normal coordinate x,~ has been chosen in a neighborhood Xc of the boundary X ' such t h a t the points are represented as x = (x', xn) there with x' E X', xn E [0, c[, the Ei are isomorphic to the pullbacks of the E l there, and there is a normal derivative 0 ~ . X ' is provided with the volume element v( x', O)dx' induced by v( x', x~ )dx' dx~ on Xc, which we view as Xc : X ' x [0, c[. D can be represented on Xc as D = a( -o~. ° + A1),

for [A[ ~ co, uniformly in dosed subsectors o f F . Consequently, one has tor the trace:

Tr S ( A - M ) -~ ~ 2-.,x--'oo c~a""+~-~ -~ j=0

(37)

where a is a homeomorphism from E1 [xc to E2[x¢ and A1 is a first order differential operator that acts in the x' variable at x~ -- 0. A1 [~n=0 has the principal symbol a°(x ', ~'). With the notation

oc

+ Z ( c ~ log A + cg)A - s - k .

(35)

k=0

The proof uses (15); details axe given in [GH01].

~/ju = (-iO~)Ju[~n=o,

(38)

we have Green's formula for D, for the sections u of E1 and w of E2 in the Sobolev space H 1,

(Du, w ) x - ( u , D * w ) x = -(a~/ou,~/ow)x,.

(39)

3. O p e r a t o r s o n m a n i f o l d s w i t h b o u n d a r y 3.1. F i r s t - o r d e r d i f f e r e n t i a l o p e r a t o r s manifolds with boundary

on

Resolvents and other operator families can also be studied for operators on manifolds with boundary. In the case of elliptic differential operators with differential boundary conditions, there are the fundamental studies of Seeley [$69], [$69'] and Greiner [G71] that show how trace expansions of the resolvent or heat operator can be obtained with purely power terms (no logarithms), and accordingly just simple poles ~n the expansion corresponding to (33). Again, the problem is more complicated when pseudodifferential elements enter. A prominent case that has been the focus of much attention is the Atiyah-Patodi-Singer problem, that we shall briefly recall: Let X be a compact C °o n-dimensional manifold with boundary X ' and let D be a first-order elliptic differential operator on X;

D : Coo(X, E1) --* COO(X, E2),

(36)

D e f i n i t i o n 3.1. 1 ° We say that D is "of D i r a c t y p e " when a is a unitary morphism, and

AI : A + XnVl q- Po,

(40)

where A is an elliptic first-order differential operator in Coo( E~ ) which is seIfadjoint with respect to the Hermitian metric in El, and the Pj are differential operators of order _< j . 2 ° The p r o d u c t c a s e is the case where D is of Dirac-type and, moreover, v(x)dx = v(x', O)dx~dxn on Xc, a is constant in Xn, and P1 = Po = 0 . As explained in [G92], p. 2036, unitarity of a in (37) can be obtained by a simple homotopy near X', whereas the assumption on A1 in 1° is an essential restriction in comparison with arbit r a r y first-order elliptic systems; it means that the principal symbol a°(x ', ~') of A1 at xn = 0 is Hermitian symmetric. P1 and P0 can be taken arbitrary near X ' , but for larger xn, P1 is subject to the requirement that D be 'elliptic.

G. Grubb /Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

When 1° holds, a°(x ', (') equals the principal symbol a°(x',~ ') of A. Along with A one considers the orthogonal projections H>, H>, H<, H< and II~ onto the closed spaces V>, V>, V<, V< and Vx spanned by the eigenvalues of A in L2(E~) that are > 0 , > 0 , < 0 , < 0resp. = A. (Since A is selfadjoint and elliptic of order 1, it has a discrete spectrum consisting of eigenvalues of finite multiplicity going to +c¢.) These projections are classical Cdo's of order 0; Hx is of order -c¢. Atiyah, Patodi and Singer considered in [APS75] the product case. This case is also studied e.g., in [GS96], [BW93], [BL99], [W99], whereas the case where only 1° holds is studied in [G92], [GS95] and other works. Cases where not even 1° holds are studied systematically in [G99]. The problem considered in [APS75] was the boundary value problem

