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On hearing the shape of a general multi-connected vibrating membrane in R2 with piecewise smooth positive functions in the Robin boundary conditions
Applied Mathematics and Computation 135 (2003) 361–375 www.elsevier.com/locate/amc
On hearing the shape of a general multi-connected vibrating membrane in 2 R with piecewise smooth positive functions in the Robin boundary conditions E.M.E. Zayed Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
Abstract P The asymptotic expansion of the trace of the heat kernel HðtÞ ¼ 1 J ¼1 expðtkJ Þ as t ! 0þ has been derived for a variety of domains, where fkJ g are the eigenvalues of the negative Laplace operator 2 2 X o D ¼ oxi i¼1 in the ðx1 ; x2 Þ-plane. The dependence of HðtÞ on the connectivity of domains and the boundary conditions are analyzed. Particular attention is given for a general multiply connected bounded domain in R2 together with a finite number of piecewise Robin boundary conditions, where the coefficients in the boundary conditions are piecewise smooth positive functions. Some applications of an ideal gas enclosed in the general multiply connected domain are given. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Inverse problem; Heat kernel; Eigenvalues; Hearing the shape of multi-connected domain; Classical ideal gas
E-mail address: [email protected] (E.M.E. Zayed). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 3 3 7 - X
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1. Introduction Let X be a given arbitrary simply connected bounded domain in R2 with a smooth boundary oX. Consider the Robin problem o Du ¼ ku in X; þ c u ¼ 0 on oX; ð1:1Þ on where o=on denotes the differentiation along the outward pointing normal to Þ. oX, where the coefficient c is a positive smooth function and u 2 C 2 ðXÞ \ CðX Denote its eigenvalues counted according to multiplicity by 0 < k1 6 k2 6 k3 6 6 kJ 6 ! 1
as J ! 1:
ð1:2Þ
The basic problem is that of determining some geometric quantities associated with the bounded domain X from complete knowledge of the eigenvalues (1.2) using the asymptotic expansion of the trace of the heat kernel HðtÞ ¼
1 X
expðtkJ Þ
as t ! 0þ :
ð1:3Þ
J ¼1
The Robin problem (1.1) has been investigated by many authors (see, for example, [2,5–7,10,16]) in the following special cases: (i) Case 1. c ¼ 0 (the Neumann problem) Z jXj joXj 7 t 1=2 HðtÞ ¼ þ þ a þ K 2 ðzÞ dz þ OðtÞ as t ! 0þ : 0 4pt 8ðptÞ1=2 256 p oX ð1:4Þ (ii) Case 2. c ! 1 (the Dirichlet problem) jXj joXj 1 t 1=2 þ a þ HðtÞ ¼ 0 4pt 8ðptÞ1=2 256 p
Z
K 2 ðzÞ dz þ OðtÞ as t ! 0þ :
oX
ð1:5Þ In these formulae, jXj is the area of X, joXj is the total length of oX and KðzÞ is the curvature of oX, where z is the arc length of the counterclockwise-oriented boundary oX. The constant term a0 has geometric significance, e.g., if X is smooth and convex, then a0 ¼ 1=6 and if X is permitted to have a finite number of smooth convex holes ‘‘H’’, then a0 ¼ ð1 H Þ=6: We merely note that aspects of the question of Kac [6], namely, can one hear the shape of a drum? have been discussed by Sleeman and Zayed [12] for the Robin problem (1.1) when c is a positive constant, and by Zayed [22] when c is a positive smooth function. Further, the Robin problem (1.1) has been investigated by Hsu [5] in the general situation where X is a compact Riemannian
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363
manifold of n-dimensions with smooth boundary, and has determined the first four terms of HðtÞ as t ! 0þ . Thus, in the mathematical terms, Kac’s question becomes: Does the boundary condition define the spectrum uniquely? The proof of Gordon et al. [3] uses drums made by piecing together with a few (identical) basic shapes, for example, triangles. Gordon et al. [3] have proved that different shaped drums can posses identical spectra, that is they are ‘‘isospectral’’. The examples of [1,17] also show that one cannot always hear the shape of a domain in Rn for n P 3. Also, Milnor [8] has constructed two non-congruent 16-dimensional tori whose Laplace Beltrami operators have precisely the same eigenvalues. The object of this paper is to discuss the following more general inverse problem: Let X be a general multiply connected bounded domain in R2 surrounding internally by simply connected bounded domains Xj with smooth boundaries oXj ðj ¼ 1; . . . ; n 1Þ and externally by a simply connected bounded domain Xn with a smooth boundary oXn . Suppose that the eigenvalues (1.2) are given for the Helmholtz equation Du ¼ ku
ð1:6Þ
in X
together with the piecewise smooth Robin boundary conditions
o þ ci u ¼ 0 oni
on Ci ;
ð1:7Þ
where the boundaries oXj ðj ¼ 1; . . . ; nÞ consist of a finite number Skj of piecewise smooth components Ci ði ¼ 1 þ kj1 ; . . . ; kj Þ such that oXj ¼ i¼1þk Ci where j1 k0 ¼ 0 and ci are piecewise smooth positive functions defined on Ci , while o=oni denote differentiations along the outward normals to Ci . The basic problem is to determine some geometric quantities associated with the general multiply connected bounded domain X from complete knowledge of its eigenvalues (1.2) using the asymptotic expansion of HðtÞ for small positive t. Note that the problem (1.6), (1.7) has been discussed recently by Zayed [25] in the case ci are piecewise smooth positive constants.
2. Statement of results Theorem 2.1. Suppose that the boundaries oXj ðj ¼ 1; . . . ; nÞ of the general multiply connected domain X are given locally by the equations xa ¼ y a ðzj Þ ðj ¼ 1; . . . ; nÞ and a ¼ 1; 2 in which zj are the arc lengths of the counterclockwise-oriented boundaries oXj and y a ðzj Þ 2 C 1 ðoXj Þ. Let Li be the lengths of the components Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ 1; . . . ; r; r þ 1; . . . ; nÞ, respectively. Let Kj ðzj Þ be the curvatures of the boundaries oXj , respectively. Then, the
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result of our main problem (1.6), (1.7) can be summarized in the following form: a1 a2 HðtÞ ¼ þ 1=2 þ a3 þ a4 t1=2 þ OðtÞ as t ! 0; ð2:1Þ t t where a1 ¼
jXj ; 4p
0 1 kj n X 1 X @ a2 ¼ 1=2 L i A; 8p i¼1þk j¼1 j1
8 2 3 Z kj r X ð2 nÞ 1 < X 4 þ a3 ¼ ci ðzj Þ dzj 5 6 2p : j¼1 i¼1þk C i j1 2 39 Z kj = n X X 4 ci ðzj Þ dzj 5 ; ; Ci i¼1þk j¼rþ1 j1
a4 ¼
7 256p1=2
8 9 = Z kj n < X X 32 Kj2 ðzj Þ ðci ðzj ÞKj ðzj Þ 2c2i ðzj ÞÞ dzj : : ; 7 C i i¼1þk j¼1 j1
From this theorem, we note that formula (2.1) is in agreement with Zayed’s result [25] when ci are piecewise smooth positive constants. With reference to formula (1.4) and [22–24], the asymptotic expansion (2.1) may be interpreted as follows: (i) X is a general multiply connected domain in R2 and we have the piecewise smooth Robin boundary conditions (1.7) where ci are piecewise smooth positive functions. (ii) For the first four terms, X is a general multiply connected domain in R2 with 8 2 3 Z kj r X 3
holes, and has area jXj and the components Ci of the boundaries oXj are of lengths Li and curvatures
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365
½Kj2 ðzj Þ 327ðci ðzj ÞKj ðzj Þ 2c2i ðzj ÞÞ1=2 together with the Neumann boundary conditions, provided ‘‘H’’ is a positive integer. We close this section with the following two questions: (i) Can one construct two non-congruent general multiply connected domains (with the Dirichlet or Neumann or Robin boundary conditions) whose Laplace operators have precisely the same eigenvalues? (ii) What is the interpretation of X if H is not integer? These are two open problems, which have been left to the interested readers.
