Concrete method for recovering the Euler characteristic of quantum graphs
CConcrete method for recovering the Euler characteristic ofquantum graphs ∗ Corentin L´ena † and Andrea Serio † Abstract
Trace formulas play a central role in the study of spectral geometry and in particularof quantum graphs. The basis of our work is the result by Kurasov which links the Eulercharacteristic χ of metric graphs to the spectrum of their standard Laplacian. These ideaswere shown to be applicable even in an experimental context where only a finite number ofeigenvalues from a physical realization of quantum graph can be measured.In the present work we analyse sufficient hypotheses which guarantee the successfulrecovery of χ . We also study how to improve the efficiency of the method and in particularhow to minimise the number of eigenvalues required. Finally, we compare our findings withnumerical examples—surprisingly, just a few dozens of eigenvalues can be enough. Keywords : quantum graphs, trace formula, Euler characteristic.
Schr¨odinger operators on metric graphs have long been studied as simplified models of complexquantum mechanical systems, notably large molecules [1, 2]. More recently, under the name of quantum graphs , they have been widely used to describe nano-wires, wave-guides and networks[3, Chapter 8]. They have also been investigated by mathematicians as a simplified setting inwhich to understand complex problems in spectral theory, such as quantum chaos [4], traceformulas [5, 4, 6, 7, 8] and nodal patterns [9]. From this point of view, they stand midway be-tween one-dimensional and multi-dimensional Schr¨odinger operators. Indeed, metric graphs areone-dimensional objects with a non-trivial geometry. The central theme in studying quantumgraphs is therefore the interaction of the spectrum with the geometry of the underlying metricgraph.
We consider in this work a finite, compact and connected metric graph Γ. Following [10], wedescribe it by a set of N edges { e , . . . , e N } , with e n = [ x n − , x n ], and a set of M vertices ∗ Accepted by “Journal of Physics A: Mathematical and Theoretical” (DOI: 10.1088/1751-8121/ab95c1). Workunder Creative Commons Attribution 4.0 licence . † Department of Mathematics, Stockholm University, 10691 Stockholm, Sweden.Emails: [email protected] , [email protected] . a r X i v : . [ m a t h . SP ] J un v , . . . , v M } , which is a partition of the set of endpoints { x p : 1 ≤ p ≤ N } . The associatedquantum graph is an operator in the Hilbert space L (Γ) = ⊕ Nn =1 L ( e n ) . More precisely, we define the standard Laplacian L Γ on Γ, which is the object of our study, inthe following way. Its domain Dom( L Γ ) consists of functions f ∈ L (Γ) satisfying1. for every 1 ≤ n ≤ N , f | e n ∈ W ( e n );2. for every 1 ≤ m ≤ M ,(a) f ( x p ) = f ( x q ) whenever the endpoints x p and x q belong to the vertex v m (continuitycondition),(b) (cid:80) x p ∈ v m ∂f ( x p ) = 0 (balance condition).In the balance condition, ∂f ( x p ) is the derivative along the edge with endpoint x p , taken towardsthe inside of the edge. The conditions in (ii) are called the standard vertex conditions. Theaction of L Γ on f ∈ Dom( L Γ ) is then given by the differential expression L Γ f ( x ) = − f (cid:48)(cid:48) ( x ) forall n ∈ { , . . . , N } and x in the interior of e n .It is known that the operator L Γ is self-adjoint, non-negative and has compact resolvent.Its spectrum therefore consists of a sequence of non-negative eigenvalues ( λ j ) j ≥ of finite mul-tiplicity tending to + ∞ (we repeat them according to their multiplicities). Let us note that λ is always 0 and that the hypothesis that Γ be connected implies that it is simple, with theassociated eigenfunction proportional to the constant function 1. For the sake of convenience,we will work in the rest of the paper with