Du=fonX,

H>70u=0onX',

(41)

with D of Dirac-type, in the product case. This boundary condition (and slight modifications of H> by finite dimensional eigenprojections) is often called the spectral boundary condition. One can also replace H> by the exact Calderdn projector C + associated with D, as e.g. in [BW93], [W99]; it differs from H> by a ¢do in X' of order - c ¢ in the product case ([G99], Prop. 4.1). As shown in [G99] (with reference to [$69"]), on may replace H> by much more general classical pseudodifferential 0-order projections B in E~, adapted to a general elliptic first-order D, such that the problem

Du= f onX,

BTou=OonX',

(42)

still defines a Fredholm operator DB (the socalled well-posed boundary conditions). Typically, B can be of the form B -- H> + II>QH< with a zero-order Cdo Q, and lower-order perturbations can be added. In all these cases (most generally in [G99]) we have obtained trace expansions of the form

Tr@e--tDs*DB) "~

~

akt k/2

--n
+ ~--~(ak logt + a'k)t W2, for t --* O, k=O

(43)

79

with the associated expansions of resolvents and of power operators; here ~ is a morphism in El. (Note that the notation differs slightly from that in (32); there is only one infinite series.) ' Let us give a quick explanantion of the general strategy. One can write the resolvent in the form

(D*BDB -- AI) -1 = Q~,+ + K~S~T~, where Qx is a Cdo representing ( D ' D - A I ) -1 on an open manifold )( D X, Q~,+ is its truncation to X, K~ is a Poisson operator going from X' to X and T~ is a trace operator going from X to X' (more about such operators in Section 3.2 below), and S~ is a Cdo on X' containing the particular information connected with the boundary condition. The resolvent derivatives O~(D*BDB - A I ) -1 have a similar form (el. (29)). Then (for r so large that the operator is trace-class) one finds by circular permutation

Tr x@O~ (D*BDB - AI) -1) = Trx(~O~Q~,~_) + Trx@O[(K~S~Tx)) -- Trx(~O~Q~,+) + Trx,O~(T~K~S~),

(44)

where the difficult term is turned into the trace of a Cdo on the boundary. The trace of ~oO~Q;~,+ is well-known (has an expansion like in (31) without the log-terms), and the operator on X' is treated by use of our weakly parametric calculus for ~bdo's, giving an expansion with powers and logs. Taken together, they lead to (43). An important question in this connection is what the value of the coefficients are, and how they reflect the geometry of the problem. In [GS96], the coefficients in the product case with spectral boundary condition were described in terms of the zeta and eta functions of A. In the non-product case in a natural setting with spectral boundary condition, the first three terms in (43) for n _> 3 were determined by Dowker, Gilkey and Kirsten in [DGK99] (moreover, the latter two authors have recently determined the next term in the case n _> 4). A basic information is of course whether some of the coefficients in (43) vanish under suitable circumstances. Here are some results to that effect:

gO

G. Grubb/Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

It is known, first from [G92] for spectral boundary conditions in the case of Definition 3.1 1°, and then for increasingly general cases (including the condition with B = C + in the product case [G99], for odd n [W99], see moreover [G01D, that a0 vanishes when ~a = I. The coefficient a~ "behind" a0 is of great interest; it is global and contains the eta invaxiant of A. For the product case with a spectral boundary condition, much more was shown in [GS96]: When n is odd, all the log-coefficients ak with k > 0 vanish. When n is even, the log-coefficients with k even > 0 vanish (so there are only higher order log-terms of the form ct~+½ logt, l E N). The question of whether the remaining logcoefficients are generically nonzero was answered in the affirmative by Gilkey and Grubb in [GG98]. We have recently (in [G01"]) considered perturbations of the product case, in order to find out to what extent the logarithmic terms preserve their properties. The first half-integer log-term al tl/2 logt in (43) has in the product case, when ~(x', x~) = ~°(x') on Xc, the coefficient

al ---- --7r-lcl (~0, A 2)

(45)

according to [GS96], where Cl is the coefficient of t 1/2 in the heat trace expansion for A 2 on X':