3. Construction of the results With reference to [5,6,18,21–24], it is easily seen that HðtÞ associated with the main problem (1.6), (1.7) is given by Z Z HðtÞ ¼ Gc ðt; x; xÞ dx; ð3:1Þ X
X ; which satisfies where the heat kernel Gc ðt; x; yÞ is defined on ð0; 1Þ X , and c is a function of ci ði ¼ 1 þ the following: For fixed x 2 X kj1 ; . . . ; kj ; j ¼ 1; . . . ; r; r þ 1; . . . ; nÞ, Gc ðt; x; yÞ satisfies the heat equation in t; y: o ð3:2Þ Dy Gc ðt; x; yÞ ¼ 0 ot subject to the Robin boundary conditions o þ ci ðyÞ Gc ðt; x; yÞ ¼ 0 on Ci oniy
ð3:3Þ
and the initial condition lim Gc ðt; x; yÞ ¼ dðx yÞ;
t!0þ
ð3:4Þ
where dðx yÞ is the Dirac delta function located at the source point x ¼ y: Note that in (3.2), (3.3) the subscript ‘‘y’’ means that the derivatives are taken in y-variable. By the superposition principle of the heat equation, we write Gc ðt; x; yÞ ¼ GN ðt; x; yÞ þ ,c ðt; x; yÞ;
ð3:5Þ
where GN ðt; x; yÞ is the Neumann heat kernel on X which satisfies the heat equation (3.2) and the Neumann boundary conditions o=oniy GN ðt; x; yÞ ¼ 0 on
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Ci and the initial condition (3.4), while ,c ðt; x; yÞ satisfies the heat equation (3.2), the boundary conditions o ,c ðt; x; yÞ ¼ ci ðyÞGc ðt; x; yÞ oniy
ð3:6Þ
on Ci
and the initial condition lim ,c ðt; x; yÞ ¼ 0:
ð3:7Þ
t!0þ
Now, the solution ,c ðt; x; yÞ which satisfies (3.2), (3.6) and (3.7) (see, for example, [5,14,15,22]) can be written in the form
,c ðt; x; yÞ ¼
8 Z kj r < X X
Z
t
ds : i¼1þk 0 j1 8 Z t kj n < X X
j¼rþ1
: i¼1þk
j1
GN ðt s; x; zj Þci ðzj ÞGc ðs; zj ; yÞ dzj Ci
j¼1
; 9 =
Z ds
0
9 =
Ci
GN ðt s; x; zj Þci ðzj ÞGc ðs; zj ; yÞ dzj : ; ð3:8Þ
From (3.5) and (3.8) we have the following integral equation: Gc ðt; x; yÞ ¼ GN ðt; x; yÞ þ
8 Z kj r < X X j¼1
: i¼1þk
8 Z kj n < X X j¼rþ1
: i¼1þk
0
j1
j1
Z
t
ds Ci
0
Z
t
GN ðt s; x; zj Þci ðzj ÞGc ðs; zj ; yÞdzj
ds Ci
9 = ;
9 =
GN ðt s; x; zj Þci ðzj ÞGc ðs; zj ; yÞdzj : ; ð3:9Þ
On applying the iteration method (see, for example, [19,20]) to the integral equation (3.9) we obtain an infinite convergent series 1 X
m
ð3:10Þ
F0 ðt; x; yÞ ¼ GN ðt; x; yÞ
ð3:11Þ
Gc ðt; x; yÞ ¼
ð1Þ Fm ðt; x; yÞ;
m¼0
where
and
E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375
Fm ðt; x;yÞ ¼
8 Z kj n < X X
: i¼1þk
j¼1
j1
GN ðt s;x;zj Þci ðzj ÞFm1 ðs; zj ;yÞdzj
ds
; 9 Z = t ds GN ðt s;x;zj Þci ðzj ÞFm1 ðs;zj ;yÞdzj ; ; Ci 0
: i¼1þk 0 j1 8 Z kj r < X X
j¼rþ1
Z
t
367
9 =
Ci
ð3:12Þ where m ¼ 1; 2; 3; . . . We will often use the following simple estimate for the Neumann heat kernel (see, for example, [5]): There exist positive constants t0 ; c1 such that for all X ; we get t < t0 , ðx; yÞ 2 X ( ) 2 jx yj 1 GN ðt; x; yÞ 6 c1 t exp : ð3:13Þ c1 t
Lemma 3.1. We have that 1 Z Z X m¼3
jFm ðt; x; xÞj dx ¼ OðtÞ as t ! 0þ :
ð3:14Þ
X
Proof. With reference to the article [7], we use the convolution property of the Gaussian kernel to verify by induction that Fm ðt; x; yÞ has an estimate of the form
jFm ðt; x; yÞj 6 c2 cm3
( ) 1 mþ1 jx yj2 ðm2Þ=2 C t exp ; 2 c1 t
ð3:15Þ
where c2 and c3 are positive constants and C is the gamma function. Summing over m and using (3.10) we see that there exist positive constants c4 ; t0 such that X , we get for all t < t0 and ðx; yÞ 2 X ( Gc ðt; x; yÞ 6 c4 t
1
exp
jx yj2 c1 t
) :
ð3:16Þ
Let gcm ¼ ð1Þm Fm ðt; x; yÞ, with the boundary functions ci ¼ kci k1 . Then by the recursive relation (3.12) of gcm ðt; x; yÞ, we get
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E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375 1 X
gcm ðt; x; yÞ 6
m¼2
8 kj n < X X : i¼1þk
j¼1
2
kci k1
Z
Z
s1
0
Ci
GN ðs1 s2 ; z0j ; z00j Þ
ds2 0
GN ðt s1 ; x; z0j Þ dz0j
ds1
j1
Z
Z
t
Ci
1 X
9 =
gci m ðs2 ; z00j ; yÞ dz00j : ; m¼0 ð3:17Þ
But it is clear from (3.10) and (3.12) that 1 X
gci m ðs; z; yÞ 6 Gkci k1 ðs; z; yÞ:
m¼0
Hence the right-hand side of (3.17) can be estimated by the Gaussian type upper bounds (3.13) and (3.16) of the heat kernels GN ðt; x; yÞ and Gkci k1 ðt; x; yÞ, and we get ( ) 2 1 X jx yj gcm ðt; x; yÞ 6 c5 exp : ð3:18Þ c6 t m¼2 where c5 and c6 are positive constants. From the recursive relation of gcm again, we have Z Z gcm ðt; x; xÞ dx X
¼
8 kj n < X X j¼1
: i¼1þk
2 kci k1
Z
j1
Z
t
ðt uÞ du
Z
0
dyj Ci
GN ðt u; yj ; zj Þgðm2Þci ðu; zj ; yj Þ dzj Ci
6
8 kj n < X X j¼1
¼
: i¼1þk
j1
8 kj n < X X j¼1
2 kci k1 t
: i¼1þk
j1
2
kci k1 t
Z
du 0
Z
Z
t
dyj Ci
9 = ;
GN ðt u; yj ; zj Þgðm1Þci ðu; zj ; yj Þ dzj
Ci
9 =
Z Ci
gðm1Þci ðt; yj ; yj Þ dyj : ;
9 = ;
ð3:19Þ
Summing over m from 3 to infinity and using (3.18) we obtain (3.14) from the inequality jFm j 6 gcm . This proves Lemma 3.1.
