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Asymptotics of the heat equation with ‘exotic’ boundary conditions or with time dependent coefficients
UCLEAR PHYSIC~
ELSEVIER
PROCEEDINGS SUPPLEMENTS Nuclear PhysicsB (Proc. Suppl.) 104 (2002) 63-70
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Asymptotics of the heat equation with 'exotic' boundary conditions or with time dependent coefficients Peter B. Gilkey a, Klaus Kirsten b, JeongHyeong Park c and Dmitri Vassilevich d aMathematics Department, University of Oregon, Eugene Or 97403 USA emall:[email protected] bDepartment of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester UK M13 9PL UK email: [email protected] CMathematics Department, Honam University, Kwangju 506-714 South Korea emall: j [email protected] dInstitute for Theoretical Physics, University of Leipzig, Augustusplatz 10, 04109 Leipzig, Germany. email: [email protected] The heat trace asymptotics are discussed for operators of Laplace type with Dirichlet, Robin, spectral, D/N, and transmittal boundary conditions. The heat content asymptotics are discussed for operators with time dependent coefficients and Dirichlet or Robin boundary conditions.
1. I n t r o d u c t i o n Standard Dirichlet and Neumann boundary conditions appear in numerous physical applications, some of which are nicely described at this Conference. In certain cases physics requires consideration of more 'exotic' boundary value problems. For example, divergences of the Casimir energy in non-static, but reasonably slow varying, external fields are related to the asymptotics of the Schr5dinger equation with a time dependent Hamiltonian. After the Wick rotation the latter are defined by the heat trace asymptotics for operators with time dependent coefficients. It is easy to imagine a physical experiment when temperature of a part of the surface of a body is kept constant while the heat flow from the other part to the outside is negligible. Such physical experiments are described by the D / N boundary value problem. Transmittal boundary conditions appear in the case of semi-transparent surfaces or when the geometry of the manifold is not smooth. The most fashionable example (and the closest to the topic of the present Conference) of the nonsmooth geometries is given by the brane world
scenario [33]. Spectral boundary conditions are of relevance in one-loop quantum cosmology and supergravity [15,16]. Furthermore, given their nice transformation properties under chiral rotations and supertranslations there is little doubt that study of spectral boundary conditions is also useful. Let D be an operator of Laplace type acting on the space of smooth sections to a vector bundle V over a compact Riemannian manifold M of dimension m with smooth boundary OM. Let D s be the realization of D with Dirichlet or Robin boundary conditions; we will consider more 'exotic' boundary conditions presently. Let e -tDs be the fundamental solution of the heat equation; u := e - t D s ¢ is determined by the equations: u(x;O) = ¢, B u = O, and (0t + D)u = O. Let f be a smooth localizing or smearing function. We define the smeared heat trace: a(f; D, B)(t) := Tr p ( f e - t D B ) . As t ~ 0 there is an asymptotic series [25,26,34] a ( f ; D , B ) ,~ ~ , > _ o t ( n - m ) / ~ a n ( f , D , B ) .
0920-5632/02/$ - see front matter © 2002 ElsevierScienceB.V. All rights reserved. PII S0920-5632(01)01595-X
(1)
PB. Gilkey et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
64
The asymptotic heat trace coefficients m a y be decomposed as the sum of an interior and a boundary contribution:
an(f,D,B)=aM(f,D)+a
OM n (f,D,B).
The invariants a M and a °M are computable as integrals of local geometric invariants. Let ¢ be an auxiliary section to V defined over the b o u n d a r y OM and let the potential p measure internal heat sources and sinks. Let u be the t e m p e r a t u r e distribution defined by the inhomogeneous equations:
u(x;O) = ¢, Bu = ¢, and (0t + D)u = p . With Dirichlet b o u n d a r y conditions, we keep the b o u n d a r y at constant t e m p e r a t u r e ~; with Neum a n n b o u n d a r y conditions, we p u m p heat into M at a rate defined by ¢ to control the heat flow in the normal direction. Let p be the specific heat of the manifold. The total heat content
~ ( p , ¢ , ¢ , p ; D , B ) ( t ) := f M u P
In Section 5, we discuss spectral boundary conditions. In contrast to Dirichlet, Robin and transmittal b o u n d a r y conditions, spectral boundary conditions are non-local. In T h e o r e m 5.1, we give formulas for the b o u n d a r y correction terms if n < 3 [13,21]. A p a r t from normalizing factors involving powers of 4~r, the formulas of Theorems 2.1, 3.1, and 4.1 involve coefficients which are independent of the dimension m of the underlying manifold. In contrast, the formulae of Theorem 5.1 are very dimension dependent. This is one of the notable features of spectral boundary conditions. In Section 6, we consider a time dependent family l) -- :Dr of operators of Laplace type. The heat t e m p e r a t u r e distribution is defined by:
u(x; O) = ¢, Bu -- 0, and (Or + :Dt)u = O. The m a p ¢ -~ u is described by a smooth kernel function K with the property that:
u(x; t) = fM K(t, x, y, :D, B)¢(y)dy.
has an asymptotic expansion as t ~ 0
~ E~>o fin(p, ¢, ¢, P; D, B)t ~/2. The heat eontent asymptotics ~n can be decomposed as the sum of an interior and a b o u n d a r y contribution given by locally computable invariants. The coefficients an and fin encode spectral information a b o u t the global geometry of the manifold. In Section 2 we discuss the interior invariants a M. These invariants vanish if n is odd. In Theorem 2.1, we give formulas [18] for the invariants an for n = 0, 2, 4; formulas for the invariants a6 [18] and as [1,3] are known. In Section 3, we define the Dirichlet and Robin boundary operator - see equation (2). In Theorem 3.1, we give formulas [9,27,30] for the associated b o u n d a r y correction termsa°nM if n < 4; formulas for a5 are known [11]. In Section 4, we define transmittal b o u n d a r y conditions - see equation (3). In Theorem 4.1, we give formulas [8,23,29] for the boundary correction t e r m s a~ if n < 3; formulas for a4 are known
[23].
The heat trace asymptotics are then defined not by the heat trace but directly in terms of the kernel function:
a(f, :D, B)(t) :-- fM f T r K(t, x, x, :D, B)
~-]nt(~-m)/2a~(f,
:D, B).
In Theorem 6.1 we give formulas for the interior invariants. We define b o u n d a r y conditions in equation (4) which are time dependent. In Theorem 6.2, we give formulas for the boundary correction in the heat equation asymptotics. In Section 7, we give a brief discussion of the D / N problem [4,12,14]. Here, in contrast to other boundary conditions, there is not a classical asymptotic expansion at the a3 level. In Section 8, we discuss the heat content asymptotics. In T h e o r e m 8.1, we give formulae [5,7,20,24] for the invariants /3,~ for n _< 4 for Dirichlet or Robin b o u n d a r y conditions. The coefficients which appear do not depend on the dimension m. In the static setting, partial results are available for j35 [5,6].