Let us finally mention that when only the boundary condition is perturbed, there is the following phenomenon, formulated explicitly in [G01] on the basis of results from [G99]: T h e o r e m 3.3. When B is replaced by B ' = B + S with S of order - J < - 1 , then the coefficients ak in (43) are the same for DB and DB, for k ~_ J-n. There are similar results on trace expansions for DBe -tDB*DB, corresponding to the eta function Tr(DB (DB*DB)-8). 3.2. P s e u d o d i f f e r e n t i a l b o u n d a r y operators To address the same questions for more general boundary value problems connected with pseudodifferential operators, we begin by recalling some elements of the Boutet de Monvel calculus IBM71] of boundary problems. First let us describe the model case, concerned with operators on the half-axis R+. The variable is denoted xn. We denote by r + the restriction operators from distributions on R to distributions on R + , and we denote by e + the "extension by zero" operator (applied to functions on R+): e÷ f ( x n ) = f ( x n ) if xn > 0,

OO

T~(~°e-tA~)~ ~

ck(~°,A2)tk/~

(46)

e+f(x,~) = 0 if xn < 0;

(47)

k=l--n

( c k ( ~ ° , A 2) = 0 for k - n + l o d d ) . Whennis odd, Cl(~°,A 2) -- 0, and when n is even, it is generically nonzero (cf. [GG98]). In [G01"] we show the following new result:

T h e o r e m 3.2. Let D O be as in Definition 3.1 2° and let D be a perturbation where A is replaced by A + x~P1 near X ' , t)1 a first-order tangential differential operator. Let B ° -- H> and B = H> + S with S a classical Cdo of order < - n - 1. Let ~o(x',xn) = ~°(x') for x~ e [0, c[. Then the coefficient al in (43) is the same for D°o and DB. When n is odd, it is zero; when n is even, it is as in (45), (46), generically nonzero. For perturbations A + x~P1 + Po with Po ~ 0 there does not seem to be a similar stability.

with a corresponding definition of e - . Out of the Schwartz space £ ( R ) of rapidly decreasing C °O functions we construct the spaces S~ = S ( R ± ) = r ± S ( R ) , = e+£++e-£-4C[5'];

(48)

here C[5'] indicates the "polynomials" in 5', i.e., linear combinations of derivatives of 5, ~'~o
~-/-- = ~ ' ( e - S _ 4 C [ 5 ' ] ) ----~-~:14C[~n], = ~s

= ~+4~-.

(49)

G. Grubb/Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

A constant-coefficient Odo P -- OPn(p(~n)) of integer order m on R is said to satisfy the transmission condition when P(~n) E 7-/. Another description is that p has an asymptotic series development P(~n) ~ ~m>_j>-o¢ Srn--J~Jn, in the sense that rn~j>m--N

is O((~n) m+*-k-N) for all indices k, l, N E N. A third description is that rmp(T -1) extends from T > 0 and ~- < 0 to be C °° at T = 0. Cf. e.g. [G96], Sect. 2.2. When p is of this kind, the truncated ~bdo P+, defined on functions on R + by

P+u = r+ Pe+u,

(50)

maps S+ into S+ (no singularities arise at xn --

0). Besides the standard trace operators (38), we introduce trace operators of class 0; for functions u on R + they are mappings T = O P T s ( t ) : u(x~) ~

t(x~)u(x~) dx~ (51)

with t(xn) E 8+; more generally, a trace operator of class r > 0 is an operator of the form

Tlu =

~

si~/3u+ Tu

(52)

81

A general s.g.o. G1 of class r _~ 0 is of the form

Glu =

~

g3~/ju + Gu,

(56)

O<_j
where the K j are Poisson operators and G is a singular Green operator of class 0 as defined in (55). These are the basic ingredients in the calculus. In more generality, one allows xn-dependence in the symbol p of P , and one defines operators relative to R + by letting the operators act in the tangential variables x I -- ( x x , . . . , x n - 1 ) as pseudodifferential operators and in the normal variable xn as described above. Then the functions p, t, k, ~ are taken to depend moreover on (xl, ~t) E R 2(n-1). For our purposes, we furthermore let the functions depend on a parameter #. Denoting the Cdo definition with respect to x ~ by OP' (cf (_5)), we then get operators defined relative to Truncated ~do's: OP (p(x, ~, #))+ = OP'OP~(p(x, ~, #))+, trace operators of class 0: O P T ( t ( x ' , x~, ~',/~)) = OP'OPWn (t(x', xn, (', 1~)), Poisson operators:

O
with T as in (51). Such operators map functions on R + into numbers. There is another (dual) type of mappings going the other way; the Poisson operators (in some texts called potential operators) g = OPK~(k) : v ~ k(x~)v

(54)

R 2 + = It+ x It+. A singular Green operator of class 0 is an integral operator with kernel

~(z., y~) ~ &+, Gu = OPGn(.~)u =

= O P ' O P K ~ ( k ( x ' , x~, (1, #)), s.g.o.s of class 0: OPG(~(x', xn, y~, ~', #))

(57)

(53)

with k(x~) E S+. Finally, there is an operator type from functions on R + to functions on I t + , called a singular Green operator (s.g.o.). Denote

s++ = s ( ~ + ) = r~+ ry+ S ( i t ),

O P K ( k ( x ' , x~, (', ]~))

~(xn, yn)u(y,~) dyn. (55)

= O P ' O P G ~ ( ~ ( x ' , xn, y~, ~', #)). The definition of trace operators and s.g.o.s is extended to class > 0 as in (52), (56). E x a m p l e 3.4. As a simple and important example defining a Poisson operator, we mention /¢(x~,~',#) = e -I(~',u)lx~ with # E R+; here OPK(e-I(~',u)lx-) is the solution operator for the Dirichlet problem (/~2 _ A)u(x) = 0 on R ~ ,

u(x',O) = v(x'). (58)

G. Grubb/Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

82

A crucial question for a parameter-dependent calculus is now how to prescribe the dependence on all symbolic variables x', ~', xn, yn, # taken together, in order to get operators with the properties we want. We shall here only deal with 1polyhomogeneity (the case a = 1 in Definition 2.2), where # enters in homogeneity considerations with the same power as ~'. The calculus may be explained in another way (the "complex formulation") where the Fourier transform in xn plays a great role. This involves some complicated spaces of holomorphic funtions of ~n, defined in terms of 7/ and 7/± (49), that we want to keep out of the presentation here. On the other hand, in the complex explanation, ~, enters in the homogeneities of symbols in the same way as the other variables ~', #, whereas in the present "real formulation", x , and y , enter in the functions {, ~:, ~} (called the symbol-kernels) in a quasi-homogeneous way oppposite to that of (fi', #). (For example, the symbol k = Jc= _-.~.e+k of the operator described in Example 3.4 is k ( ( , # ) = (1(~',#)1 + i ~ ) - 1 , which is homogeneous of degree - 1 , whereas the symbol-kernel k(x,~, ~', #) = e-I(~',,)l=, has a certain quasi-homogeneity.) We shall now let the symbol spaces defined in Definition 2.2 be based on x' E R n-1 and take values in a Banach space B, consisting e.g. of functions of x , 6 R + . We let a = 1 and leave it out of the notation, we denote the parameter by #, and we replace 6 by an integer d. Now we need to take powers of I((', #)1 into the definition. To smooth out the behavior near ((', #) = 0, it is convenient to replace I(~', #)1 by [(~', #)]; here [x] denotes a Coo function of x E R g satisfying: [x] = l x I for lxl _>l, [ x ] e [ ½ , 1 l f o r Ixl<__l. Note that [ ( ~ ' , 1 ) ] : 1 ( I)1 = = I~t

(~'/~)

~,,1) ; I forlzl

= I.I = Izl-l(=~'),

(59)

Coo functions p(x',~',#) vahied in B which sat-

is ,

with

= z,

O(izlp(-,-, I ) E S m + J ( R " - l x R " - I , B )

for !z e r ,

with uniform estimates for Izl < 1, !Z in closed subsectors of F.

(62)

Moreover, we define s m ' d : ( R ~-1 x R ~-1, F, B)

=/~d[((',/~)]'Sm,°,°(R"-lxR"-l,r,B).

(63)

The indication ( R " - l x R " - I , F , B) will often be abbreviated to (F, B). Keeping the identification of # with ;1 in mind, we shall also say that p(x',~', I) lies in s m ' d : ( r , B ) . We have left the requirement of being holomorphic in # E F ° out of the definition since = [~', #] is not so (one could instead work with a variant of ~ that is holomorphic on suitable sectors, as in [G96], (A.2")-(A.2'")). The symbol is assumed C °o in # E F considered as a subset of R 2. Moreover, we write 01zI (instead of 0z), since it is a control of the radial derivative that is needed (uniformly when the argument of z runs in a compact interval), and we shall in the following let Izl enter as a real parameter. To define symbol-kernels for the boundary operators, we use the cases B = Loo(R+) and --2 B = Loo(R++), with variables xn or un, resp. (x~, y~) or (u~, vn); these variables will then be mentioned in the detailed description of the function.