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369
From (3.1), (3.10), (3.11) and (3.14) we deduce for t ! 0þ that HðtÞ ¼ HN ðtÞ
Z Z
F1 ðt; x; xÞ dx þ
X
Z Z
F2 ðt; x; xÞ dx þ OðtÞ;
ð3:20Þ
X
where HN ðtÞ ¼
Z Z
GN ðt; x; xÞ dx:
ð3:21Þ
X
On the other hand, the asymptotic expansion of HN ðtÞ as t ! 0þ for the Neumann problems (i.e., ci ¼ 0) is well known (see, for example, [25]) and is given by 0 1 kj n t 1=2 X X jXj 1 @ A þ ð2 nÞ þ 7 þ HN ðtÞ ¼ L i 4pt 8ðptÞ1=2 j¼1 i¼1þk 6 256 p j1 8 9 Z kj = n < X X Kj2 ðzj Þ dzj þ OðtÞ as t ! 0: : i¼1þk ; Ci j¼1
ð3:22Þ
j1
The problem now is to calculate the integrals of Fa ðt; x; xÞ ða ¼ 1; 2Þ over the general multiply connected domain X as follows: Lemma 3.2. If X is a general multiply connected bounded domain in R2 , then we have Z Z
F1 ðt; x; xÞ dx
X
8 2 3 Z kj r X 1
i¼1þkj1
Ci
ð3:23Þ
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E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375
Proof. The definition of F1 ðt; x; xÞ and the Chapman–Kolmogorov equation of the heat kernel (see [5,9]) imply 8 2 3 Z Z Z kj
With reference to [8,22,23] we deduce for t ! 0þ that if zj 2 Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ 1; . . . ; rÞ, then 1 1 1=2 1 ðptÞ Kj ðzj Þ þ Oð1Þ; GN ðt; zj ; zj Þ ¼ ð3:25Þ 2pt 4 while if zj 2 Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ r þ 1; . . . ; nÞ, then 1 1 1=2 1 þ ðptÞ Kj ðzj Þ þ Oð1Þ: GN ðt; zj ; zj Þ ¼ 2pt 4
ð3:26Þ
On inserting (3.25) and (3.26) into (3.24) we arrive at the proof of Lemma 3.2. Lemma 3.3. If X is a general multiply connected bounded domain in R2 , then we have 8 9 Z Z Z kj = n < X 1 t 1=2 X F2 ðt; x; xÞdx ¼ c2i ðzj Þdzj þ OðtÞ as t ! 0þ : : i¼1þk ; 4 p Ci j¼1 X
j1
ð3:27Þ Proof. From the definition of F2 ðt; x; xÞ and with the help of the expression of F1 ðt; x; xÞ, we deduce that 8 Z Z Z t Z kj n < X X F2 ðt; x; xÞ dx ¼ ðt uÞ du c2i ðzj Þ dzj : C 0 i i¼1þkj1 j¼1 X 9 Z = GN ðt u; zj ; yj Þ ci ðyj Þ GN ðu; yj ; zj Þ dyj : ; Ci ð3:28Þ We replace ci ðyj Þ in the above integral by ci ðzj Þ þ Oðjzj yj jÞ and split the integral into two integrals accordingly. On using the estimate (3.13) we deduce that
E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375
Z
371
jzj yj jGN ðt u; yj ; zj ÞGN ðu; zj ; yj Þ dyj Ci
6 c1 ½uðt uÞ
1
(
Z
jyj j exp R1
c2 tjyj j2 uðt uÞ
) dyj :
ð3:29Þ
Since the integral in the right-hand side of (3.29) is bounded by c7 t1 where c7 is a positive constant, we deduce as t ! 0þ that 8 9 Z Z Z kj n < X = X ð3:30Þ F2 ðt; x; xÞ dx ¼ c2i ðzj Þgðt; zj Þ dzj þ OðtÞ; : i¼1þk ; Ci j¼1 j1
X
where gðt; zj Þ ¼
Z 0
t
ðt uÞ du
Z
GN ðt u; yj ; zj ÞGN ðu; zj ; yj Þ dyj :
ð3:31Þ
Ci
The right-hand side of (3.31) can be computed by taking the first term in the series expansion of the Neumann heat kernel GN ðt u; yj ; zj Þ ¼ 2qðt u; yj ; zj Þ, GN ðu; zj ; yj Þ ¼ 2qðu; zj ; yj Þ, where ( ) 2 jyj zj j 1 qðt; yj ; zj Þ ¼ ð4ptÞ exp : 4t The explicit computation can be carried out with the help of a suitably chosen local coordinates system and the localization principle (see [5]). We leave the details of this computation to the interested reader and we content ourselves with the statement that the leading term gðt; zj Þ is equal to the same integral in the Euclidean plane. Thus, as t ! 0þ we get ( ) Z Z t 1 du jzj yj j2 jzj yj j2 exp gðt; zj Þ ¼ 2 dyj þ OðtÞ: 4p 0 u R1 4ðt uÞ 4u ð3:32Þ After some calculations, we deduced as t ! 0þ that Z t 1 t u 1=2 1 t 1=2 gðt; zj Þ ¼ 1=2 3=2 du þ OðtÞ ¼ þ OðtÞ: 2t p u 4 p 0
ð3:33Þ
On inserting (3.33) into (3.30) we arrive at the proof of Lemma 3.3. Now, our main result (2.1) follows immediately from the formulae (3.20)– (3.22) and Lemmas 3.2 and 3.3.
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4. An application of HðtÞ to an ideal gas Following Gutierrez and Yanez [4], we are interested in examining how the thermodynamic properties of an ideal gas are influenced by the geometry of its container. Thermodynamic properties of an ideal gas can be extracted from the partition function ZðbÞ ¼
zN ðbÞ ; N!