the eigenfrequencies k j = (cid:112) λ j .Our goal is to recover information about the geometry of the graph from the knowledge ofsome of its eigenfrequencies. This is a variation on Kac’s famous problem “Can one hear theshape of a drum?” [11], with some specificities that we will address below. The analogue toKac’s problem in our setting would be showing that two graphs with the same spectrum areisometric. There are partial results in that direction. For instance, the eigenfrequencies of aquantum graph satisfy Weyl’s law: as j → + ∞ , k j ∼ πj/ L , where L is the total length of thegraph, i.e. the sum of the lengths of all its edges (see for instance [12]). The total length ofthe graph can therefore be read off the complete spectrum. Furthermore, the answer to Kac’sproblem is positive provided all edge-lengths of the graph are rationally independent [6, 7].Nevertheless, if this last condition is not satisfied, we can find graphs with the same spectrumthat are not isometric [6, 13, 14, 15].We will restrict ourselves to recovering the main topological invariant of a connected graph:its Euler characteristic χ , defined by χ = M − N . It is related to β , the number of independentcycles in the graph, by β = 1 − χ . Let us point out that β can also be recovered from thenumber of zeros of the eigenfunctions, as shown in [16]. These numbers are however less easilyaccessible to experiments. We focus on a situation that can be realized experimentally. A microwave network can bedescribed to a good approximation by a quantum graph, whose eigenfrequencies correspond toresonances of the system. In [17] the authors describe the measurement of about 50 resonancesof such networks and use this experimental result to recover χ . Contrary to expectations, theyfind this relatively small number of eigenfrequencies to be sufficient. Their computation is basedon a trace formula as we will see below in more details. Extending the work in this paper, we2ant to check the correctness of the method and modify it in order to improve the efficiency,or namely to reduce the number of eigenfrequencies required.The total length of the graph is a continuous parameter and an infinite number of eigen-frequencies are needed to compute it using Weyl’s law, as described above. By contrast, χ isinteger valued and we can hope to compute it with only a finite number of them. However,some assumptions on the graph are needed. Indeed, by attaching a very small loop to someedge in an existing graph, we decrease χ by 1 while altering the first 50 eigenfrequencies by anarbitrarily small amount. We come back to this point at the end of Section 5. Therefore, ourgoal will also be to determine a reasonable set of assumptions that guarantees the success ofour methods. All our methods for computing the Euler characteristic are based on the trace formula in [7, 8](building on previous work in [5, 4, 6]). This formula connects the spectrum of L Γ , defined inthe previous section, with the set of periodic orbits in Γ, to be described below. More precisely,the spectrum is described by the distribution u ( k ) := 2 δ ( k ) + (cid:88) k j > (cid:16) δ ( k − k j ) + δ ( k + k j ) (cid:17) , (1)which is a tempered distribution of order 0. We denote by P the set of closed continuous pathsin Γ that are allowed to turn back on themselves only at vertices, which we call periodic orbits ,or orbits . If p ∈ P , we denote by prim( p ) any shortest orbit such that p can be obtained byrepeating prim( p ). Thus any two such orbits have the same length, hence (cid:96) (prim( p )) is welldefined. The following quantity will play a crucial role in our methods: (cid:96) min := min { (cid:96) ( p ) : p ∈ P} . (2) Observation . Notice that (cid:96) min is given by the minimum between the length of the shortestloop and twice the length of the shortest edge.