<_l, a n d t h a t

1)1 when ~ = I"

(60)

[(~', ~)] is also written [~", ~]; it will in the following be denoted ~ (as in [G96]), so from now on,

= [~', I] = [~', #], with # __ - ;1.

D e f i n i t i o n 3.5. Let m E R, d and s E Z. Then Sm'°'°(R ~-1 x R ~-1, F, B) consists of the

(61)

D e f i n i t i o n 3.6. Let m E R, d and s E Z. 1 ° The space s m ' d ' s ( R n - l x R n - I , F , S + ) (briefly denoted sm'd:(F,S+)) consists of the functions ](x',xn,~',#) in Coo(R '~-1 x R + x R "-1 x F ) satisfying, for 411 l,l' E N, l l, _ , (z~i )l--l , unO~,~f(x, lzlu

. . .E . 'i)

E S " d ' s + l ( R ' ~ - l x R ' ~ - l , r , Loo,,,.(R+))

(64)

G. Grubb/Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

(equivalently, u,O~,.f(x I ~' - , , Izlu,, ~,,}) belongs to • S m'd+l-l',~+l.l+l' (r, Loo,,. (R+))). 2 ° The space sm'd'~(R"-I x R " - I , F, S++) (briefly denoted Sm'¢8(F,S++)) consists of the

functions ](z',

y.,

in

x

R n-~ x F) satisfying, for all l, l', k, k' • N , 1) t l--l ' + k - k ' l l~ k k ' - r (Z~ ) UnO~u. VnO~v.f(x ' Izl un' Izl vn' ~" z

• sm'd's+2(Rn-1XR n - l , I", Loo .... . . (R~_+)).(65) In detail, the statement in (64) means that for all j,

[l~=l(z d ~- s - 1 •

!

p

r

I

/ / 1 (z~) / - I u.O" f(z,lzlu.,~',~))ll -

"< (~')"+J,

Lo¢

(66)

with similar estimates for derivatives 0~,0~, ~ /~ with m replaced by m - [~[. There is a related explanation of (65). We here use "_ to indicate "<_ a constant times"; also > will be used, and - is used when both < and > hold. The third index s is included to keep track of factors a in a manageable way. When s -- 0, we may lave it out of the notation. For the trace formulas later on, it is important to know that we always have inclusions (in view of (14)): S m'd's C S mTs'd'O rl S m,d+s,O if s < 0,

S m'd'~ C S m+~'d'° + S m'd+~'° if s >__0,

(67)

for S- as well as for S-spaces. In applications, the symbols will often be assumed to be holomorphic in /~ for # • F ° with I(~', #)[ >- z (some ~ > 0); such symbols are called holomorphic in #, and this property is preserved in compositions. D e f i n i t i o n 3.7. 1° The functions in sm'd'~(F,$+) are the P o i s s o n s y m b o l - k e r n e l s and t r a c e s y m b o l k e r n e l s o f class 0, of degree m + d + s, in the paxametrized calculus. 2 ° The functions in Sm,d,8(F, S++) are the sing u l a r G r e e n s y m b o l - k e r n e l s o f class 0 and degree m + d + s in the parametrized calculus.