ð4:1Þ
where N is the number of particles and zðbÞ is given by X zðbÞ ¼ expðbEJ Þ;
ð4:2Þ
J
where b ¼ ðkB T Þ1 , kB is the Boltzmann constant, and T is the absolute temperature. The energy level (eigenvalues) EJ of one particle are obtained from the stationary states W ¼ u expfiEt= hg of the time-dependent Schr€ odinger equation
2 h oW DW þ V ðxÞW ¼ ih ot 2M
with V ðxÞ ¼ 0;
ð4:3Þ
where M is the mass and h is the Planck constant. Thus, u satisfies the Helmholtz equation Du ¼ ku with k ¼ 2ME=h2 . Therefore, we deduce that the asymptotic expansion of the sum (1.3) of HðtÞ as t ! 0þ which formally is the same as the one-particle function (4.2) of zðbÞ as b ! 0þ . The purpose of this section is to use our main result (2.1) to derive a general expression for the corrections to the thermodynamic quantities, particularly, the energy of an ideal gas enclosed in the general multiply connected domain X in R2 . Following the discussions of Section 2, we can obtain information about the shape of the general multiply connected domain X, by studying the asymptotic expansion of the sum (4.2) when b ! 0 (i.e., T ! 1, the ideal gas case). Noting that the eigenvalue problem of the Schr€ odinger equation is the same as the eigenvalue problem of the wave equation, we can use directly the asymptotic expansion (2.1) of HðtÞ replacing t by ðh2 =2MÞb. Let us now consider the general partition function (4.1) in two-dimensions. On using formula (2.1) for the Robin problem, (4.2) gives
2M zðbÞ ¼ h2
1=2 2 1=2 a1 2M a2 h þ þ a3 þ a4 b1=2 þ OðbÞ as b ! 0þ : 2 1=2 b 2M h b ð4:4Þ
We set out to apply formula (4.4) to the thermodynamic quantities such that the internal energy U ¼ ½ðo=obÞ ln ZðbÞV ;N ; the pressure P ¼ b1 ½ðo=oV Þ
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373
ln ZðbÞT ;N and the specific heat C ¼ ðoU =oT ÞN ;V , among others (see [4]), where ‘‘V’’ denotes the area of the domain X. Thus, in the case of the internal energy, we get ) ( 1=2 2 1=2 o 2M a1 2M a2 h 1=2 U ¼ N ln þ a3 þ a4 b þ OðbÞ : þ ob b 2M h2 h2 b1=2 1
Now, differentiating, expanding in powers of b ¼ ðkB T Þ and using the defi1=2 nition of the thermal wave length KðT Þ ¼ ð2ph2 =MkB T Þ we deduce, after some reduction, that the internal energy U ðT Þ has the asymptotic form: ( " # 2 a2 1 a2 2a3 2 pffiffiffi KðT Þ þ U ðT Þ ¼ NkB T 1 K ðT Þ 8p a1 a1 4a1 p ) " # 3 1 a2 3a2 a3 3a4 3 4 as T ! 1: 2 þ K ðT Þ þ O½K ðT Þ 16p3=2 a1 a1 a1 ð4:5Þ Similar expressions hold for the pressure and the specific heat. Note that Gutierrez and Yanez [4] have recently constructed a formula similar to (4.5) but for a simply connected bounded domain with the Dirichlet boundary conditions by using formula (1.5) of Section 1. When transforming the sum (4.2) into an integral, one introduces the density of states function qðEÞ in the form Z 1 zðbÞ ¼ ebE qðEÞ dE; ð4:6Þ 0
where qðEÞ denotes the number of eigenvalues of Eq. (1.6) lying in a range of unit width around a specific value E of the particle energy. From (4.4) and (4.6) we deduce when X R2 that the density of states formula has the form 2M 2M 0 E þ ð4:7Þ qðEÞ ¼ a a2 E1=2 þ 1 h2 p h2 Let us now examine an ideal Bose gas confined to the general multiply connected domain X with the piecewise smooth Robin boundary conditions (1.7). To this end we start with the following expression for the number of particles ‘‘N’’ in the system: N¼
X E
eaþbE 1
1
¼
1 XX E
ekakbE ;
ð4:8Þ
k¼1
where a ¼ bl , and l is the chemical potential of the system (see [11]). Replacing the summation over E in (4.8) by an integration and employing expression (4.6) we obtain the formula
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E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375
N¼
1 X
eka zðkbÞ:
ð4:9Þ
k¼1
From (4.4) and (4.9) we deduce that N¼
4pa1 2p1=2 a2 a4 g1=2 ðaÞ þ a3 g0 ðaÞ þ 1=2 KðT Þg1=2 ðaÞ þ ; ðaÞ þ g 1 2 KðT Þ 2p K ðT Þ ð4:10Þ P1
where gt ðaÞ ¼ k¼1 k t eka is the familiar Bose–Einstein functions. Our main results (4.5), (4.7) and (4.10) show, in principle, that an ideal gas could feel some aspects of the shape of the general multiply connected bounded domain X R2 because its thermodynamic quantities depend on some geometric properties of X. But, we note that an ideal gas, even with all terms in the expansion of the partition function completely known, is not able to discriminate between two different shapes. This theoretical result was experimentally verified in [13] by employing thin microwave cavities shaped in the form of two different domains known to be isospectral. Of course, with reference to the articles [3,13], there are domains where although different in shapes, the thermodynamic properties of an ideal gas will be exactly the same. From these discussions, we deduce that the answer of either Kac’s question [6] or Gutierrez and Yanez’s question [4] is negative.
References [1] P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier Grenoble 36 (1986) 167–192. [2] H.P.W. Gottlieb, Eigenvalues of the Laplacian with Neumann boundary conditions, J. Austral. Math. Soc. Ser. B 26 (1985) 193–309. [3] C. Gordon, D.L. Webb, S. Wolpert, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. 27 (1992) 134–138. [4] G. Gutierrez, J.M. Yanez, Can an ideal gas feel the shape of its container? Amer. J. Phys. 65 (1997) 739–743. [5] P. Hsu, On the H-function of a Riemannian manifold with boundary, Trans. Amer. Math. Soc. 333 (1992) 643–671. [6] M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73 (1966) 1–23. [7] H.P. McKean, I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 11 (1967) 43–69. [8] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964) 542. [9] V.G. Papanicolaou, Trace formulas and the behaviour of large eigenvalues, SIAM J. Math. Anal. 26 (1995) 218–237. [10] A. Pleijel, A study of certain Green’s functions with applications in the theory of vibrating membranes, Ark. Mat. 2 (1954) 553–569. [11] R. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, New York, 1965. [12] B.D. Sleeman, E.M.E. Zayed, An inverse eigenvalue problem for a general convex domain, J. Math. Anal. Appl. 94 (1983) 78–95.
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[13] S. Sridhar, A. Kudrolli, Experiments on Not hearing the shape of drums, Phys. Rev. Lett. 72 (1994) 2175–2178. [14] I. Stakgold, in: Boundary Value Problems of Mathematical Physics, vol. 2, Macmillan, New York, 1968. [15] I. Stakgold, Green’s Functions and Boundary Value Problems, Wiley, New York, 1979. [16] K. Stewartson, R.T. Waechter, On hearing the shape of a drum: further results, Proc. Camb. Philos. Soc. 69 (1971) 353–363. [17] H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. Ecole Norm. Sup. 15 (1982) 441–456. [18] E.M.E. Zayed, Eigenvalues of the Laplacian for the third boundary value problem, J. Austral. Math. Soc. Ser. B 29 (1987) 79–87. [19] E.M.E. Zayed, Hearing the shape of a general convex domain, J. Math. Anal. Appl. 142 (1989) 170–187. [20] E.M.E. Zayed, Heat equation for an arbitrary doubly connected region in R2 with mixed boundary conditions, Z. Angew. Math. Phys. 40 (1989) 339–355. [21] E.M.E. Zayed, On hearing the shape of an arbitrary doubly connected region in R2 , J. Austral. Math. Soc. Ser. B 31 (1990) 472–483. [22] E.M.E. Zayed, Short-time asymptotics of the heat kernel of the Laplacian of a bounded domain with Robin boundary conditions, Houston J. Math. 24 (1998) 377–385. [23] E.M.E. Zayed, On hearing the shape of a bounded domain with Robin boundary conditions, IMA J. Appl. Math. 64 (2000) 95–108. [24] E.M.E. Zayed, An inverse problem for a general annular drum with positive smooth function in the Robin boundary conditions, Collect. Math. 51 (2000) 261–275. [25] E.M.E. Zayed, Short-time asymptotics of the heat kernel of the Laplacian for a multiplyconnected domain in R2 with Robin boundary conditions, Appl. Anal. 77 (2001) 177–194.