Proposition 2.2 (Theorem 1 from [7, 8], also [10]) . We have u ( k ) = χδ ( k ) + L π + 1 π (cid:88) p ∈P (cid:96) ( prim ( p )) S V ( p ) cos k(cid:96) ( p ) , (3) where S V ( p ) is the product of all the vertex scattering coefficients along the path p . Theconvergence of the series occurs in the space S (cid:48) ( R ) of tempered distributions.Remark . The terms S V ( p ) will ultimately drop out of our formulas due to our choice of testfunctions. Since they will not play a role in our analysis we just refer to [7] for their definition. Remark . Taking the Fourier transform on both sides of Equation (3), we obtainˆ u ( (cid:96) ) = χ + 2 L δ ( (cid:96) ) + (cid:88) p ∈P (cid:96) (prim( p )) S V ( p ) (cid:16) δ ( (cid:96) − (cid:96) ( p )) + δ ( (cid:96) + (cid:96) ( p )) (cid:17) , (4)which is the form of the Trace Formula that we will be using in the rest of the paper.3n all the paper, we define the Fourier transform so that ˆ f ( k ) = (cid:82) + ∞−∞ e ik(cid:96) f ( (cid:96) ) d(cid:96) for any f ∈ L ( R ). By definition of the Fourier transform of a tempered distribution, for ϕ in theSchwartz class S ( R ), ˆ u [ ϕ ] = u [ ˆ ϕ ], which means, using Equations (1) and (4), χ (cid:90) + ∞−∞ ϕ ( (cid:96) ) d(cid:96) + 2 L ϕ (0) + (cid:88) p ∈P (cid:96) (prim( p )) S V ( p ) ( ϕ ( − (cid:96) ( p )) + ϕ ( (cid:96) ( p )))= 2 ˆ ϕ (0) + (cid:88) k j > (cid:16) ˆ ϕ ( k j ) + ˆ ϕ ( − k j ) (cid:17) . (5)Equation (5) also holds for some functions not in S ( R ). For our purpose, we will consider theset T of functions ϕ satisfying the following assumptions:1. ϕ is continuous on R and nonnegative;2. supp( ϕ ) ⊆ [0 , T ] for some T > (cid:82) + ∞−∞ ϕ ( (cid:96) ) d(cid:96) = 1;4. the function F ϕ is integrable on (0 , + ∞ ), where F ϕ ( k ) := sup y ≥ k |(cid:60) ( ˆ ϕ ( y )) | .Let us make some comments on Condition (iv). The function F ϕ can be seen as the smallestnon-increasing majorant of |(cid:60) ( ˆ ϕ ) | on (0 , + ∞ ). Its integrability is used both to express χ as thesum of a convergent series and to prove error estimates. Proposition 2.5. If ϕ ∈ T , χ + (cid:88) p ∈P (cid:96) ( prim ( p )) S V ( p ) (cid:16) ϕ ( − (cid:96) ( p )) + ϕ ( (cid:96) ( p )) (cid:17) = 2 ˆ ϕ (0) + 2 (cid:88) k j > (cid:60) ( ˆ ϕ ( k j )) . (6)We prove the above proposition in A. Let us note that whenever ϕ is in T , so is ϕ t ( l ) := tϕ ( t(cid:96) ), for any t > Proposition 2.6.
For any ϕ ∈ T such that T = 1 in (ii) then χ = 2 ˆ ϕ t (0) + 2 (cid:88) k j > (cid:60) ( ˆ ϕ t ( k j )) ∀ t ≥ /(cid:96) min , (7) where (cid:96) min is defined in (2).Proof. Since t ≥ /(cid:96) min , the support of ϕ t is contained in [0 , (cid:96) min ]. Equation (7) then followsimmediately from Equation (6).In the rest of the paper, we assume that t ≥ /(cid:96) min . When we will analyse our method, itwill accordingly be important to know a lower bound of (cid:96) min in order to guarantee a successfulestimate of χ . On the other hand, from the above proposition we also have the followingcorollary, cf. [18]. Corollary 2.7.