83

One also defines subspaces of these symbolkernel spaces, consisting of the functions that are series of terms with decreasing m-index and quasi-homogeneity in (xn, ~',/~) or (x~, y~, ~', #) (the Fourier transformed terms will be truly homogeneous in (~n, ~', #) resp. (~n, ~/n, ~', #) for [~'[ >_ c > 0). These symbol-kernels are called (weakly) polyhomogeneous. R e m a r k 3.8. By Fourier transformation in x , of the functions extended by 0 for x , < 0, ,.£m'd's(F,S+) is carried over into the space sm'd,s(r,~-{ +) of P o i s s o n s y m b o l s of degree m + s + d, and co-Fourier transformation (formula (4) with minus replaced by plus) gives the space sm'd's(F, ~ - 1 ) of t r a c e s y m b o l s o f class 0 and degree m + s + d. By Fourier transformation in x , and coFourier transformation in y , of the functions in sm'd's(F,$++) extended by 0 for x~ < O, yn < O, we get the space sm'd's(F, 7~+~7-{_-1) of s i n g u l a r G r e e n s y m b o l s o f class 0 and degree m-t-d+ s. The explanation for the +1 resp. +2 in the sindex in (64) resp. (65) is that with this choice, m + d + s is consistent with the top degree of homogeneity in the Fourier transformed situation for polyhomogeneous symbols, where the scalings in x~ and y~ lead to shifts in the indices. Further details in [G01']. R e m a r k 3.9. To motivate the scaling in x~ in the above definition of parameter-dependent boundary symbols, consider the symbol-kernel in the basic Example 3.4, k(x~, ~', #)_= e -I(¢',v)l~. Setting # - 7, 1 we find for k(xn,~', !)z = e - } (ze')~" (cf. (60)), by use of the formula (with cj = jJe - j ) sup x~e -ax" = cja -3 for a > 0,

(68)

z,~kO

that sup=.> 0 IO~k(x,~,~', ¼)l behaves like z-J for each ~. This does not comply with the uniform estimates in z < 1 required in (62) (with B = Lo~(R+)). But if we replace x~ by zun, we find using (68) that the resulting function kl(u~,~',z) = k(zun,~', 1) = e -(z¢%" has sup~.ko [O~e-~"(z~'>[ <_ (~'}J, j • N, which fits well with (62). (It is easy to check a few steps by

G. Grubb /Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

84

hand calculation; a systematic proof is given in [G01'], Wh. 3.2.) On the other hand, because of the scaling, the example e-W]~- (constant in #) does not fit into the calculus. This is an unpleasant difficulty in the applications: #-independent ¢do's on the boundary R n - t are included, but #-independent Poisson, trace and singular Green operators are in general not so; Theorem 2.4 does not extend to these new symbol families. We also remark that the generalization of Theorem 2.5 to the new s.g.o.s is not really convenient. It is easy to show that composition of the operators in the list (57) leads to other operators of these types. The basic step is the composition with respect to the variable xn, denoted on. Here we have for example:

5°

~ on ~' =

~(x=, zn, ~)~'(.., yn, ~) dz= ~0 ~

8m",d",¢'+i(r, & + ) . Here 1° is a simple product result. For 2 °, we use that, according to (66),

izd~-s-l~l ~_ <~,>m,

izd'~-¢-l(1 +

(z~,)2u2)~q "< (¢,)m',

with x= = [z[un. Then we have on each ray # = re i°, for I~1 > 1:

z~+d'~ -~-~'-~ fo ~ ~(x~)~'(xn) dxn _~ (~,)m ~o °° z d' ~-~ ' k'(Izlun) ~

Z dun

P r o p o s i t i o n 3.10. Let

[ l ( x ' , x n , y n , ~ ' , # ) E sm'd's(F,S++),

(69)

~(~',~n,~',~), Tc(x',~n,~',~) c s~,d,s(r,s+),

(70)

Then (suppressing the variables (x',~') in the formulas)

1°

~ on ~' = ~(zn, ~)~'(yn,

~)

• Sm",d",¢' (r, 8++), 2°

~On k' =

/7 ~(Xn,.)~'(Xn,.)dx.~

• smH'd"'s"+l(r, C ) ,

3°

~On~' =

/7 ~(Xn,~)~'(xn,y,~,~)dz,~

• ,s,~",~",¢'+~(r, & ) , 4°

0 On ~' =

/7

~(Zn, Yn, ~)k'(~n, ~) dyn

e S m H'd ~t' s,,--1 + ( F , S . ).

/7

(1 + (z~t)2u2)-ldun

___ <~,)m+~',

and let ~', t' and ~¢, be given similarly with m, d, s replaced by m I, d t, and d . Define m " = m + m', d" = d + d', s" = s + s'.

< (~')m+m'lz~]

since Iz~l --
(trn~)(x',~',.) ;

/7 ~(x',~n,~.,~',.)d~.