65
P.B. Gilkey et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
2. Interior H e a t Trace Coefficients We introduce the following notational conventions to describe the interior heat trace coefficients a M. Let Greek indices p, u range from 1 to m and index a local coordinate frame. Let Latin indices i,j, k,l range from 1 to m and index an o r t h o n o r m a l frame. We adopt the Einstein convention and sum over repeated indices. The operator D determines a connection V and an endomorphism E so t h a t D = - ( T r V 2 + E).
3. H e a t Trace A s y m p t o t i c s for R o b i n and Dirichlet B o u n d a r y C o n d i t i o n s Near the boundary, let R o m a n indices a, b range from 1 to m - 1 and index a local orthonormal frame {e~} for the tangent bundle of OM. We let em be the inward unit normal. Let
Lab := (V~oeb, em) be the second fundamental form. Decompose the b o u n d a r y of M as the disjoint union of two closed (possibly empty) sets:
If we express
OM = C N (3 C D.
D = -(gW'O~O~, + a~O~ + b) relative to a local system of coordinates, then the connection 1 form co and the endomorphism E are given by:
co~ = ½gv~(a~"+ g'~F,~,~'I), and E = b - g~'~'(O~,co, + w~,w, - c o ~ F v j ) . If D = 6d is the scalar Laplacian, then the connection V is trivial and the endomorphism E vanishes. More generally, if D = (d6 + 5d)p is the Laplacian on p forms, then V is the Levi-Civita connection and E is given by the Weitzenb6ck formulas [19]. If D is the spin Laplacian, then V is the spin connection and with our sign convention E = --~T 1 where ~- is the scalar curvature. Let ';' denote multiple covariant differentiation with respect to the connection on V and the LeviCivita connection of M . Let
PO := Rikkj and T : : Pii be the Ricci tensor and the scalar curvature. Let 12 be the curvature of the connection V. If ,4 is a scalar invariant, we let Tr (,4) := Tr (,41). The invariants a M vanish if n is odd. If n is even and if n < 4, then we have [18]:
T h e o r e m 2.1
1. aM(f, D) = (47r) - m / 2 fM Tr (f). 2. aM(f,D) = (47r) - m / 2 1 f u f T r ( T -t- 6E). 3. a M ( f , D ) = (a~r~ 1 - - , - m / 2 596 IM f T r {60E;kk + 6 0 r E + 180E 2 + 3 0 ~ i j f ~ i j -5 12T;kk +5T 2 -- 2Ipl 2 + 21RI2}.
Let u;m be the covariant derivative of u with respect to the inward unit normal using the natural connection defined by D. Let the b o u n d a r y operator B u := ulc ~ ~ (u;m + Su)Ic ~
(2)
define Dirichlet b o u n d a r y conditions on Co and Robin b o u n d a r y conditions on CN. Let ':' denote multiple covariant differentiation of tensors defined on OM with respect to the connection on V and the Levi-Civita connection of the boundary. Note t h a t ';' and ':' differ by the second fundamental form. We have [9,27,30]:
T h e o r e m 3.1
1. agM(f,D,B) = O.
2. aO1M(f,D, B) = -(41r)(1-m)/2¼
f e d Tr (f)
+(47r)(1-m)/2¼ feN Tr (f). 3. aOM(f, D, B) = (47r) - m / 2 1 fc~ Tr {2fLaa - 3 f ; m } + (4~r) - m / 2 1 fcN Tr { 2 f i ~ + 1 2 f S + 3f;m}.
4. aO3U(f, D, B) = -(47r)(1-m)/23~4 fed Tr { f ( 9 6 E + 16T -- 8prom + 7LaaLbb --lOLabLab) -- 30f;mLaa + 24f;mm} +(47r)(1-m)/2 3-~4fcN Tr { f ( 9 6 E + 16T --8prom + 13LaaLbb + 2LabL~,b + 96SLaa + 1 9 2 S 2) + f ; m ( 6 L ~ + 96S) + 24f;mm}.
P.B. Gilkey et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
66
r . aO4M(f,D,B) = (4~r~-m/21--!-/ 360 JCD Tr { f ( - 1 2 0 E ; m + 120ELa. - 18~-;m
normals of E C M i ; v = v + = - ~ , - . We let ~ . := V + - V ~ - and L.~b := :t:(V~aeb, v ).
+ 2 0 T L a a -- 4 p m m Lbb -- 1 2 R ~ m b m L a b
+4R.b~bLa¢ + 24L..:bb + ~ L..LbbL~ SSL.bLabL~ + 32~L.bLb~L.~) + f ; m ( - l S 0 E - 30T - L~L~.Lbb +~L.bL.b) + 24f;mmLa. - 30f;iim} +(47r)-m/23--~0 fc N 23:{f(240E;m + 1 2 0 E L . . + 42T;m + 24L.a:bb + 20TL.. --4pmmLbb -- 12R.mb~L~b + 4R~b~bLa~ + ~ La.LbbL~ + 8L~bL.bL~ +~L.bLb~La~ + 720SE + 120S~+144SLa.Lbb + 48SL.bL.b + 480S2L.. +480S 3 + 120S:..) + f;m(lSOE + 7 2 S L . . + 240S 2 + 307" -~- 12LaaLbb +12LabL~D) + f;mm(120S + 2 4 L . . )
The tensor w. measures the failure of the connections V ± to agree on E; it is a chiral tensor - if we interchange the roles of + and - , then this tensor changes sign. The tensors L + are the second fundamental forms of the inclusions E C M ±. We refer to [23] for the proof of the following theorem; see also related work in [8,29]. T h e o r e m 4.1 1.
aZo(f , D, E) = O.
2. alz (f, D, E) = 0.
+30f#im}. 3. a~(f, D, E) = (47r)-m/2} f z Tr {2f(L+a + L ~ ) - 6rE}.
4. T r a n s m i t t a l boundary conditions Let OM be empty. We suppose given a hypersurface E which divides M into two smooth components M +. We also suppose given operators of Laplace type D ± on M ±. Let v be the inward normal of E C M +. For ¢ = (¢+, ¢ - ) , we define: B e := {¢+1~. - ¢ - I z } -
.
a~a(f, D, E) = (47r)(1-m)/2~4 fE 3 + + + L2aLbb + 2 L + L f b ) Tr {~y(Z.aLbb + + _ + +3f(L~,bLab + L.~,Lab + 2L.bL.b) + 9 ( L + + Lza)(f~+ + f ; ; - ) + 4 8 f E 2 + 24fWaWa - 24f(L+. + L~-a)E -24(/++ + f;;-)~}.