On hearing the shape of a general multi-connected vibrating membrane in 2 R with piecewise smooth positive functions in the Robin boundary conditions E.M.E. Zayed Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
Abstract P The asymptotic expansion of the trace of the heat kernel HðtÞ ¼ 1 J ¼1 expðtkJ Þ as t ! 0þ has been derived for a variety of domains, where fkJ g are the eigenvalues of the negative Laplace operator 2 2 X o D ¼ oxi i¼1 in the ðx1 ; x2 Þ-plane. The dependence of HðtÞ on the connectivity of domains and the boundary conditions are analyzed. Particular attention is given for a general multiply connected bounded domain in R2 together with a finite number of piecewise Robin boundary conditions, where the coefficients in the boundary conditions are piecewise smooth positive functions. Some applications of an ideal gas enclosed in the general multiply connected domain are given. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Inverse problem; Heat kernel; Eigenvalues; Hearing the shape of multi-connected domain; Classical ideal gas
E-mail address: [email protected] (E.M.E. Zayed). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 3 3 7 - X
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1. Introduction Let X be a given arbitrary simply connected bounded domain in R2 with a smooth boundary oX. Consider the Robin problem o Du ¼ ku in X; þ c u ¼ 0 on oX; ð1:1Þ on where o=on denotes the differentiation along the outward pointing normal to Þ. oX, where the coefficient c is a positive smooth function and u 2 C 2 ðXÞ \ CðX Denote its eigenvalues counted according to multiplicity by 0 < k1 6 k2 6 k3 6 6 kJ 6 ! 1
as J ! 1:
ð1:2Þ
The basic problem is that of determining some geometric quantities associated with the bounded domain X from complete knowledge of the eigenvalues (1.2) using the asymptotic expansion of the trace of the heat kernel HðtÞ ¼
1 X
expðtkJ Þ
as t ! 0þ :
ð1:3Þ
J ¼1
The Robin problem (1.1) has been investigated by many authors (see, for example, [2,5–7,10,16]) in the following special cases: (i) Case 1. c ¼ 0 (the Neumann problem) Z jXj joXj 7 t 1=2 HðtÞ ¼ þ þ a þ K 2 ðzÞ dz þ OðtÞ as t ! 0þ : 0 4pt 8ðptÞ1=2 256 p oX ð1:4Þ (ii) Case 2. c ! 1 (the Dirichlet problem) jXj joXj 1 t 1=2 þ a þ HðtÞ ¼ 0 4pt 8ðptÞ1=2 256 p
Z
K 2 ðzÞ dz þ OðtÞ as t ! 0þ :
oX
ð1:5Þ In these formulae, jXj is the area of X, joXj is the total length of oX and KðzÞ is the curvature of oX, where z is the arc length of the counterclockwise-oriented boundary oX. The constant term a0 has geometric significance, e.g., if X is smooth and convex, then a0 ¼ 1=6 and if X is permitted to have a finite number of smooth convex holes ‘‘H’’, then a0 ¼ ð1 H Þ=6: We merely note that aspects of the question of Kac [6], namely, can one hear the shape of a drum? have been discussed by Sleeman and Zayed [12] for the Robin problem (1.1) when c is a positive constant, and by Zayed [22] when c is a positive smooth function. Further, the Robin problem (1.1) has been investigated by Hsu [5] in the general situation where X is a compact Riemannian
E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375
363
manifold of n-dimensions with smooth boundary, and has determined the first four terms of HðtÞ as t ! 0þ . Thus, in the mathematical terms, Kac’s question becomes: Does the boundary condition define the spectrum uniquely? The proof of Gordon et al. [3] uses drums made by piecing together with a few (identical) basic shapes, for example, triangles. Gordon et al. [3] have proved that different shaped drums can posses identical spectra, that is they are ‘‘isospectral’’. The examples of [1,17] also show that one cannot always hear the shape of a domain in Rn for n P 3. Also, Milnor [8] has constructed two non-congruent 16-dimensional tori whose Laplace Beltrami operators have precisely the same eigenvalues. The object of this paper is to discuss the following more general inverse problem: Let X be a general multiply connected bounded domain in R2 surrounding internally by simply connected bounded domains Xj with smooth boundaries oXj ðj ¼ 1; . . . ; n 1Þ and externally by a simply connected bounded domain Xn with a smooth boundary oXn . Suppose that the eigenvalues (1.2) are given for the Helmholtz equation Du ¼ ku
ð1:6Þ
in X
together with the piecewise smooth Robin boundary conditions
o þ ci u ¼ 0 oni
on Ci ;
ð1:7Þ
where the boundaries oXj ðj ¼ 1; . . . ; nÞ consist of a finite number Skj of piecewise smooth components Ci ði ¼ 1 þ kj1 ; . . . ; kj Þ such that oXj ¼ i¼1þk Ci where j1 k0 ¼ 0 and ci are piecewise smooth positive functions defined on Ci , while o=oni denote differentiations along the outward normals to Ci . The basic problem is to determine some geometric quantities associated with the general multiply connected bounded domain X from complete knowledge of its eigenvalues (1.2) using the asymptotic expansion of HðtÞ for small positive t. Note that the problem (1.6), (1.7) has been discussed recently by Zayed [25] in the case ci are piecewise smooth positive constants.
2. Statement of results Theorem 2.1. Suppose that the boundaries oXj ðj ¼ 1; . . . ; nÞ of the general multiply connected domain X are given locally by the equations xa ¼ y a ðzj Þ ðj ¼ 1; . . . ; nÞ and a ¼ 1; 2 in which zj are the arc lengths of the counterclockwise-oriented boundaries oXj and y a ðzj Þ 2 C 1 ðoXj Þ. Let Li be the lengths of the components Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ 1; . . . ; r; r þ 1; . . . ; nÞ, respectively. Let Kj ðzj Þ be the curvatures of the boundaries oXj , respectively. Then, the
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result of our main problem (1.6), (1.7) can be summarized in the following form: a1 a2 HðtÞ ¼ þ 1=2 þ a3 þ a4 t1=2 þ OðtÞ as t ! 0; ð2:1Þ t t where a1 ¼
jXj ; 4p
0 1 kj n X 1 X @ a2 ¼ 1=2 L i A; 8p i¼1þk j¼1 j1
8 2 3 Z kj r X ð2 nÞ 1 < X 4 þ a3 ¼ ci ðzj Þ dzj 5 6 2p : j¼1 i¼1þk C i j1 2 39 Z kj = n X X 4 ci ðzj Þ dzj 5 ; ; Ci i¼1þk j¼rþ1 j1
a4 ¼
7 256p1=2
8 9 = Z kj n < X X 32 Kj2 ðzj Þ ðci ðzj ÞKj ðzj Þ 2c2i ðzj ÞÞ dzj : : ; 7 C i i¼1þk j¼1 j1
From this theorem, we note that formula (2.1) is in agreement with Zayed’s result [25] when ci are piecewise smooth positive constants. With reference to formula (1.4) and [22–24], the asymptotic expansion (2.1) may be interpreted as follows: (i) X is a general multiply connected domain in R2 and we have the piecewise smooth Robin boundary conditions (1.7) where ci are piecewise smooth positive functions. (ii) For the first four terms, X is a general multiply connected domain in R2 with 8 2 3 Z kj r X 3
holes, and has area jXj and the components Ci of the boundaries oXj are of lengths Li and curvatures
E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375
365
½Kj2 ðzj Þ 327ðci ðzj ÞKj ðzj Þ 2c2i ðzj ÞÞ1=2 together with the Neumann boundary conditions, provided ‘‘H’’ is a positive integer. We close this section with the following two questions: (i) Can one construct two non-congruent general multiply connected domains (with the Dirichlet or Neumann or Robin boundary conditions) whose Laplace operators have precisely the same eigenvalues? (ii) What is the interpretation of X if H is not integer? These are two open problems, which have been left to the interested readers.