(71)

to a singular Green symbol-kernel ~ of class 0; this gives the trace of the operator in the xn-variable defined by ~. It is a Cdo symbol belonging to our calculus: T h e o r e m 3.11. Let [1 be a singular Green symbol-kernel on R~+ :

[1(x',x~,yn,~',#) E s m ' d ' ~ - l ( F , S + + ) .

(72)

G. GrubblNuclear Physics

B (Proc.

Then the normal trace of 3 is a +do symbol on R”- 1 satisfying tr,+‘,

t’, p) E bs+qr,

C).

(73)

We refer to [GOl’] for the detailed proof. When G = OPG(i), the $do with symbol trni will be denoted tr,G. There are further composition rules, involving also a parameter-dependent pseudodifferential operator on R”, truncated to R”, and satisfying a suitable transmission condition at z12 = 0 taking the parameter into account. The basic property needed here is that when Q is a pdependent $do on R”, the trace operator T and the Poisson operator K defined by

Tu = yoQ+u,

Ku = T+&(v(z’) @ 6(x,)),

(74)

should have symbols of the type introduced in Definition 3.6. This restricts considerably the generality of the @do’s that are allowed. Resolvents of differential operators, and strongly polyhomogeneous $do’s (with symbols stemming from $do’s in one more cotangent variable which is replaced by p) enter, but $do’s that are constant in I_Lwill in general not be included, since they produce p-independent Poisson and trace operators by (74), cf. Remark 3.9. The precise definition of the appropriate transmission condition, and the proof of the full composition rules, make heavy use of the complex formulation of the theory and will be omitted here. It is found that when A and A’ are systems of operators belonging to the calculus, so is their composition A”,

S >(

P;+G’ T’

Suppl.)

104

71-88

85

boundary X’. (Suitable noncompact manifolds may be included too, cf. e.g. [G96].) Now let us consider trace expansions. When A is a p-dependent system as in (75)) of order m < -n and class 0, going from C”(X, E)xCOO(X’, F) to itself (where E and F are vector bundles over X resp. X’), then

For TrxP+ and Trx,S we have results as in Theorem 2.9 and Corollary 2.10, based on the theory for boundaryless manifolds. But TrxG demands new results. It is here that Theorem 3.11 is extremely useful, because it allows us to write, in each local coordinate patch at the boundary, TrxG = Trx, (tr,G):Here tr,G is a $do on the boundary, which is itself a boundaryless manifold, and the symbol of tr,G belongs to the weakly parametric calculus of [GS95], so Theorem 2.9 and Corollary 2.10 can be applied in dimension n - 1 to give a trace expansion for this term. One finds for example: Let G be a singular Green Theorem 3.12. operator of class 0 on X, with polyhomogeneous symbol-kernel lying in Sm~d~-l(Rn-l x R”-‘, P, S++) in local coordinates (whereby tr,G (R+‘, r, C) in local coordihas symbol in Sm~d~o nates on Xl), holomorphic in p. Assume moreover that the homogeneous terms in the symbol of tr,G lying in Sm-j~d~O with m - j 2 -n are integrable in c’. Then G is trace-class, and the trace of G has an asymptotic expansion

~ G N 2

K’ S’ >

(2002)

CjCLm+d+n-l-j

j=O

(75)

+

ccc;log j.J+ &d-k

(76)

k=O

with resulting symbol classes where the indices are added and modified like in Proposition 3.10. The operators behave in a standard way under coordinate transformations, so they can be defined to act on the sections of smooth vector bundles over smooth compact manifolds X with

for ,LL + co in closed subsectors of I?. The coefficients cj and CL are determined from the homogeneous terms in the symbol. The use of tr, here replaces the circular permutation used in a special case in (44).