(3) -
Thus ¢ satisfies transmittal boundary conditions if ¢ is continuous and if the normal derivatives of ¢+ match to the normal derivatives of ¢ - modulo the impedance transmission term E. We let D~ be the realization of D = (D+,D -) with these boundary conditions. Let f = ( f + , f - ) be smooth on M ± and continuous on E; we impose no matching condition on the normal derivatives of f . Let
a(f, D, B) = Tr L~(f e-tDB) "~ Y~>_otf~-m)/2a~(f, D, B). We can decompose the invariants an in the form:
a.(f,D,B) = a .M+( f , D ) + a M - ( f , D ) +a~(f, D, B). The invariants a .M=L can be computed using the formulas of Theorem 2.1. Let 9 ± be the inward
5. Spectral boundary conditions Let P : C°°(E1) ~ C~(E2) be an elliptic complex of Dirac type; this means that the associated second order operators P*P and PP* are of Laplace type. Since such an elliptic complex does not in general admit local boundary conditions [2], we impose spectral boundary conditions. Let 7 be the leading symbol of the operator P. Then V + 3"* defines a unitary Clifford module structure on E1 (9 E2. Let V = V1 (9 V2 be a compatible unitary connection [10]. This means that: V(3" + 3'*) = 0, and (Vs, ~) + (s, V~) = d(s, ~). In general this auxiliary connection will not coincide with the connections associated to the Laplacians A1 = P*P and A2 = PP*. We expand P = 3"iV i + ¢
P.B. Gilkey et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
where ~b is a smooth linear map from E~ to E~.. We parallel translate frames for E along the normal geodesic rays defined by the inward unit normal. Relative to such a gauge, we have Vm = Ore. We set x m = 0 to define the tangential operator B on C°°(El[oM): B := ( , m ) - ~ { ~ V .
4. aa(f, D, B) = (4~) -(m-1)/2 fOM f T r {gS( 1 1 -- ~m_2)(¢~b ^^ + ~*¢*)
+ ~ ( s - 2m + 7 - sCm(~m' =) )' V ~ V > m _ = +~(2rn - { - ~ (11
+ ¢}.
3--2m ^ T ^* q- --Gc-~_2 C(m))'ygT ¢')'a ¢
1 (rn-lCimi _
+~g(l
Let B* be the adjoint of B relative to the structures on the boundary and let 0 be an auxiliary self-adjoint operator. We define
~,
/
1)7
4m-l°C'm'" I, ))Prom
ff.~(17+Sm
23-2m-4m2 C(m.l~ rn--2 ~ //
48(m-b 1) ~ L L
4 Jr17+7rn 2
+~(--
+4rn3-11m2+5rn-lC(m) )
B+B* A.---+O 2
m--2
+~c(~)(oo
Let B denote projection on the span of the eigenspaces corresponding to the non-negative spectrum of A. Let P~ be the associated realization of P and let DB := (PB)*PB. Results of Grubb and Seeley [17] show that there is an asymptotic series as t ~ 0 of the form: TrL2{fe -tD" }
Y~0 4 so the terms an for n < 3 are well defined. Let
1 T O 3oTOx~ + a-~-13o J;
...~_Lo,~f.m / 5 m _ - - 7
s,~-gC(m))Tr{i}
+~(2C(m)
- 1)f;mmTr {I}.
S(m-3) t
S
6. T i m e d e p e n d e n t c o e f f i c i e n t s Previously, we have considered static operators. Let l) be an operator of Laplace type where the coefficients are time dependent. We expand Du :-- D U + Y~r>0 t r {gr, iju;ij + .~#u# + E~u} and consider time dependent Dirichlet and Robin boundary conditions:
Bu : =
U]CD (U;m + Su + t(Tau;a + Slu))lc,.
(4)
We consider the heat equation:
"/~--n1¢, and rn
- 3 - 2m2--6mtS('~'/'~rn~m--2 k"~//
•(~yVS~y~ + ~ T ¢^* ~ T ~^* ) 48 ~ m - 2
:=
67
1 --1
c(m):=r(v)r(~)
rn+l
r( ~ )
--1
We refer to [13,21] for the proof of the following theorem: T h e o r e m 5.1 We have
1. ao(f, D, B) = (4~r)-m/2 fM Tr {f}.
u(x; 0) = ¢, Bu = 0, (or + ~ ) u = 0. There is a smooth kernel function K(t, x, y, 7), 13) so that we may express:
u(x; t) =- fM K(t, x, y, 1), B)¢(y). We take the fiber trace to define
2. al (f, D, B)
= (47r)~(m-1)/2¼(C(m) - 1) fOM Tr {fit. 3.
a2(f,D, B) = aM(f,D) -t-(47r)-m/2 fOM f"~ (½[~b+ ~*] ~ ( 1
- lrrC(m))Tr {f;m}.
This agrees with the previous definition i f / ) is static. We refer to [22] for the proof of the following two results which give the additional terms in the asymptotic expansion arising from the time dependent nature of the coefficients:
RB. Gilkey et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
68 T h e o r e m 6.1
1. aM(f, T9) = aM(f,D). 2. aM(f,T )) = aM(/,D)
iM 3. aM(f, T9) = aM(f,D) + (47r)-"/23--~ofM f T r ('-4-~l,ii~l,jj 45 -}- "45~ l , i j ~ l , i j + 60~2,ii --180~Cl -}- 15~l,iiRjkkj -- 30~l,ijRikkj +90Gl,iiE + 609vl,i;i + 15Gl,ii;jj --30~l,ij;ij). Let B0 denote the associated static boundary conditions. We have:
complementary spherical caps about the north and south poles of the ball which intersect in a circle of latitude. The setting where E is not empty is known in the literature as the N / D problem. It has been investigated extensively from the functional analytic point of view [28,31,32,35]. It is natural to conjecture the asymptotic expansion described in (1) could be generalized to this setting by adding an extra integral over ~. of some suitably chosen local invariant. Preliminary computations [4,12] suggest the additional correction term for n -- 2 is given by: =
T h e o r e m 6.2
1. aOM(f, T9,13) = a°M(f,D, Bo) for n < 2. 2. aOM(f , 79, 13) = a03M(f , D, Bo)
-F(47r)(l-m)/2 3--~4fCD fTr (--24~l,aa)
fCN 3. aOM(f,V, 13) = a°M(f,D, Bo)
fc.