3. Construction of the results With reference to [5,6,18,21–24], it is easily seen that HðtÞ associated with the main problem (1.6), (1.7) is given by Z Z HðtÞ ¼ Gc ðt; x; xÞ dx; ð3:1Þ X
X ; which satisfies where the heat kernel Gc ðt; x; yÞ is defined on ð0; 1Þ X , and c is a function of ci ði ¼ 1 þ the following: For fixed x 2 X kj1 ; . . . ; kj ; j ¼ 1; . . . ; r; r þ 1; . . . ; nÞ, Gc ðt; x; yÞ satisfies the heat equation in t; y: o ð3:2Þ Dy Gc ðt; x; yÞ ¼ 0 ot subject to the Robin boundary conditions o þ ci ðyÞ Gc ðt; x; yÞ ¼ 0 on Ci oniy
ð3:3Þ
and the initial condition lim Gc ðt; x; yÞ ¼ dðx yÞ;
t!0þ
ð3:4Þ
where dðx yÞ is the Dirac delta function located at the source point x ¼ y: Note that in (3.2), (3.3) the subscript ‘‘y’’ means that the derivatives are taken in y-variable. By the superposition principle of the heat equation, we write Gc ðt; x; yÞ ¼ GN ðt; x; yÞ þ ,c ðt; x; yÞ;
ð3:5Þ
where GN ðt; x; yÞ is the Neumann heat kernel on X which satisfies the heat equation (3.2) and the Neumann boundary conditions o=oniy GN ðt; x; yÞ ¼ 0 on
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Ci and the initial condition (3.4), while ,c ðt; x; yÞ satisfies the heat equation (3.2), the boundary conditions o ,c ðt; x; yÞ ¼ ci ðyÞGc ðt; x; yÞ oniy
ð3:6Þ
on Ci
and the initial condition lim ,c ðt; x; yÞ ¼ 0:
ð3:7Þ
t!0þ
Now, the solution ,c ðt; x; yÞ which satisfies (3.2), (3.6) and (3.7) (see, for example, [5,14,15,22]) can be written in the form
,c ðt; x; yÞ ¼
8 Z kj r < X X
Z
t
ds : i¼1þk 0 j1 8 Z t kj n < X X
j¼rþ1
: i¼1þk
j1
GN ðt s; x; zj Þci ðzj ÞGc ðs; zj ; yÞ dzj Ci
j¼1
; 9 =
Z ds
0
9 =
Ci
GN ðt s; x; zj Þci ðzj ÞGc ðs; zj ; yÞ dzj : ; ð3:8Þ
From (3.5) and (3.8) we have the following integral equation: Gc ðt; x; yÞ ¼ GN ðt; x; yÞ þ
8 Z kj r < X X j¼1
: i¼1þk
8 Z kj n < X X j¼rþ1
: i¼1þk
0
j1
j1
Z
t
ds Ci
0
Z
t
GN ðt s; x; zj Þci ðzj ÞGc ðs; zj ; yÞdzj
ds Ci
9 = ;
9 =
GN ðt s; x; zj Þci ðzj ÞGc ðs; zj ; yÞdzj : ; ð3:9Þ
On applying the iteration method (see, for example, [19,20]) to the integral equation (3.9) we obtain an infinite convergent series 1 X
m
ð3:10Þ
F0 ðt; x; yÞ ¼ GN ðt; x; yÞ
ð3:11Þ
Gc ðt; x; yÞ ¼
ð1Þ Fm ðt; x; yÞ;
m¼0
where
and
E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375
Fm ðt; x;yÞ ¼
8 Z kj n < X X
: i¼1þk
j¼1
j1
GN ðt s;x;zj Þci ðzj ÞFm1 ðs; zj ;yÞdzj
ds
; 9 Z = t ds GN ðt s;x;zj Þci ðzj ÞFm1 ðs;zj ;yÞdzj ; ; Ci 0
: i¼1þk 0 j1 8 Z kj r < X X
j¼rþ1
Z
t
367
9 =
Ci
ð3:12Þ where m ¼ 1; 2; 3; . . . We will often use the following simple estimate for the Neumann heat kernel (see, for example, [5]): There exist positive constants t0 ; c1 such that for all X ; we get t < t0 , ðx; yÞ 2 X ( ) 2 jx yj 1 GN ðt; x; yÞ 6 c1 t exp : ð3:13Þ c1 t
Lemma 3.1. We have that 1 Z Z X m¼3
jFm ðt; x; xÞj dx ¼ OðtÞ as t ! 0þ :
ð3:14Þ
X
Proof. With reference to the article [7], we use the convolution property of the Gaussian kernel to verify by induction that Fm ðt; x; yÞ has an estimate of the form
jFm ðt; x; yÞj 6 c2 cm3
( ) 1 mþ1 jx yj2 ðm2Þ=2 C t exp ; 2 c1 t
ð3:15Þ
where c2 and c3 are positive constants and C is the gamma function. Summing over m and using (3.10) we see that there exist positive constants c4 ; t0 such that X , we get for all t < t0 and ðx; yÞ 2 X ( Gc ðt; x; yÞ 6 c4 t
1
exp
jx yj2 c1 t
) :
ð3:16Þ
Let gcm ¼ ð1Þm Fm ðt; x; yÞ, with the boundary functions ci ¼ kci k1 . Then by the recursive relation (3.12) of gcm ðt; x; yÞ, we get
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E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375 1 X
gcm ðt; x; yÞ 6
m¼2
8 kj n < X X : i¼1þk
j¼1
2
kci k1
Z
Z
s1
0
Ci
GN ðs1 s2 ; z0j ; z00j Þ
ds2 0
GN ðt s1 ; x; z0j Þ dz0j
ds1
j1
Z
Z
t
Ci
1 X
9 =
gci m ðs2 ; z00j ; yÞ dz00j : ; m¼0 ð3:17Þ
But it is clear from (3.10) and (3.12) that 1 X
gci m ðs; z; yÞ 6 Gkci k1 ðs; z; yÞ:
m¼0
Hence the right-hand side of (3.17) can be estimated by the Gaussian type upper bounds (3.13) and (3.16) of the heat kernels GN ðt; x; yÞ and Gkci k1 ðt; x; yÞ, and we get ( ) 2 1 X jx yj gcm ðt; x; yÞ 6 c5 exp : ð3:18Þ c6 t m¼2 where c5 and c6 are positive constants. From the recursive relation of gcm again, we have Z Z gcm ðt; x; xÞ dx X
¼
8 kj n < X X j¼1
: i¼1þk
2 kci k1
Z
j1
Z
t
ðt uÞ du
Z
0
dyj Ci
GN ðt u; yj ; zj Þgðm2Þci ðu; zj ; yj Þ dzj Ci
6
8 kj n < X X j¼1
¼
: i¼1þk
j1
8 kj n < X X j¼1
2 kci k1 t
: i¼1þk
j1
2
kci k1 t
Z
du 0
Z
Z
t
dyj Ci
9 = ;
GN ðt u; yj ; zj Þgðm1Þci ðu; zj ; yj Þ dzj
Ci
9 =
Z Ci
gðm1Þci ðt; yj ; yj Þ dyj : ;
9 = ;
ð3:19Þ
Summing over m from 3 to infinity and using (3.18) we obtain (3.14) from the inequality jFm j 6 gcm . This proves Lemma 3.1.
