The number of eigenstates: counting function and heat kernel
aa r X i v : . [ m a t h - ph ] M a r Preprint typeset in JHEP style - HYPER VERSION
The number of eigenstates: counting function andheat kernel
Wu-Sheng Dai and Mi Xie
Department of Physics, Tianjin University, Tianjin 300072, P. R. ChinaLiuHui Center for Applied Mathematics, Nankai University & Tianjin University,Tianjin 300072, P. R. ChinaE-mail: [email protected]
E-mail: [email protected]
Abstract:
The main aim of this paper is twofold: (1) revealing a relation between thecounting function N ( λ ) (the number of the eigenstates with eigenvalue smaller than a givennumber) and the heat kernel K ( t ), which is still an open problem in mathematics, and(2) introducing an approach for the calculation of N ( λ ), for there is no effective methodfor calculating N ( λ ) beyond leading order. We suggest a new expression of N ( λ ) whichis more suitable for practical calculations. A renormalization procedure is constructed forremoving the divergences which appear when obtaining N ( λ ) from a nonuniformly conver-gent expansion of K ( t ). We calculate N ( λ ) for D -dimensional boxes, three-dimensionalballs, and two-dimensional multiply-connected irregular regions. By the Gauss-Bonnettheorem, we generalize the simply-connected heat kernel to the multiply-connected case;this result proves Kac’s conjecture on the two-dimensional multiply-connected heat ker-nel. The approaches for calculating eigenvalue spectra and state densities from N ( λ ) areintroduced. Keywords:
Differential and Algebraic Geometry, Boundary Quantum Field Theory. ontents
1. Introduction 12. An expression for the counting function: an alternative definition for-mula 43. The relation between counting function and heat kernel 54. Calculating N ( λ ) from an asymptotics of K ( t ) N ( λ ) 74.2 Renormalization 7
5. Comparison of the two approaches for calculating N ( λ ) : D -dimensionalboxes 86. Calculating N ( λ ) from K ( t ) : three-dimensional balls 97. Multiply-connected cases: two-dimensional K ( t ) and N ( λ ) N ( λ ) and K ( t ) in two dimensions 107.2 K ( t ) and N ( λ ) in Multiply-connected regions 11
8. Calculating spectrum from N ( λ ) N ( λ ) and K ( t ) 139.2 Calculating the state density from the spectrum 14
10. Conclusions 14
1. Introduction
A problem stemming from physics and soon becoming an important mathematical problemis to recover geometry of a manifold from the knowledge of the eigenvalues of a naturaldifferential operator. This problem originates in the theory of radiation (how to determinethe state density of the electromagnetic wave in a given cavity) and is formulated by Kacas ”Can one hear the shape of a drum?” [1].What the original problem asked is that how many eigenvalues are smaller than a givennumber λ . This problem is formulated as to seek the so-called counting function N ( λ ).For a given spectrum { λ n } , the counting function is directly defined to be N ( λ ) = the number of eigenvalues (with multiplicity) smaller than λ , (1.1)– 1 –.e., N ( λ ) = X λ n <λ . (1.2)In other words, the counting function represents the number of the eigenstates whoseeigenvalues are smaller than λ . Nevertheless, the counting function N ( λ ) is very difficultto calculate and there is no general method for calculating N ( λ ) in mathematics [2].This is because when calculating N ( λ ), one often encounters some unsolved problems innumber theory. For example, when calculating the counting function for the spectrum ofthe Laplace operator on a tori, one encounters the Gauss circle problem in number theory.The Hardy-Littlewood-Karamata Tauberian theorem gives the first term of the asymptoticexpansion of N ( λ ), but does not provide any information beyond the first-order term [2].For a given spectrum { λ n } , one also introduces the heat kernel, K ( t ) = X n e − λ n t , (1.3)which is an another important function describing the relation between the eigenvalues ofthe operator and the geometrical property of the manifold. The heat kernel is relativelyeasy to calculate and some methods for calculating heat kernels have been developed [3]. Inspecial, Elizalde et al. developed a very effective approach for calculating K ( t ) [4, 5, 6, 7, 8].There is no doubt that there must exist a relation between the counting function N ( λ )and the heat kernel K ( t ), since they are both determined by the same spectrum { λ n } .Nevertheless, the relation between N ( λ ) and K ( t ) is still an open problem in mathematics[2]. Though the definition of counting function N ( λ ), eq. (1.1) or (1.2), is comprehensiveand intuitive, other than the definition of heat kernel K ( t ), eq. (1.3), the definition of N ( λ )is not suitable for practical calculations. In fact, if starting from this definition to calculatethe counting function, one may always encounter the mathematical difficulty mentionedabove. In this paper, we will propose an expression for N ( λ ), which has a similar formwith the definition formula of K ( t ), eq. (1.3), and is more operable than the definition(1.2). This expression can be regarded as an alternative definition formula for N ( λ ).As a main result of this paper, we provide a relation between the counting function N ( λ ) and the heat kernel K ( t ). This problem interests us from both purely mathematicalviewpoint and practical viewpoint. This relation allows us to calculate N ( λ ) from K ( t )which, as mentioned above, is often relatively easier to calculate.The relation between the counting function N ( λ ) and the heat kernel K ( t ) presentedin this paper is an integral transformation. However, in practice the heat kernel is oftengiven in the form of a series expansion and in most cases only the first several expansioncoefficients can be obtained [3]. This requires us to integrate term by term. Nevertheless,the series expansion of K ( t ) is not convergent uniformly. As a result, the integral of someterms will be divergent. The counting function N ( λ ) is of course finite; the divergences arecaused by illegally integrating a nonuniformly convergent series term by term. For dealingwith this problem, we provide a renormalization procedure to remove the divergences.– 2 –n fact, the results of this paper provide two approaches for calculating the countingfunction N ( λ ): (1) by the expression for N ( λ ) presented in this paper, one can calculate N ( λ ) directly, and (2) by the relation between N ( λ ) and K ( t ), one can first calculatethe heat kernel K ( t ) and then calculate N ( λ ) from K ( t ). As a comparison, we calculate N ( λ ) for a D -dimensional box by both these two approaches, respectively.As an example of the calculation of counting functions by the relation between N ( λ )and K ( t ), we calculate N ( λ ) from K ( t ) for three-dimensional balls.As another example, we calculate the counting function from the heat kernel for theminus Laplace operator in a two-dimensional region with irregular shape and nontrivialtopology. For two-dimensional heat kernels, there are two known results: the heat kernelfor a simply-connected region [9] and a hypothesis made by Kac on the heat kernel for themultiply-connected case [1]. In the present paper, we first generalize the heat kernel fora simply-connected region given in [9] to the case of a multiply-connected region which isbounded by a smooth but irregular curve and with some holes. This result proves Kac’sconjecture. Then, we calculate the counting function for the two-dimensional multiply-connected region from the heat kernel by the relation between N ( λ ) and K ( t ).As applications, we discuss the problem that how to calculate the asymptotic expres-sions for eigenvalue spectra and state densities from the counting function N ( λ ). By N ( λ ), we construct an equation for eigenvalues and then provide an approximate solutionin which the eigenvalue is expressed as a function of heat kernel coefficients. Moreover, wealso provide two expressions for state densities by the relation between N ( λ ) and K ( t )and by the expression of N ( λ ) provided in the present paper.The counting function and the heat kernel are interesting in both mathematics andphysics:In mathematics, the relation between the spectrum of the Laplace operator on a Rie-mannian manifold and the geometry of this Riemannian manifold is an important subject[2, 10, 11, 12], and the problem of spectral asymptotics is one of the central problemsin the theory of partial differential operators [13]. One of the main problems is to seekthe asymptotic expansions of the counting function N ( λ ) and the heat kernel K ( t ). Thegeneral relation