{30al,oonbb
--60Gl,mmLbb + 30Gl,abLab + 30GX,mm;,~ --30~l,aa;m + 0~l,~m;~ -- 30~-1,m} +f;mTr {-45G1,~ + 4561,mm} +(47r)-m/23-~0 fen fTr {3061,~Lbb +120Gl,mmLbb -- 150G~,~bL~b --60~l,mm;m q- 60~l,aa;m + 150~'1,m -q-180S~l,aa -- 180S~l,mm q- 360S1} +f;mTr {4561,aa -- 45Gl,mm}. 7. T h e D / N P r o b l e m In Section 3, we assumed that CN fq CD was empty to define the boundary operator B of equation (2). This meant that the Neumann and Dirichlet components did not interact. In this section, we suppose that E := Co n CN is a n o n empty smooth submanifold of OM of dimension m--2. We can motivate this more generalized setting with a physical example. Let M be a solid ball which floats in ice water. The part of the boundary of the ball which is in air satisfies Neumann conditions and the part which is in the water satisfies Dirichlet conditions. Thus B is defined by
(f).
However, it has been shown [14] that the asymptotic expansion does not exist with locally computable coefficients at the 43 level. Thus probably either log terms arise or non-local terms arise; it is also possible, of course, that no asymptotic expansion exists. 8. H e a t C o n t e n t A s y m p t o t i c s Let 79 be a time dependent family of operators of Laplace type. Let ¢ ( y ; t ) be a smooth section to V defined over OM. On the Neumann boundary component C/v, we use a Neumann heat pump to pump heat into M at a rate defined by ¢; in this setting, the parameter S controls the coupling between the heat transfer and the temperature difference on the Neumann component. On the Dirichlet component we use a Dirichlet heat pump to keep the temperature at ¢. Let p be a heat source. The temperature distribution u = Up,¢,¢(x; t) which is defined by these data is the solution to the equations:
(Or + 79)u ----p, u(x; 0) ----¢, and Bu = g:. Let p be the specific heat; we regard p as a section to the dual bundle V* and let (., .) denote the dual pairing between V and V*. The total heat energy content fl is defined by/~(t) := fM up. We expand/3 in an asyptotic series as t I 0 to define the associated heat content asymptotics:
13 ,., ~-~.ntn/2&(p,¢,¢,p;D,B).
PB. Gilkey et aL /Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
Let /) and /~ be the dual operator and dual boundary condition on the dual bundle V*. We summarize results of [5,7,20,24]: 8.1
Theorem 1.
~o(p,¢,¢,p;D,B) = IM{¢,P)"
e. Zdp, ¢, ¢, p; 7~, B) _
2 ¢ ~fco{< -
Co, p>}.
S. fl2(p, ¢, ¢, p; 79, B)
--fM{ - {Po, P>} + f e d {<½L,o(¢ - ~P0),P> -<(¢ - ¢0), p;,,>} =
+fc~
{<(~¢ - ¢0), p>}.
4. Z3(p, ¢, ¢, p;7~,B) _ 2 - - - ~2f c ~ { ~ - ~ < D e , p) -~<(¢ - ¢0), bp> + ½<(¢ - ¢0):o, p:o> <¢,, p) + <(-½E + ~LooLbb 1L ab L ab + 1R -~ 6 amain -- ~l,rnrn)
+~-~z~Ic~
5. Z4(p, ¢, ~, p; v , B) = ½fM{(Pl, P) -- (Dpo, p) + {De, Dp}
-<(~1,o¢#j + ~5,~¢# + E1¢), p>} + f e d { ~1Loo - } - ~1L aa(¢a,p) + ½(%bl,p;m> +½{(D¢);m,p) + ½{(¢ - ¢0), (/gp);m) 1 -¼
-} +~<~om(¢ - ¢0), p:o)
--~-K~'l,m) <(¢ -- I/J0) , fl> --~6 ~l,am<({ b -- ¢0):a, P> " ~}
+{(½S + ¼L~a)(B¢ - ¢o),/~p) ½~l,mm((S~b - - 1/30), P > } .
We have presented explicit combinatorial formulas for both the heat content and the heat trace asymptotics. One of our motivations in computing these invariants was to see if there was a d i r e c t combinatorial link between the invariants; there does not seem to be one immediately evident although techniques used in the computation of both the heat content and the heat trace asymptotics share certain common features and in principle there are methods which might permit both to be computed simultaneously. Another example of an asymptotic formula involving geometric data arises from expanding the volume of a tube of radius r around a submanifold N embedded in an ambient manifold, see for example [36]. Again, there does not seem to be any direct combinatorial link between these asymptotic formulae and those we have presented here. Acknowledgements
It is a pleasant task to thank J.S. Dowker for helpful comments regarding this paper. The research of P. Gilkey was partially supported by the NSF (USA) and the MPI (Leipzig, Germany). The research of K. Kirsten was partially supported by the MPI (Leipzig, Germany) and by EPSRC under Grant No. GR/M45726. The research of JH. Park was partially supported by Korea Research Foundation Grant (KRF-2000-015DS0003). The research of D. Vassilevich was partially supported by the DFG project Bo 1112/111 (Germany) and by the ESI (Austria). REFERENCES
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69
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P.B. Gilkeyet aL/Nuclear PhysicsB (Proc. Suppl.) 104 (2002) 63-70 boundary value problem Workshop on Spectral Geometry, Bristol 2000. M. van den Berg, S. Desjardins and P.B. Gilkey, in: Topological Methods in Nonlinear Analysis, 2 (1993) 147-162. M. van den Berg and P.B. Gilkey, Commun. Analysis and Geometry 7 (1999) 279-294. --, Proc. of Royal Society Edinburgh 130A (2000) 307-312. M. Bordag and D.V. Vassilevich, J. Phys. A32 (1999) 8247 [hep-th/9907076]. T.P. Branson and P.B. Gilkey, Commun. Part. Diff. Eq. 15 (1990) 245-272. --, J. Funct. Anal. 108 (1992) 47-87. T.P. Branson, P.B. Gilkey, K. Kirsten and D.V. Vassilevich, Nucl. Phys. B563 (1999) 603-626. J.S. Dowker, The N U D problem, preprint (hep-th/0007127). J.S. Dowker, P.B. Gilkey and K.Kirsten, Contemporary Math. 242 (1999) 107-124. --, On properties of the asymptotic expansion of the heat trace for the N/D problem, to appear International Journal of Mathematics. P.D. D'Eath and G. Esposito, Phys. Rev. D44 (1991) 1713. G. Esposito and A.Yu. Kamenshchik, Phys. Rev. D54 (1996) 3869. G.Grubb and R.T. Seeley, Invent. Math. 121 (1995) 481-529; see also J. Geom. Anal. 6 (1996) 31-77. P.B. Gilkey, J. Diff. Geom. 10 (1975) 601 618. P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem (2nd edition), CRC Press, Boca Raton, 1995. --,J. Phys., A32 (1999) 2825-2834. P.B. Gilkey and K. Kirsten, Heat asymptotics with spectral boundary conditions II, preprint. P.B. Gilkey, K. Kirsten and JH. Park, Heat Trace Asymptotics of a time dependent process, J. Phys. A34 (2001) 1153-1168. P.B. Gilkey, K. Kirsten and D.V. Vassilevich, Heat trace asymptotics with transmittal boundary conditions and quantum braneworld scenario, to appear in Nuclear Physics B, hep-th/0101105.