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369
From (3.1), (3.10), (3.11) and (3.14) we deduce for t ! 0þ that HðtÞ ¼ HN ðtÞ
Z Z
F1 ðt; x; xÞ dx þ
X
Z Z
F2 ðt; x; xÞ dx þ OðtÞ;
ð3:20Þ
X
where HN ðtÞ ¼
Z Z
GN ðt; x; xÞ dx:
ð3:21Þ
X
On the other hand, the asymptotic expansion of HN ðtÞ as t ! 0þ for the Neumann problems (i.e., ci ¼ 0) is well known (see, for example, [25]) and is given by 0 1 kj n t 1=2 X X jXj 1 @ A þ ð2 nÞ þ 7 þ HN ðtÞ ¼ L i 4pt 8ðptÞ1=2 j¼1 i¼1þk 6 256 p j1 8 9 Z kj = n < X X Kj2 ðzj Þ dzj þ OðtÞ as t ! 0: : i¼1þk ; Ci j¼1
ð3:22Þ
j1
The problem now is to calculate the integrals of Fa ðt; x; xÞ ða ¼ 1; 2Þ over the general multiply connected domain X as follows: Lemma 3.2. If X is a general multiply connected bounded domain in R2 , then we have Z Z
F1 ðt; x; xÞ dx
X
8 2 3 Z kj r X 1
i¼1þkj1
Ci
ð3:23Þ
370
E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375
Proof. The definition of F1 ðt; x; xÞ and the Chapman–Kolmogorov equation of the heat kernel (see [5,9]) imply 8 2 3 Z Z Z kj
With reference to [8,22,23] we deduce for t ! 0þ that if zj 2 Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ 1; . . . ; rÞ, then 1 1 1=2 1 ðptÞ Kj ðzj Þ þ Oð1Þ; GN ðt; zj ; zj Þ ¼ ð3:25Þ 2pt 4 while if zj 2 Ci ði ¼ 1 þ kj1 ; . . . ; kj ; j ¼ r þ 1; . . . ; nÞ, then 1 1 1=2 1 þ ðptÞ Kj ðzj Þ þ Oð1Þ: GN ðt; zj ; zj Þ ¼ 2pt 4
ð3:26Þ
On inserting (3.25) and (3.26) into (3.24) we arrive at the proof of Lemma 3.2. Lemma 3.3. If X is a general multiply connected bounded domain in R2 , then we have 8 9 Z Z Z kj = n < X 1 t 1=2 X F2 ðt; x; xÞdx ¼ c2i ðzj Þdzj þ OðtÞ as t ! 0þ : : i¼1þk ; 4 p Ci j¼1 X
j1
ð3:27Þ Proof. From the definition of F2 ðt; x; xÞ and with the help of the expression of F1 ðt; x; xÞ, we deduce that 8 Z Z Z t Z kj n < X X F2 ðt; x; xÞ dx ¼ ðt uÞ du c2i ðzj Þ dzj : C 0 i i¼1þkj1 j¼1 X 9 Z = GN ðt u; zj ; yj Þ ci ðyj Þ GN ðu; yj ; zj Þ dyj : ; Ci ð3:28Þ We replace ci ðyj Þ in the above integral by ci ðzj Þ þ Oðjzj yj jÞ and split the integral into two integrals accordingly. On using the estimate (3.13) we deduce that
E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375
Z
371
jzj yj jGN ðt u; yj ; zj ÞGN ðu; zj ; yj Þ dyj Ci
6 c1 ½uðt uÞ
1
(
Z
jyj j exp R1
c2 tjyj j2 uðt uÞ
) dyj :
ð3:29Þ
Since the integral in the right-hand side of (3.29) is bounded by c7 t1 where c7 is a positive constant, we deduce as t ! 0þ that 8 9 Z Z Z kj n < X = X ð3:30Þ F2 ðt; x; xÞ dx ¼ c2i ðzj Þgðt; zj Þ dzj þ OðtÞ; : i¼1þk ; Ci j¼1 j1
X
where gðt; zj Þ ¼
Z 0
t
ðt uÞ du
Z
GN ðt u; yj ; zj ÞGN ðu; zj ; yj Þ dyj :
ð3:31Þ
Ci
The right-hand side of (3.31) can be computed by taking the first term in the series expansion of the Neumann heat kernel GN ðt u; yj ; zj Þ ¼ 2qðt u; yj ; zj Þ, GN ðu; zj ; yj Þ ¼ 2qðu; zj ; yj Þ, where ( ) 2 jyj zj j 1 qðt; yj ; zj Þ ¼ ð4ptÞ exp : 4t The explicit computation can be carried out with the help of a suitably chosen local coordinates system and the localization principle (see [5]). We leave the details of this computation to the interested reader and we content ourselves with the statement that the leading term gðt; zj Þ is equal to the same integral in the Euclidean plane. Thus, as t ! 0þ we get ( ) Z Z t 1 du jzj yj j2 jzj yj j2 exp gðt; zj Þ ¼ 2 dyj þ OðtÞ: 4p 0 u R1 4ðt uÞ 4u ð3:32Þ After some calculations, we deduced as t ! 0þ that Z t 1 t u 1=2 1 t 1=2 gðt; zj Þ ¼ 1=2 3=2 du þ OðtÞ ¼ þ OðtÞ: 2t p u 4 p 0
ð3:33Þ
On inserting (3.33) into (3.30) we arrive at the proof of Lemma 3.3. Now, our main result (2.1) follows immediately from the formulae (3.20)– (3.22) and Lemmas 3.2 and 3.3.
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4. An application of HðtÞ to an ideal gas Following Gutierrez and Yanez [4], we are interested in examining how the thermodynamic properties of an ideal gas are influenced by the geometry of its container. Thermodynamic properties of an ideal gas can be extracted from the partition function ZðbÞ ¼
zN ðbÞ ; N!