between N ( λ ) and K ( t ) is still unknown. Especially, there is no generalmethod for calculating the asymptotic expansion of the counting function N ( λ ).In physics, the spectrum of the Laplace operator on a Riemannian manifold can bedirectly applied to boundary quantum fields. Moreover, reconstructing the geometricalproperty of a system from an eigenproblem is an interesting and important problem. Forexample, Aurich et al. reconstruct the shape of the universe from the result of the eigen-problem [14]. The counting function is directly related to the Casimir effect, the spectrumproblem, and the state density, etc. Moreover, there are many studies on heat kernels.In quantum field theory, it is of crucial importance to evaluate the one-loop divergences,and the control of the ultra-violet divergences can be achieved by using heat kernel reg-ularization methods [3, 15, 16]. The heat kernel expansion becomes a standard tool inthe calculations of vacuum energies [17], the Casimir effect [18, 19], and quantum anoma-lies [20]. Now a large amount of research has been devoted to quantum gravity based onthe heat kernel expansion, including semiclassical approaches [21], black hole thermody-– 3 –amics [22], wormhole physics [23], heat kernels in curved space [24], and supergravitytheories [25, 26]. In addition, the heat kernel expansion is also used to study string theory[27, 28, 29] and noncommutative field theory [30, 31, 32]. The relation between the countingfunction and the heat kernel allows one to introduce the concept of the counting functioninto these fields. Moreover, there exist some indirect ways to construct the heat kernelwithout solving the heat equation, and the heat kernel coefficients for various manifoldshave been discussed [3, 6, 33, 34], which makes the heat kernel expansion a very convenientand effective tool. By the relation provided in the present paper, such results of the heatkernel can be directly applied to the problem of the counting function.In section 2, we provide an expression for the counting function N ( λ ), which is moreoperable compared to the definition (1.2) and can be regarded as an alternative defini-tion formula for the counting function. In section 3, we point out a relation between thecounting function N ( λ ) and the heat kernel K ( t ). In section 4, we provide a renormal-ization procedure for removing the divergences appearing in the calculation of N ( λ ) froma nonuniformly convergent series expansion of K ( t ). In section 5, we compare the twoapproaches for calculating N ( λ ) using a D -dimensional box as an example. In section6, we calculate the counting function of a three-dimensional ball from the heat kernel.In section 7, for two-dimensional cases, we first generalize the heat kernel in a simply-connected region to a multiply-connected region, and then calculate the counting functionin a multiply-connected region from the heat kernel. The result of the heat kernel in themultiply-connected region obtained in the present paper proves Kac’s hypothesis. In sec-tions 8 and 9, as applications of the counting function, we discuss how to calculate theasymptotic expressions for spectra and state densities from N ( λ ). The conclusions aresummarized in section 10.
2. An expression for the counting function: an alternative definition for-mula
The definition formula for the counting function N ( λ ), eq. (1.2), is not suitable for practi-cal calculations. In fact, starting from this definition to calculate N ( λ ), one may encountermany difficulties, e.g., the unsolved problems in number theory [2]. Contrarily, the defini-tion formula for the heat kernel, eq. (1.3), is more operable. In this section, we present anexpression for N ( λ ), which has a similar form with that of the heat kernel K ( t ), eq. (1.3).Such an expression can be regarded as an operable definition formula for N ( λ ). Theorem 1 N ( λ ) = lim β →∞ X n e β ( λ n − λ ) + 1 . (2.1) Proof.
Observing that lim β →∞ e β ( λ n − λ ) + 1 = ( , when λ n < λ, , when λ n > λ, – 4 –e have lim β →∞ X n e β ( λ n − λ ) + 1 = X λ n <λ N ( λ ) . Comparing with eq. (1.2), eq. (2.1) is obviously more operable, since eq. (2.1) convertsthe partial sum in eq. (1.2), P λ n <λ , into a sum over all possible values, P λ n < ∞ . Thiswill, of course, make the calculation easier.