24. P.B. Gilkey and JH. Park, Mod. Phys. Lett. A15 (2000) 1165-1179. 25. P. Greiner, Global Analysis, Amer. Math. Soc. Proc. Symp. Pure Math. 16 (1970) 133137. 26. G. Grubb, Functional Calculus of Pseudo Differential Boundary Problems, Progress in Math. 65, Birkh£user, Boston, 1986. 27. G. Kennedy , R. Critchley, and J.S. Dowker, Ann. Phys. 125 (1980) 346-400. 28. J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value problems and Applications, Die Grundlehren, vol. 181 SpringerVerlag, New York, 1972. 29. I.G. Moss, Phys. Lett. B491 (2000) 203-206 [hep-th/0007185]. 30. I.G. Moss and J.S. Dowker, Phys. Lett. B229 (1989) 261-263. 31. J. Peetre, Ann. Seuola. Norm. Sup. Pisa. 15 (1963) 337-353. 32. A.J. Pryde, J. Math. Anal. Appl. 80 (1981) 203-244. 33. L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690; Phys. Rev. Lett. 83 (1999) 3370. 34. R.T. Seeley, Amer. J. Math. 91 (1969) 963983. 35. S.R. Simanca, Ill. J. Math. 32 (1988) 99-114. 36. A. Gray, Tubes, Addison-Wesley Publishing Company, 1990 [ISBN 0-201-15676-8].
ELSEVIER
PROCEEDINGS SUPPLEMENTS Nuclear PhysicsB (Proc. Suppl.) 104 (2002) 63-70
www.elsevier,com/locate/npe
Asymptotics of the heat equation with 'exotic' boundary conditions or with time dependent coefficients Peter B. Gilkey a, Klaus Kirsten b, JeongHyeong Park c and Dmitri Vassilevich d aMathematics Department, University of Oregon, Eugene Or 97403 USA emall:[email protected] bDepartment of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester UK M13 9PL UK email: [email protected] CMathematics Department, Honam University, Kwangju 506-714 South Korea emall: j [email protected] dInstitute for Theoretical Physics, University of Leipzig, Augustusplatz 10, 04109 Leipzig, Germany. email: [email protected] The heat trace asymptotics are discussed for operators of Laplace type with Dirichlet, Robin, spectral, D/N, and transmittal boundary conditions. The heat content asymptotics are discussed for operators with time dependent coefficients and Dirichlet or Robin boundary conditions.
1. I n t r o d u c t i o n Standard Dirichlet and Neumann boundary conditions appear in numerous physical applications, some of which are nicely described at this Conference. In certain cases physics requires consideration of more 'exotic' boundary value problems. For example, divergences of the Casimir energy in non-static, but reasonably slow varying, external fields are related to the asymptotics of the Schr5dinger equation with a time dependent Hamiltonian. After the Wick rotation the latter are defined by the heat trace asymptotics for operators with time dependent coefficients. It is easy to imagine a physical experiment when temperature of a part of the surface of a body is kept constant while the heat flow from the other part to the outside is negligible. Such physical experiments are described by the D / N boundary value problem. Transmittal boundary conditions appear in the case of semi-transparent surfaces or when the geometry of the manifold is not smooth. The most fashionable example (and the closest to the topic of the present Conference) of the nonsmooth geometries is given by the brane world
scenario [33]. Spectral boundary conditions are of relevance in one-loop quantum cosmology and supergravity [15,16]. Furthermore, given their nice transformation properties under chiral rotations and supertranslations there is little doubt that study of spectral boundary conditions is also useful. Let D be an operator of Laplace type acting on the space of smooth sections to a vector bundle V over a compact Riemannian manifold M of dimension m with smooth boundary OM. Let D s be the realization of D with Dirichlet or Robin boundary conditions; we will consider more 'exotic' boundary conditions presently. Let e -tDs be the fundamental solution of the heat equation; u := e - t D s ¢ is determined by the equations: u(x;O) = ¢, B u = O, and (0t + D)u = O. Let f be a smooth localizing or smearing function. We define the smeared heat trace: a(f; D, B)(t) := Tr p ( f e - t D B ) . As t ~ 0 there is an asymptotic series [25,26,34] a ( f ; D , B ) ,~ ~ , > _ o t ( n - m ) / ~ a n ( f , D , B ) .
0920-5632/02/$ - see front matter © 2002 ElsevierScienceB.V. All rights reserved. PII S0920-5632(01)01595-X
(1)
PB. Gilkey et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
64
The asymptotic heat trace coefficients m a y be decomposed as the sum of an interior and a boundary contribution:
an(f,D,B)=aM(f,D)+a
OM n (f,D,B).
The invariants a M and a °M are computable as integrals of local geometric invariants. Let ¢ be an auxiliary section to V defined over the b o u n d a r y OM and let the potential p measure internal heat sources and sinks. Let u be the t e m p e r a t u r e distribution defined by the inhomogeneous equations:
u(x;O) = ¢, Bu = ¢, and (0t + D)u = p . With Dirichlet b o u n d a r y conditions, we keep the b o u n d a r y at constant t e m p e r a t u r e ~; with Neum a n n b o u n d a r y conditions, we p u m p heat into M at a rate defined by ¢ to control the heat flow in the normal direction. Let p be the specific heat of the manifold. The total heat content
~ ( p , ¢ , ¢ , p ; D , B ) ( t ) := f M u P
In Section 5, we discuss spectral boundary conditions. In contrast to Dirichlet, Robin and transmittal b o u n d a r y conditions, spectral boundary conditions are non-local. In T h e o r e m 5.1, we give formulas for the b o u n d a r y correction terms if n < 3 [13,21]. A p a r t from normalizing factors involving powers of 4~r, the formulas of Theorems 2.1, 3.1, and 4.1 involve coefficients which are independent of the dimension m of the underlying manifold. In contrast, the formulae of Theorem 5.1 are very dimension dependent. This is one of the notable features of spectral boundary conditions. In Section 6, we consider a time dependent family l) -- :Dr of operators of Laplace type. The heat t e m p e r a t u r e distribution is defined by:
u(x; O) = ¢, Bu -- 0, and (Or + :Dt)u = O. The m a p ¢ -~ u is described by a smooth kernel function K with the property that:
u(x; t) = fM K(t, x, y, :D, B)¢(y)dy.