ð4:1Þ
where N is the number of particles and zðbÞ is given by X zðbÞ ¼ expðbEJ Þ;
ð4:2Þ
J
where b ¼ ðkB T Þ1 , kB is the Boltzmann constant, and T is the absolute temperature. The energy level (eigenvalues) EJ of one particle are obtained from the stationary states W ¼ u expfiEt= hg of the time-dependent Schr€ odinger equation
2 h oW DW þ V ðxÞW ¼ ih ot 2M
with V ðxÞ ¼ 0;
ð4:3Þ
where M is the mass and h is the Planck constant. Thus, u satisfies the Helmholtz equation Du ¼ ku with k ¼ 2ME=h2 . Therefore, we deduce that the asymptotic expansion of the sum (1.3) of HðtÞ as t ! 0þ which formally is the same as the one-particle function (4.2) of zðbÞ as b ! 0þ . The purpose of this section is to use our main result (2.1) to derive a general expression for the corrections to the thermodynamic quantities, particularly, the energy of an ideal gas enclosed in the general multiply connected domain X in R2 . Following the discussions of Section 2, we can obtain information about the shape of the general multiply connected domain X, by studying the asymptotic expansion of the sum (4.2) when b ! 0 (i.e., T ! 1, the ideal gas case). Noting that the eigenvalue problem of the Schr€ odinger equation is the same as the eigenvalue problem of the wave equation, we can use directly the asymptotic expansion (2.1) of HðtÞ replacing t by ðh2 =2MÞb. Let us now consider the general partition function (4.1) in two-dimensions. On using formula (2.1) for the Robin problem, (4.2) gives
2M zðbÞ ¼ h2
1=2 2 1=2 a1 2M a2 h þ þ a3 þ a4 b1=2 þ OðbÞ as b ! 0þ : 2 1=2 b 2M h b ð4:4Þ
We set out to apply formula (4.4) to the thermodynamic quantities such that the internal energy U ¼ ½ðo=obÞ ln ZðbÞV ;N ; the pressure P ¼ b1 ½ðo=oV Þ
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ln ZðbÞT ;N and the specific heat C ¼ ðoU =oT ÞN ;V , among others (see [4]), where ‘‘V’’ denotes the area of the domain X. Thus, in the case of the internal energy, we get ) ( 1=2 2 1=2 o 2M a1 2M a2 h 1=2 U ¼ N ln þ a3 þ a4 b þ OðbÞ : þ ob b 2M h2 h2 b1=2 1
Now, differentiating, expanding in powers of b ¼ ðkB T Þ and using the defi1=2 nition of the thermal wave length KðT Þ ¼ ð2ph2 =MkB T Þ we deduce, after some reduction, that the internal energy U ðT Þ has the asymptotic form: ( " # 2 a2 1 a2 2a3 2 pffiffiffi KðT Þ þ U ðT Þ ¼ NkB T 1 K ðT Þ 8p a1 a1 4a1 p ) " # 3 1 a2 3a2 a3 3a4 3 4 as T ! 1: 2 þ K ðT Þ þ O½K ðT Þ 16p3=2 a1 a1 a1 ð4:5Þ Similar expressions hold for the pressure and the specific heat. Note that Gutierrez and Yanez [4] have recently constructed a formula similar to (4.5) but for a simply connected bounded domain with the Dirichlet boundary conditions by using formula (1.5) of Section 1. When transforming the sum (4.2) into an integral, one introduces the density of states function qðEÞ in the form Z 1 zðbÞ ¼ ebE qðEÞ dE; ð4:6Þ 0
where qðEÞ denotes the number of eigenvalues of Eq. (1.6) lying in a range of unit width around a specific value E of the particle energy. From (4.4) and (4.6) we deduce when X R2 that the density of states formula has the form 2M 2M 0 E þ ð4:7Þ qðEÞ ¼ a a2 E1=2 þ 1 h2 p h2 Let us now examine an ideal Bose gas confined to the general multiply connected domain X with the piecewise smooth Robin boundary conditions (1.7). To this end we start with the following expression for the number of particles ‘‘N’’ in the system: N¼
X E
eaþbE 1
1
¼
1 XX E
ekakbE ;
ð4:8Þ
k¼1
where a ¼ bl , and l is the chemical potential of the system (see [11]). Replacing the summation over E in (4.8) by an integration and employing expression (4.6) we obtain the formula
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N¼
1 X
eka zðkbÞ:
ð4:9Þ
k¼1
From (4.4) and (4.9) we deduce that N¼
4pa1 2p1=2 a2 a4 g1=2 ðaÞ þ a3 g0 ðaÞ þ 1=2 KðT Þg1=2 ðaÞ þ ; ðaÞ þ g 1 2 KðT Þ 2p K ðT Þ ð4:10Þ P1
where gt ðaÞ ¼ k¼1 k t eka is the familiar Bose–Einstein functions. Our main results (4.5), (4.7) and (4.10) show, in principle, that an ideal gas could feel some aspects of the shape of the general multiply connected bounded domain X R2 because its thermodynamic quantities depend on some geometric properties of X. But, we note that an ideal gas, even with all terms in the expansion of the partition function completely known, is not able to discriminate between two different shapes. This theoretical result was experimentally verified in [13] by employing thin microwave cavities shaped in the form of two different domains known to be isospectral. Of course, with reference to the articles [3,13], there are domains where although different in shapes, the thermodynamic properties of an ideal gas will be exactly the same. From these discussions, we deduce that the answer of either Kac’s question [6] or Gutierrez and Yanez’s question [4] is negative.
References [1] P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier Grenoble 36 (1986) 167–192. [2] H.P.W. Gottlieb, Eigenvalues of the Laplacian with Neumann boundary conditions, J. Austral. Math. Soc. Ser. B 26 (1985) 193–309. [3] C. Gordon, D.L. Webb, S. Wolpert, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. 27 (1992) 134–138. [4] G. Gutierrez, J.M. Yanez, Can an ideal gas feel the shape of its container? Amer. J. Phys. 65 (1997) 739–743. [5] P. Hsu, On the H-function of a Riemannian manifold with boundary, Trans. Amer. Math. Soc. 333 (1992) 643–671. [6] M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73 (1966) 1–23. [7] H.P. McKean, I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 11 (1967) 43–69. [8] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964) 542. [9] V.G. Papanicolaou, Trace formulas and the behaviour of large eigenvalues, SIAM J. Math. Anal. 26 (1995) 218–237. [10] A. Pleijel, A study of certain Green’s functions with applications in the theory of vibrating membranes, Ark. Mat. 2 (1954) 553–569. [11] R. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, New York, 1965. [12] B.D. Sleeman, E.M.E. Zayed, An inverse eigenvalue problem for a general convex domain, J. Math. Anal. Appl. 94 (1983) 78–95.
E.M.E. Zayed / Appl. Math. Comput. 135 (2003) 361–375
375
[13] S. Sridhar, A. Kudrolli, Experiments on Not hearing the shape of drums, Phys. Rev. Lett. 72 (1994) 2175–2178. [14] I. Stakgold, in: Boundary Value Problems of Mathematical Physics, vol. 2, Macmillan, New York, 1968. [15] I. Stakgold, Green’s Functions and Boundary Value Problems, Wiley, New York, 1979. [16] K. Stewartson, R.T. Waechter, On hearing the shape of a drum: further results, Proc. Camb. Philos. Soc. 69 (1971) 353–363. [17] H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. Ecole Norm. Sup. 15 (1982) 441–456. [18] E.M.E. Zayed, Eigenvalues of the Laplacian for the third boundary value problem, J. Austral. Math. Soc. Ser. B 29 (1987) 79–87. [19] E.M.E. Zayed, Hearing the shape of a general convex domain, J. Math. Anal. Appl. 142 (1989) 170–187. [20] E.M.E. Zayed, Heat equation for an arbitrary doubly connected region in R2 with mixed boundary conditions, Z. Angew. Math. Phys. 40 (1989) 339–355. [21] E.M.E. Zayed, On hearing the shape of an arbitrary doubly connected region in R2 , J. Austral. Math. Soc. Ser. B 31 (1990) 472–483. [22] E.M.E. Zayed, Short-time asymptotics of the heat kernel of the Laplacian of a bounded domain with Robin boundary conditions, Houston J. Math. 24 (1998) 377–385. [23] E.M.E. Zayed, On hearing the shape of a bounded domain with Robin boundary conditions, IMA J. Appl. Math. 64 (2000) 95–108. [24] E.M.E. Zayed, An inverse problem for a general annular drum with positive smooth function in the Robin boundary conditions, Collect. Math. 51 (2000) 261–275. [25] E.M.E. Zayed, Short-time asymptotics of the heat kernel of the Laplacian for a multiplyconnected domain in R2 with Robin boundary conditions, Appl. Anal. 77 (2001) 177–194.