3. The relation between counting function and heat kernel
In this section, we provide a relation between the counting function N ( λ ) and the heatkernel K ( t ). This relation is an interesting mathematical result and is useful for practicalcalculations. As mentioned above, the calculation of the counting function N ( λ ) is moredifficult than that of the heat kernel K ( t ). With the relation between N ( λ ) and K ( t ) givenin the following, one can first calculate K ( t ) and then calculate N ( λ ) from the result of K ( t ).The relation between N ( λ ) and K ( t ) is as follows. Theorem 2 K ( t ) = t Z ∞ N ( λ ) e − λt dλ. (3.1) Proof.
The generalized Abel partial summation formula reads X u <λ n ≤ u b ( n ) f ( λ n ) = B ( u ) f ( u ) − B ( u ) f ( u ) − Z u u B ( u ) f ′ ( u ) du, (3.2)where λ i ∈ R , λ ≤ λ ≤ · · · ≤ λ n ≤ · · · , and lim n →∞ λ n = ∞ . f ( u ) is a continuously dif-ferentiable function on [ u , u ] (0 ≤ u < u , λ ≤ u ), b ( n ) ( n = 1 , , , · · · ) are arbitrarycomplex numbers, and B ( u ) = P λ n ≤ u b ( n ). We apply the generalized Abel partial sum-mation formula, eq. (3.2), with f ( u ) = e − u ( s − s ) and b ( n ) = a n e − λ n s , where s , s ∈ C .Then A ( u , s ) − A ( u , s ) = A ( u , s ) e − u ( s − s ) − A ( u , s ) e − u ( s − s ) +( s − s ) Z u u A ( u, s ) e − u ( s − s ) du, (3.3)where A ( u, s ) = X λ n ≤ u a n e − λ n s . (3.4)Setting a n = 1 in eq. (3.4), we find A ( λ,
0) = X λ n ≤ λ N ( λ ) , the counting function, and A ( ∞ , t ) = X n e − λ n t = K ( t ) , – 5 –he heat kernel. By eq. (3.4), we also have A (0 , t ) = 0. Then, by eq. (3.3), we have K ( t ) = A ( ∞ , t ) − A (0 , t ) = t Z ∞ N ( λ ) e − λt dλ. (3.5)This is just eq. (3.1). Theorem 3 N ( λ ) = 12 πi Z c + i ∞ c − i ∞ K ( t ) e λt t dt, c > lim n →∞ ln nλ n . (3.6) Proof.
By the Perron formula, we have X µ n
4. Calculating N ( λ ) from an asymptotics of K ( t ) In principle, we can calculate the counting function N ( λ ) from the heat kernel K ( t ) orcalculate K ( t ) from N ( λ ) by the relation (3.6) or (3.1) directly. However, the relationgiven above is an integral transformation. Therefore, when the heat kernel K ( t ) is inthe form of a series expansion, practically, the above relation is available only when thisintegral transformation can be applied to each term of the series, or, in other words, thisseries must can be integrated term by term. If the series is not uniformly convergent,even though the integral of the sum function is convergent, the integrals of some of theterms may be divergent. In this case, we need a renormalization procedure to remove thedivergences. – 6 – .1 An asymptotics for N ( λ )Concretely, the expansion of heat kernel K ( t ) can be expressed as [6] K ( t ) = (4 πt ) − D/ ∞ X k =0 , , , ··· B k t k , (4.1)where B k is the heat kernel coefficient, D is the dimension of space. In fact, in practiceonly the first several heat kernel coefficients can be obtained, so we have to integrate termby term. Substituting eq. (4.1) into eq. (3.6) and exchanging the order of integration andsummation gives N ( λ ) = (4 π ) − D/ ∞ X k =0 , , , ··· B k I D ( k ) , (4.2)where I D ( k ) = 12 πi Z c + i ∞ c − i ∞ t − D/ k e λt t dt. (4.3)Nevertheless, for a power series, unless the series is uniformly convergent, it is not per-missible to integrate term by term, i.e., the order of integration and summation cannotbe exchanged. When calculating an integral of a series that is not uniformly convergentterm by term, one may find that the integrals of some terms diverge. In our case, it can bedirectly seen that only when k < D/ I D ( k ) is convergent. That is to say, the termwith k ≥ D/ The integral (4.3) is divergent when k ≥ D/ N ( λ ) from the series (4.2), we need to make sense of these divergent integrals. In otherwords, we need a renormalization procedure to remove the divergences.It can be directly seen that eq. (4.3) is the inverse Laplace transformation of thefunction t − D/ k − . Thus for k < D/ I D ( k ) = 12 πi Z c + i ∞ c − i ∞ t − D/ k − e λt dt = 1Γ (1 + D/ − k ) λ D/ − k , ( k < D/ . (4.4)The definition domain for the gamma function Γ (1 + D/ − k ) is k < D/ k ≥ D/ − D/ k − δ ( − D/ k − ( λ ) [41], where δ ( n ) ( λ ) is the n th derivative of δ ( λ ).When − D/ k − x ) = x Γ ( x + 1)[36]. Concretely, when defining the gamma function as Γ ( x ) = x Γ ( x + 1), the definitiondomain of the gamma function is changed to k < D/ N ( λ ) in two parts: N ( λ ) = N T ( λ ) + N L ( λ ) , (4.5)where N T ( λ ) = (4 π ) − D/ D + X k =0 , , , ··· B k I D ( k ) (4.6)is the convergent part and N L ( λ ) = (4 π ) − D/ ∞ X k = D +1 , D + , ··· B k I RD ( k ) (4.7)is the renormalized divergent part, where I RD ( k ) denotes the renormalized result. N L ( λ )is only a higher-order contribution to the counting function and is often negligible.After renormalization, eqs. (4.6) and (4.7) can be written as N ( λ ) = ∞ X k =0 , , , ··· C k λ D/ − k + ∞ X l =0 , , , ··· (4 π ) − D/ B D/ l δ ( l ) ( λ )= D X k =0 , , , ··· C k λ D/ − k + ∞ X k = D + , D + , ··· C k λ D/ − k + ∞ X k = D +1 , D +2 , ··· (4 π ) − D/ B k δ ( k − ( D +1 )) ( λ )(4.8)with the counting function coefficient C k = (4 π ) − D/ B k Γ (1 + D/ − k ) . (4.9)It should be emphasized that the delta-function terms in eq. (4.8) will contribute thezero-point energy; this is, such contributions will play an essentially important role in thecalculation of the Casimir effect.
5. Comparison of the two approaches for calculating N ( λ ) : D -dimensionalboxes The results presented in sections 2 and 3 show that there are two approaches for calculatingthe counting function N ( λ ). The first is to directly calculate N ( λ ) by eq. (2.1), theoperable expression of N ( λ ) presented in this paper; the second is to first calculate theheat kernel K ( t ) and then to calculate N ( λ ) from K ( t ) by the relation between N ( λ )and K ( t ). In this section, as an example, we calculate N ( λ ) by the above two approachesrespectively for a rectangular D -dimensional box.The spectrum of the minus Laplace operator of a D -dimensional rectangle box reads– 8 – n ,n , ··· ,n D = π (cid:18) n L + n L + · · · + n D L D (cid:19) , n i = 1 , · · · , (5.1)where L i is the side length.First, we directly calculate the counting function N ( λ ) from eq. (2.1): N ( λ ) = lim β →∞ X n ,n , ··· ,n D e β ( λ n ,n , ··· ,nD − λ ) + 1 . (5.2)By the Euler-MacLaurin formula ∞ P n =0 F ( n ) = R ∞ F ( n ) dn + F (0) − B F ′ (0) − B F ′′′ (0)+ · · · , where B , B , · · · are Bernoulli numbers, we achieve N ( λ ) = (4 π ) − D/ " V Γ ( D/ λ D/ − D X i =1 VL i √ π Γ ( D/ / λ D/ − / + D X i 6. Calculating N ( λ ) from K ( t ) : three-dimensional balls In this section, as an example, we calculate the counting function N ( λ ) from the seriesexpansion of the heat kernel K ( t ) for a three-dimensional ball. It should be emphasizedthat for calculating the heat kernel K ( t ), it is not needed to know the eigenvalue spectrumin advance [6].For a three-dimensional ball, from eq.