has an asymptotic expansion as t ~ 0
~ E~>o fin(p, ¢, ¢, P; D, B)t ~/2. The heat eontent asymptotics ~n can be decomposed as the sum of an interior and a b o u n d a r y contribution given by locally computable invariants. The coefficients an and fin encode spectral information a b o u t the global geometry of the manifold. In Section 2 we discuss the interior invariants a M. These invariants vanish if n is odd. In Theorem 2.1, we give formulas [18] for the invariants an for n = 0, 2, 4; formulas for the invariants a6 [18] and as [1,3] are known. In Section 3, we define the Dirichlet and Robin boundary operator - see equation (2). In Theorem 3.1, we give formulas [9,27,30] for the associated b o u n d a r y correction termsa°nM if n < 4; formulas for a5 are known [11]. In Section 4, we define transmittal b o u n d a r y conditions - see equation (3). In Theorem 4.1, we give formulas [8,23,29] for the boundary correction t e r m s a~ if n < 3; formulas for a4 are known
[23].
The heat trace asymptotics are then defined not by the heat trace but directly in terms of the kernel function:
a(f, :D, B)(t) :-- fM f T r K(t, x, x, :D, B)
~-]nt(~-m)/2a~(f,
:D, B).
In Theorem 6.1 we give formulas for the interior invariants. We define b o u n d a r y conditions in equation (4) which are time dependent. In Theorem 6.2, we give formulas for the boundary correction in the heat equation asymptotics. In Section 7, we give a brief discussion of the D / N problem [4,12,14]. Here, in contrast to other boundary conditions, there is not a classical asymptotic expansion at the a3 level. In Section 8, we discuss the heat content asymptotics. In T h e o r e m 8.1, we give formulae [5,7,20,24] for the invariants /3,~ for n _< 4 for Dirichlet or Robin b o u n d a r y conditions. The coefficients which appear do not depend on the dimension m. In the static setting, partial results are available for j35 [5,6].
65
P.B. Gilkey et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
2. Interior H e a t Trace Coefficients We introduce the following notational conventions to describe the interior heat trace coefficients a M. Let Greek indices p, u range from 1 to m and index a local coordinate frame. Let Latin indices i,j, k,l range from 1 to m and index an o r t h o n o r m a l frame. We adopt the Einstein convention and sum over repeated indices. The operator D determines a connection V and an endomorphism E so t h a t D = - ( T r V 2 + E).
3. H e a t Trace A s y m p t o t i c s for R o b i n and Dirichlet B o u n d a r y C o n d i t i o n s Near the boundary, let R o m a n indices a, b range from 1 to m - 1 and index a local orthonormal frame {e~} for the tangent bundle of OM. We let em be the inward unit normal. Let
Lab := (V~oeb, em) be the second fundamental form. Decompose the b o u n d a r y of M as the disjoint union of two closed (possibly empty) sets:
If we express
OM = C N (3 C D.
D = -(gW'O~O~, + a~O~ + b) relative to a local system of coordinates, then the connection 1 form co and the endomorphism E are given by:
co~ = ½gv~(a~"+ g'~F,~,~'I), and E = b - g~'~'(O~,co, + w~,w, - c o ~ F v j ) . If D = 6d is the scalar Laplacian, then the connection V is trivial and the endomorphism E vanishes. More generally, if D = (d6 + 5d)p is the Laplacian on p forms, then V is the Levi-Civita connection and E is given by the Weitzenb6ck formulas [19]. If D is the spin Laplacian, then V is the spin connection and with our sign convention E = --~T 1 where ~- is the scalar curvature. Let ';' denote multiple covariant differentiation with respect to the connection on V and the LeviCivita connection of M . Let
PO := Rikkj and T : : Pii be the Ricci tensor and the scalar curvature. Let 12 be the curvature of the connection V. If ,4 is a scalar invariant, we let Tr (,4) := Tr (,41). The invariants a M vanish if n is odd. If n is even and if n < 4, then we have [18]:
T h e o r e m 2.1
1. aM(f, D) = (47r) - m / 2 fM Tr (f). 2. aM(f,D) = (47r) - m / 2 1 f u f T r ( T -t- 6E). 3. a M ( f , D ) = (a~r~ 1 - - , - m / 2 596 IM f T r {60E;kk + 6 0 r E + 180E 2 + 3 0 ~ i j f ~ i j -5 12T;kk +5T 2 -- 2Ipl 2 + 21RI2}.
Let u;m be the covariant derivative of u with respect to the inward unit normal using the natural connection defined by D. Let the b o u n d a r y operator B u := ulc ~ ~ (u;m + Su)Ic ~
(2)
define Dirichlet b o u n d a r y conditions on Co and Robin b o u n d a r y conditions on CN. Let ':' denote multiple covariant differentiation of tensors defined on OM with respect to the connection on V and the Levi-Civita connection of the boundary. Note t h a t ';' and ':' differ by the second fundamental form. We have [9,27,30]:
T h e o r e m 3.1
1. agM(f,D,B) = O.
2. aO1M(f,D, B) = -(41r)(1-m)/2¼
f e d Tr (f)
+(47r)(1-m)/2¼ feN Tr (f). 3. aOM(f, D, B) = (47r) - m / 2 1 fc~ Tr {2fLaa - 3 f ; m } + (4~r) - m / 2 1 fcN Tr { 2 f i ~ + 1 2 f S + 3f;m}.
4. aO3U(f, D, B) = -(47r)(1-m)/23~4 fed Tr { f ( 9 6 E + 16T -- 8prom + 7LaaLbb --lOLabLab) -- 30f;mLaa + 24f;mm} +(47r)(1-m)/2 3-~4fcN Tr { f ( 9 6 E + 16T --8prom + 13LaaLbb + 2LabL~,b + 96SLaa + 1 9 2 S 2) + f ; m ( 6 L ~ + 96S) + 24f;mm}.
P.B. Gilkey et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
66
r . aO4M(f,D,B) = (4~r~-m/21--!-/ 360 JCD Tr { f ( - 1 2 0 E ; m + 120ELa. - 18~-;m
normals of E C M i ; v = v + = - ~ , - . We let ~ . := V + - V ~ - and L.~b := :t:(V~aeb, v ).