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G. Grubb/Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88

This calculus covers all cases connected with Dirac operators, and also much more general cases, where the singular Green operators are not finite sums of products of particular Poisson and trace operators. The calculus is used in [G01"]. It should be noted that the "interior" Cdo's P allowed in this calculus are not nearly as general as for example those considered in [G96] (since most/z-independent operators do not fit in). This does not exclude that an expansion like (76) can hold for other operators, since what we really need in order to show this is that we get a Cdo in the weakly parametric calculus of [GS95] or [GH01] after performing the reduction to X ' (the application of trn). In the last section we shall consider a case with somewhat different ingredients: There is a #-independent interior Cdo involved, but the parameter-dependence occurs in a factor which is a very nice operator having a rational symbol with full control over the poles: the resolvent of a Dirichlet Laplacian. In this case it is possible to make a full analysis of trace formulas by residue calculus and use of special functions. 3.3. T h e n o n - c o m m u t a t i v e r e s i d u e Wodzicki introduced in [W84] a trace functional (a linear functional vanishing on commutators) for the algebra of classical Cdo's S on a compact manifold, called the non-commutative residue res(S), and identified it with the residue at zero in the trace expansion of SA-S: res(S) = ord A . Ress=oTr(SA-8),

(77)

here A is an auxiliary elliptic operator. In the corresponding formulation for the iterated resolvent S ( A - AI) - g , res(S) is the first log-coefficient c~) in the trace expansion (31) (up to a universal factor). Wodzicki also describes res(S) by an integral formula in terms of the symbol of S. See also Guillemin [Gu85]. For operators P+ + G in the Boutet de Monvel calculus, a similar trace functional was introduced by Fedosov, Golse, Leichtnam and Schrohe [FGLS96], defined by an integral formula in terms of the symbols of P and G. However, this was not shown to be a residue as in (77), nor a logcoefficient in a resolvent trace, simply because

such trace formulas were not sufficiently developed for the Boutet de Monvel calculus. In a joint work with Schrohe [GS01] we have attacked this problem. It is interesting that in this case it is a log-term one is after, whereas the log-terms may seem to be something of a nuisance in the previous cases. We consider the product

(P+ + G)(P1,D -- .~I) - N , where P1,D stands for the Dirichlet realization of a strongly elliptic second-order differential operator P1 with scalar principal symbol near X'. The calculus of [GOI'] is not helpful, since it does not allow general interior parameter-independent factors. We use instead that Qx = (P1 - AI) -1 has a symbol at X ' consisting of rational functions of ~ with a pole i a + ( x ', ~', #) with positive imaginary part and a pole - i a - ( x ' , ~', #) with negative imaginary part, where a+ and a - behave nicely as functions of (~',#), # = (-A)½. This is coupled with Laguerre expansions of the symbols of P and G, which break the composed symbols up in small pieces that can be treated by residue calculus and summed up afterwards. The Laguerre expansions used here are expansions in the following orthogonal basis of 7-/+ (cf. (49)): ^,

(o

-

k

~ k ( ~ . , a ) = (a + i~.)k+l ,

keN,

a=(('),

and the corresponding orthogonal basis of e+S+:

~'k(x,, a) = H ( x n ) ( a - Ox,)k(x~e-~:")/k!, where H(xn) = 1R+. (One might think that the composition of a /z-independent operator with a #-dependent operator could in general be handled by expansion of the former symbol in Laguerre functions ~ ( ~ n , (~')) and the latter symbol in Laguerre functions ~5~:(~, ((~',#))), but this gives severe problems with the convergence of the resulting series.) We show in [GS01] that the normal trace of each piece belongs to the calculus of [GS95], hence defines an operator with a trace expansion as in Corollary 2.10, and moreover, that these trace expansions sum up to a similar trace expansion of

G. Grubb/Nuclear Physics B (Proc. Suppl.) 104 (2002) 71-88 the full operator. Keeping a special check of the first log-coefficient c~), we show that it does indeed satisfy the integral formula of [FGLS96] for res(P+ + G). 4. C o n c l u d i n g r e m a r k s After having recalled the definition and basic properties of pseudodifferential operators (¢do's) on R n, we explained the special problems connected with parameter-dependent ~bdo's such as the resolvent, and we introduced a symbol calculus for A-dependent operators that allows showing complete trace expansions (for operators on compact manifolds) in powers As and logarithmic terms As log A. Heat trace expansions are likewise obtained, as well as the pole structure of zeta functions. An improved calculus allows treatment of complex powers of resolvents too. We have moreover explained a similar theory for operators on manifolds with boundary, in particular Dirac operators with spectral boundary conditions. New results on the vanishing or the stability of the logarithmic terms were presented, and we ended up by showing how the noncommutative residue of a pseudodifferential boundary operator enters as a log-coefficient. REFERENCES

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