(4.9), the expansion coefficients of the countingfunction N ( λ ) can be obtained as C k = (4 π ) − / B k Γ (5 / − k ) . (6.1)– 9 –he heat kernel coefficients for Dirichlet, Neumann, and Robin boundary conditions forthree-dimensional balls are calculated explicitly in [6].Take the case of Dirichlet boundary condition as an example. Ref. [6] gives the first 21heat kernel coefficients. From this, we can obtain the first 21 counting function coefficients C k by eq. (6.1). Only taking the tree-diagram-like part into account, from eq. (4.6), wehave N ( λ ) = 29 π R λ / − R λ + 23 π Rλ / − − π Rλ / . (6.2)It can be directly checked that the tree-diagram-like part provides the main contributionto the N ( λ ). 7. Multiply-connected cases: two-dimensional K ( t ) and N ( λ ) An interesting special case is about two-dimensional counting function and two-dimensionalheat kernel, which is just the original problem formulated by Kac as ”Can one hear theshape of a drum?”. The first result of this problem is proved by Weyl [37] and then im-proved by Pleijel [38] and Kac [1]. In [1], Kac pointed out a special relation between N ( λ )and K ( t ): in two dimensions, the leading-order term of N ( λ ) is accidentally equal to theleading-order term of K ( t ). However, beyond the leading order, there is no further result.Moreover, for the multiply-connected case, Kac made a hypothesis about the topologi-cal contribution on the heat kernel K ( t ): the contribution from the nontrivial topology(connectivity) is in proportion to the Euler-Poincar´e characteristic number.In this section, (1) with the help of the relation between N ( λ ) and K ( t ) given above, wegive a proof for Kac’s result: the leading-order term of N ( λ ) equals the leading-order termof K ( t ); (2) based on the Gauss-Bonnet theorem, we first generalize the simply-connectedheat kernel K ( t ) given in [9] to the multiply-connected case (this result proves Kac’s conjec-ture: in two dimensions, the topology contribution is proportion to the Euler-Poincar´e char-acteristic number), and, then, calculate the two-dimensional N ( λ ) in a multiply-connectedregion with the help of the relation between N ( λ ) and K ( t ). N ( λ ) and K ( t ) in two dimensions Using the relation between N ( λ ) and K ( t ), we first prove that when the number of eigen-states per unit interval (the density of eigenstates) ρ ( λ ) is a constant, N ( λ ) equals K ( t )in the limit λ → ∞ or t → 0, i.e., in the limit λ → ∞ or t → N ( λ ) = K (cid:18) λ (cid:19) or N (cid:18) t (cid:19) = K ( t ) . (7.1)The proof is straightforward. In the limit λ → ∞ or t → 0, the summations can beconverted into integrals: N ( λ ) = X λ n <λ Z λ ρ (cid:0) λ ′ (cid:1) dλ ′ , (7.2) K ( t ) = X n e − λ n t = Z ∞ ρ (cid:0) λ ′ (cid:1) e − λ ′ t dλ ′ . (7.3)– 10 – (cid:13)1(cid:13) a(cid:13) a(cid:13) a(cid:13) a(cid:13) a(cid:13) a(cid:13) a(cid:13) a(cid:13) in(cid:13) l(cid:13) out(cid:13) l(cid:13) out(cid:13) l(cid:13) in(cid:13) l(cid:13) m(cid:13) s(cid:13) ((cid:13)a(cid:13))(cid:13) ((cid:13)b(cid:13))(cid:13) Figure 1: Converting a multiply-connected region to a simply-connected region. If ρ ( λ ) = C , where C is a constant, then N ( λ ) = Cλ, (7.4) K ( t ) = Ct . (7.5)This proves eq. (7.1).In two dimensions, the state density for the eigenstate of the minus Laplace opera-tor is a constant, so the leading contribution of the counting function equals the leadingcontribution of the heat kernel. This is just the case appeared in Kac’s work [1]. K ( t ) and N ( λ ) in Multiply-connected regions For achieving the counting function in a multiply-connected region by the relation between K ( t ) and N ( λ ), we need to start with the heat kernel in a multiply-connected region. Fortwo-dimensional heat kernels, there are two known results: in the simply-connected