+ 2 0 T L a a -- 4 p m m Lbb -- 1 2 R ~ m b m L a b
+4R.b~bLa¢ + 24L..:bb + ~ L..LbbL~ SSL.bLabL~ + 32~L.bLb~L.~) + f ; m ( - l S 0 E - 30T - L~L~.Lbb +~L.bL.b) + 24f;mmLa. - 30f;iim} +(47r)-m/23--~0 fc N 23:{f(240E;m + 1 2 0 E L . . + 42T;m + 24L.a:bb + 20TL.. --4pmmLbb -- 12R.mb~L~b + 4R~b~bLa~ + ~ La.LbbL~ + 8L~bL.bL~ +~L.bLb~La~ + 720SE + 120S~+144SLa.Lbb + 48SL.bL.b + 480S2L.. +480S 3 + 120S:..) + f;m(lSOE + 7 2 S L . . + 240S 2 + 307" -~- 12LaaLbb +12LabL~D) + f;mm(120S + 2 4 L . . )
The tensor w. measures the failure of the connections V ± to agree on E; it is a chiral tensor - if we interchange the roles of + and - , then this tensor changes sign. The tensors L + are the second fundamental forms of the inclusions E C M ±. We refer to [23] for the proof of the following theorem; see also related work in [8,29]. T h e o r e m 4.1 1.
aZo(f , D, E) = O.
2. alz (f, D, E) = 0.
+30f#im}. 3. a~(f, D, E) = (47r)-m/2} f z Tr {2f(L+a + L ~ ) - 6rE}.
4. T r a n s m i t t a l boundary conditions Let OM be empty. We suppose given a hypersurface E which divides M into two smooth components M +. We also suppose given operators of Laplace type D ± on M ±. Let v be the inward normal of E C M +. For ¢ = (¢+, ¢ - ) , we define: B e := {¢+1~. - ¢ - I z } -
.
a~a(f, D, E) = (47r)(1-m)/2~4 fE 3 + + + L2aLbb + 2 L + L f b ) Tr {~y(Z.aLbb + + _ + +3f(L~,bLab + L.~,Lab + 2L.bL.b) + 9 ( L + + Lza)(f~+ + f ; ; - ) + 4 8 f E 2 + 24fWaWa - 24f(L+. + L~-a)E -24(/++ + f;;-)~}.
(3) -
Thus ¢ satisfies transmittal boundary conditions if ¢ is continuous and if the normal derivatives of ¢+ match to the normal derivatives of ¢ - modulo the impedance transmission term E. We let D~ be the realization of D = (D+,D -) with these boundary conditions. Let f = ( f + , f - ) be smooth on M ± and continuous on E; we impose no matching condition on the normal derivatives of f . Let
a(f, D, B) = Tr L~(f e-tDB) "~ Y~>_otf~-m)/2a~(f, D, B). We can decompose the invariants an in the form:
a.(f,D,B) = a .M+( f , D ) + a M - ( f , D ) +a~(f, D, B). The invariants a .M=L can be computed using the formulas of Theorem 2.1. Let 9 ± be the inward
5. Spectral boundary conditions Let P : C°°(E1) ~ C~(E2) be an elliptic complex of Dirac type; this means that the associated second order operators P*P and PP* are of Laplace type. Since such an elliptic complex does not in general admit local boundary conditions [2], we impose spectral boundary conditions. Let 7 be the leading symbol of the operator P. Then V + 3"* defines a unitary Clifford module structure on E1 (9 E2. Let V = V1 (9 V2 be a compatible unitary connection [10]. This means that: V(3" + 3'*) = 0, and (Vs, ~) + (s, V~) = d(s, ~). In general this auxiliary connection will not coincide with the connections associated to the Laplacians A1 = P*P and A2 = PP*. We expand P = 3"iV i + ¢
P.B. Gilkey et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
where ~b is a smooth linear map from E~ to E~.. We parallel translate frames for E along the normal geodesic rays defined by the inward unit normal. Relative to such a gauge, we have Vm = Ore. We set x m = 0 to define the tangential operator B on C°°(El[oM): B := ( , m ) - ~ { ~ V .
4. aa(f, D, B) = (4~) -(m-1)/2 fOM f T r {gS( 1 1 -- ~m_2)(¢~b ^^ + ~*¢*)
+ ~ ( s - 2m + 7 - sCm(~m' =) )' V ~ V > m _ = +~(2rn - { - ~ (11
+ ¢}.
3--2m ^ T ^* q- --Gc-~_2 C(m))'ygT ¢')'a ¢
1 (rn-lCimi _
+~g(l
Let B* be the adjoint of B relative to the structures on the boundary and let 0 be an auxiliary self-adjoint operator. We define
~,
/
1)7
4m-l°C'm'" I, ))Prom
ff.~(17+Sm
23-2m-4m2 C(m.l~ rn--2 ~ //
48(m-b 1) ~ L L
4 Jr17+7rn 2
+~(--
+4rn3-11m2+5rn-lC(m) )
B+B* A.---+O 2
m--2
+~c(~)(oo
Let B denote projection on the span of the eigenspaces corresponding to the non-negative spectrum of A. Let P~ be the associated realization of P and let DB := (PB)*PB. Results of Grubb and Seeley [17] show that there is an asymptotic series as t ~ 0 of the form: TrL2{fe -tD" }
Y~0
1 T O 3oTOx~ + a-~-13o J;
...~_Lo,~f.m / 5 m _ - - 7
s,~-gC(m))Tr{i}
+~(2C(m)
- 1)f;mmTr {I}.
S(m-3) t
S
6. T i m e d e p e n d e n t c o e f f i c i e n t s Previously, we have considered static operators. Let l) be an operator of Laplace type where the coefficients are time dependent. We expand Du :-- D U + Y~r>0 t r {gr, iju;ij + .~#u# + E~u} and consider time dependent Dirichlet and Robin boundary conditions:
Bu : =
U]CD (U;m + Su + t(Tau;a + Slu))lc,.
(4)
We consider the heat equation:
"/~--n1¢, and rn
- 3 - 2m2--6mtS('~'/'~rn~m--2 k"~//
•(~yVS~y~ + ~ T ¢^* ~ T ~^* ) 48 ~ m - 2
:=
67
1 --1
c(m):=r(v)r(~)
rn+l
r( ~ )
--1
We refer to [13,21] for the proof of the following theorem: T h e o r e m 5.1 We have
1. ao(f, D, B) = (4~r)-m/2 fM Tr {f}.
u(x; 0) = ¢, Bu = 0, (or + ~ ) u = 0. There is a smooth kernel function K(t, x, y, 7), 13) so that we may express:
u(x; t) =- fM K(t, x, y, 1), B)¢(y). We take the fiber trace to define
2. al (f, D, B)
= (47r)~(m-1)/2¼(C(m) - 1) fOM Tr {fit. 3.
a2(f,D, B) = aM(f,D) -t-(47r)-m/2 fOM f"~ (½[~b+ ~*] ~ ( 1
- lrrC(m))Tr {f;m}.