case,ref. [9] provides a series expansion of the heat kernel; in the multiply-connected case, Kacmakes a hypothesis on the heat kernel [1]. In the following, based on the Gauss-Bonnettheorem, we first generalize the simply-connected result of heat kernel K ( t ) given in [9] tothe multiply-connected case. The result can be viewed as a proof of Kac’s hypothesis.In the simply-connected case, the heat kernel for a two-dimensional plane bounded bya smooth curve Γ s and with the Dirichlet boundary condition on Γ s reads [9] K ( t ) = S πt − L √ πt + 112 π Z Γ s k ( s ) ds + · · · , (7.6)where S is the area of the region, L is the length of Γ s , k ( s ) is the curvature of the curveΓ s at the point s .To generalize the result of the simply-connected region, eq. (7.6), to the multiply-connected case, we first convert the multiply-connected region bounded by Γ m (figure 1a)to a simply-connected one bounded by a piecewise smooth simple closed curve Γ s (figure1b). In the simply-connected region illustrated in figure 1b, the first two terms which areproportional to the area and the perimeter of the region, respectively, can be calculated– 11 –y the method in the simply-connected case directly. For the third term, by the Gauss-Bonnet theorem, we can calculate the integral of the curvature along Γ s . The Gauss-Bonnettheorem reads [39] X i ( π − a i ) + Z Γ s k ( s ) ds + Z Z Kdσ = 2 πχ, (7.7)where a i is the interior angle of Γ s at each vertex, K = 0 in the present case is the Gausscurvature, and χ is the Euler-Poincar´e characteristic number. From figure 1, since theintegrals along l ini and l outi ( i = 1 , · · · , r ) cancel each other in pairs, we achieve Z Γ m k ( s ) ds = Z Γ s k ( s ) ds, (7.8)where r is the number of holes in the region. Moreover, it is also easy to see that r X i =1 4 X α =1 ( π − a αi ) = r π. (7.9)The region bounded by Γ s in figure 1b is simply-connected, so χ = 1. Thus, from eq. (7.7),we have Z Γ s k ( s ) ds = 2 π (1 − r ) . (7.10)By the approach given by [20], we can calculate the first two terms. Then, with eqs. (7.6),(7.8), and (7.10), we have K ( t ) = S πt − L √ πt + 1 − r . (7.11)This is just the result that Kac hypothesized in [1].By the relation between N ( λ ) and K ( t ), we can directly calculate the counting func-tion in the two-dimensional multiply-connected region: N ( λ ) = S π λ − Lπ √ λ + 1 − r . (7.12) 8. Calculating spectrum from N ( λ ) From the counting function N ( λ ), we can directly obtain the asymptotic expression forthe spectrum of the system [40]. In view of the fact that one can obtain the heat kernel K ( t ) and, accordingly, obtain the counting function N ( λ ) without knowing the spectrumin advance [6], this result can serve as an approach for calculating the spectrum.The counting function is the number of eigenvalues smaller than λ , so for the n -theigenvalue λ n , we have N ( λ n ) = n. (8.1)Therefore, from eqs. (8.1) and (4.8), we have(4 π ) − D/ ∞ X k =0 , , , ··· B k Γ (1 + D/ − k ) λ D/ − kn = n. (8.2)– 12 –he eigenvalue λ n can be solved from eq. (8.2).For approximately solving eq. (8.2), we assume that λ n can be expanded as λ n = ∞ X m =0 α m n (2 − m ) /D . (8.3)Then we have λ n = 4 π Γ /D (cid:18) D (cid:19) B /D n /D − √ π Γ /D (cid:0) D + 1 (cid:1) D Γ (cid:0) D + (cid:1) B / B /D n /D + " D Γ (cid:0) D (cid:1) (cid:0) D + (cid:1) B / B − B B n + · · · . (8.4)As examples, we list some spectra obtained by this approach for various dimensionalballs with Dirichlet, Neumann, and Robin boundary conditions, respectively. The heatkernel coefficients are given in [6].For a three-dimensional ball, the spectrum for the Dirichlet and the Neumann or Robinboundary conditions is λ n = 32 (cid:0) π (cid:1) / R n / ± (cid:0) π (cid:1) / R n / + (cid:18) π − (cid:19) R , (8.5)where R is the radius. In this equation and following, the upper sign stands for the Dirichletboundary condition and the lower sign for the Neumann or Robin boundary condition.For a four-dimensional ball, the spectrum is λ n = 8 R n / ± √ 23 1 R n / − 269 1 R , (8.6)and for a five-dimensional ball, the spectrum is λ n = 12 (cid:16) √ π (cid:17) / R n / ± π 32 (3600 π ) / R n / + (cid:18) π − (cid:19) R . (8.7) 9. The state density In this section, we provide two approaches to achieve the state density based on the aboveresults. N ( λ ) and K ( t )The first approach is straightforward. The meaning of the counting function N ( λ ) is thenumber of the states whose eigenvalues are smaller than λ , so the state density reads ρ ( λ ) = dN ( λ ) dλ . (9.1)– 13 –hus, from eq. (4.8), we have ρ ( λ ) = (4 π ) − D/ ∞ X k =0 , , , ··· B k Γ ( D/ − k ) λ D/ − k − + ∞ X l =0 , , , ··· (4 π ) − D/ B D/ l δ ( l +1) ( λ ) . = (4 π ) − D/ D − X k =0 , , , ··· B k Γ ( D/ − k ) λ D/ − k − + (4 π ) − D/ ∞ X k = D + , D + , ··· B k Γ ( D/ − k ) λ D/ − k − + ∞ X k = D +1 , D +2 , ··· (4 π ) − D/ B k δ ( k − D ) ( λ ) (9.2)From this result, one can obtain the state density once he knows the heat kernel coefficients. An alternative way for obtaining the state density is to start with the expression of thecounting function given in section 2.Derivating both sides of eq. (2.1) by λ gives ρ ( λ ) = dN ( λ ) dλ = lim β →∞ X n βe β ( λ n − λ ) (cid:2) e β ( λ n − λ ) + 1 (cid:3) . (9.3)Or, approximately, we can convert the summation over n to an integral: ρ ( λ ) = lim β →∞ Z dn βe β ( λ n − λ ) (cid:2) e β ( λ n − λ ) + 1 (cid:3) . (9.4)Then, we can directly calculate the state density from the spectrum. 10. Conclusions In this paper, we reveal a relation between the counting function N ( λ ) and the heat kernel K ( t ) and provide an operable expression for N ( λ ). By the relation, one can calculatethe counting function from a known heat kernel, and vice versa . By the expression of thecounting function presented in this paper, one can achieve the counting function by a directcalculation.The relation between N ( λ ) and K ( t ) is an integral transformation and its inversetransformation. When calculating N ( λ ) from K ( t ) by this relation, however, one mayencounter the problem of divergence. This is because in most cases the expression of K ( t )is in the form of a power series (in fact, often only first several heat kernel coefficientscan be obtained), but the series is not uniformly convergent. As a result, when applyingthe integral transformation to the series, one needs to integrate term by term. However,to integrate term by term is not feasible: the integration of some terms will diverge. Forremoving the divergences, we develop a renormalization procedure.The results of this paper provide two approaches for calculating the counting functions:one is to calculate N ( λ ) directly from the expression of N ( λ ) given in section 2 and the– 14 –ther is based on the relation between N ( λ ) and K ( t ) given in section 3. As a comparisonbetween the two approaches, we calculate the counting function for D -dimensional boxesby these two approaches, respectively.As applications of the relation between N ( λ ) and K ( t ), we also calculate the count-ing functions for three-dimensional balls and two-dimensional multiply-connected irregularregions. 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