This agrees with the previous definition i f / ) is static. We refer to [22] for the proof of the following two results which give the additional terms in the asymptotic expansion arising from the time dependent nature of the coefficients:
RB. Gilkey et al./Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
68 T h e o r e m 6.1
1. aM(f, T9) = aM(f,D). 2. aM(f,T )) = aM(/,D)
iM 3. aM(f, T9) = aM(f,D) + (47r)-"/23--~ofM f T r ('-4-~l,ii~l,jj 45 -}- "45~ l , i j ~ l , i j + 60~2,ii --180~Cl -}- 15~l,iiRjkkj -- 30~l,ijRikkj +90Gl,iiE + 609vl,i;i + 15Gl,ii;jj --30~l,ij;ij). Let B0 denote the associated static boundary conditions. We have:
complementary spherical caps about the north and south poles of the ball which intersect in a circle of latitude. The setting where E is not empty is known in the literature as the N / D problem. It has been investigated extensively from the functional analytic point of view [28,31,32,35]. It is natural to conjecture the asymptotic expansion described in (1) could be generalized to this setting by adding an extra integral over ~. of some suitably chosen local invariant. Preliminary computations [4,12] suggest the additional correction term for n -- 2 is given by: =
T h e o r e m 6.2
1. aOM(f, T9,13) = a°M(f,D, Bo) for n < 2. 2. aOM(f , 79, 13) = a03M(f , D, Bo)
-F(47r)(l-m)/2 3--~4fCD fTr (--24~l,aa)
fCN 3. aOM(f,V, 13) = a°M(f,D, Bo)
fc.
{30al,oonbb
--60Gl,mmLbb + 30Gl,abLab + 30GX,mm;,~ --30~l,aa;m + 0~l,~m;~ -- 30~-1,m} +f;mTr {-45G1,~ + 4561,mm} +(47r)-m/23-~0 fen fTr {3061,~Lbb +120Gl,mmLbb -- 150G~,~bL~b --60~l,mm;m q- 60~l,aa;m + 150~'1,m -q-180S~l,aa -- 180S~l,mm q- 360S1} +f;mTr {4561,aa -- 45Gl,mm}. 7. T h e D / N P r o b l e m In Section 3, we assumed that CN fq CD was empty to define the boundary operator B of equation (2). This meant that the Neumann and Dirichlet components did not interact. In this section, we suppose that E := Co n CN is a n o n empty smooth submanifold of OM of dimension m--2. We can motivate this more generalized setting with a physical example. Let M be a solid ball which floats in ice water. The part of the boundary of the ball which is in air satisfies Neumann conditions and the part which is in the water satisfies Dirichlet conditions. Thus B is defined by
(f).
However, it has been shown [14] that the asymptotic expansion does not exist with locally computable coefficients at the 43 level. Thus probably either log terms arise or non-local terms arise; it is also possible, of course, that no asymptotic expansion exists. 8. H e a t C o n t e n t A s y m p t o t i c s Let 79 be a time dependent family of operators of Laplace type. Let ¢ ( y ; t ) be a smooth section to V defined over OM. On the Neumann boundary component C/v, we use a Neumann heat pump to pump heat into M at a rate defined by ¢; in this setting, the parameter S controls the coupling between the heat transfer and the temperature difference on the Neumann component. On the Dirichlet component we use a Dirichlet heat pump to keep the temperature at ¢. Let p be a heat source. The temperature distribution u = Up,¢,¢(x; t) which is defined by these data is the solution to the equations:
(Or + 79)u ----p, u(x; 0) ----¢, and Bu = g:. Let p be the specific heat; we regard p as a section to the dual bundle V* and let (., .) denote the dual pairing between V and V*. The total heat energy content fl is defined by/~(t) := fM up. We expand/3 in an asyptotic series as t I 0 to define the associated heat content asymptotics:
13 ,., ~-~.ntn/2&(p,¢,¢,p;D,B).
PB. Gilkey et aL /Nuclear Physics B (Proc. Suppl.) 104 (2002) 63-70
Let /) and /~ be the dual operator and dual boundary condition on the dual bundle V*. We summarize results of [5,7,20,24]: 8.1
Theorem 1.
~o(p,¢,¢,p;D,B) = IM{¢,P)"
e. Zdp, ¢, ¢, p; 7~, B) _
2 ¢ ~fco{< -
Co, p>}.
S. fl2(p, ¢, ¢, p; 79, B)
--fM{
+fc~
{<(~¢ - ¢0), p>}.
4. Z3(p, ¢, ¢, p;7~,B) _ 2 - - - ~2f c ~ { ~
+~-~z~Ic~
5. Z4(p, ¢, ~, p; v , B) = ½fM{(Pl, P) -- (Dpo, p) + {De, Dp}
-<(~1,o¢#j + ~5,~¢# + E1¢), p>} + f e d { ~1Loo
-}
--~-K~'l,m) <(¢ -- I/J0) , fl> --~6 ~l,am<({ b -- ¢0):a, P> " ~
+{(½S + ¼L~a)(B¢ - ¢o),/~p) ½~l,mm((S~b - - 1/30), P > } .
We have presented explicit combinatorial formulas for both the heat content and the heat trace asymptotics. One of our motivations in computing these invariants was to see if there was a d i r e c t combinatorial link between the invariants; there does not seem to be one immediately evident although techniques used in the computation of both the heat content and the heat trace asymptotics share certain common features and in principle there are methods which might permit both to be computed simultaneously. Another example of an asymptotic formula involving geometric data arises from expanding the volume of a tube of radius r around a submanifold N embedded in an ambient manifold, see for example [36]. Again, there does not seem to be any direct combinatorial link between these asymptotic formulae and those we have presented here. Acknowledgements
It is a pleasant task to thank J.S. Dowker for helpful comments regarding this paper. The research of P. Gilkey was partially supported by the NSF (USA) and the MPI (Leipzig, Germany). The research of K. Kirsten was partially supported by the MPI (Leipzig, Germany) and by EPSRC under Grant No. GR/M45726. The research of JH. Park was partially supported by Korea Research Foundation Grant (KRF-2000-015DS0003). The research of D. Vassilevich was partially supported by the DFG project Bo 1112/111 (Germany) and by the ESI (Austria). REFERENCES
7 "~-(~-~l,mm;m -- "1~ l , m m L aa
+fc~{ ½
9. R e m a r k s
10.
•(¢ - ¢o), p>} {<(Be - ¢o), ~p>}.
69
½ (¢1, p)
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