For any ϕ ∈ T then χ = lim t →∞ ϕ t (0) + 2 (cid:88) k j > (cid:60) ( ˆ ϕ t ( k j )) . (8)4he limit and summation signs in (8) cannot be exchanged since lim t →∞ (cid:60) ( ˆ ϕ t ( k )) = ˆ ϕ (0)independently of k . Hence it is clear that (8) cannot be approximated by the limit of a finite sumover the smallest, say J , eigenfrequencies. Because of Assumption (iv) on ϕ , the summation willbe shown to be absolutely converging (see Section 3). Based on that, our plan is to quantify foreach t how many eigenfrequencies J are necessary for the truncated summation to differ from(8) by less than 1 /
2, and hence the nearest integer to the sum to coincide with χ . Theorem 4.1gives an optimal (minimal) number J ∗ of required eigenfrequencies, based on the choice of t .As one can expect the worse the estimate of (cid:96) min (smaller lower bound), hence the larger t is,the larger the number J ∗ of required eigenfrequencies. Remark . The above trace formula holds for the standard Laplacian, specifically for standardvertex conditions imposed at all the vertices. In [19, 20] are presented versions of the traceformula for more general vertex conditions where in particular the constant term in ˆ u maydiffer from χ . Hence our method might not be applicable in presence of vertex conditions otherthan standard. It is even possible that χ cannot be recovered knowing the whole spectrum ofthe graph: see [13, 14, 15] for an example of two graphs with mixed vertex conditions whichhave distinct values of χ despite being isospectral. Let us apply Formula (7) to a triangular function: ψ ( (cid:96) ) := 4 (cid:96) [0 , / ( (cid:96) ) + 4(1 − (cid:96) ) [1 / , ( (cid:96) ) . Everywhere in the paper, A stands for the characteristic function of the set A . We findˆ ψ ( k ) := (cid:90) ψ ( (cid:96) ) e ik(cid:96) d(cid:96) = e ik/ (cid:18) sin( k/ k/ (cid:19) . Proposition (2.6) then gives us the following expression for χ , assuming t ≥ /(cid:96) min : χ = 2 + 2 (cid:88) k j > cos( k j / t ) (cid:18) sin( k j / t ) k j / t (cid:19) . (9)We can also apply the same formula to a truncated trigonometric function: ϕ ( (cid:96) ) := (1 − cos(2 π(cid:96) )) [0 , ( (cid:96) )We findˆ ϕ ( k ) := (cid:90) (cid:18) − e iπ(cid:96) − e − iπ(cid:96) (cid:19) e ik(cid:96) d(cid:96) = − π k ( k − π )( k + 2 π ) e ik/ sin( k/ χ , valid whenever t ≥ /(cid:96) min : χ = 2 − π (cid:88) k j > sin( k j /t ) k j /t ( k j /t − π )( k j /t + 2 π ) . (10)This formula is the one employed in [17]. 5n order to recover χ from a finite set of eigenfrequencies, we have to replace the series on theright-hand side of Formulas (9) and (10) by partial sums. Since the terms in the second series goto zero more quickly than in the first, we can expect that fewer eigenfrequencies are necessaryto find a close approximation of χ , which we will then be rounded (up or down) to the nearestinteger. Both numerical trials and the analysis of the experimental results in [17] confirm this.The more regular ϕ is, the faster ˆ ϕ decays at infinity. We can then wonder how much we candecrease the number of eigenfrequencies used to compute χ . To study this question, we willconsider the one-parameter family of test functions ϕ d ( (cid:96) ) = C d (1 − cos(2 π(cid:96) )) d [0 , ( (cid:96) ) , where d is a positive integer, which we call order , and C d is chosen so that (cid:82) + ∞−∞ ϕ d ( (cid:96) ) d(cid:96) = 1.We easily check that the function ϕ d is of class C d − on R . We show in B that C d = 2 d ( d !) (2 d )!and that the Fourier transform of ϕ d isˆ ϕ d ( k ) = ( − d ( d !) π Π dj = − d ( k/ (2 π ) + j ) e ik/ sin( k/ . Our goal is to give rigorous error estimates and to perform numerical trials for this family offunctions.
For a fixed graph Γ, we assume that only the smallest J eigenvalues of the standard Laplacianare known. We consider the distribution (1) evaluated on the Fourier transform of a genericfunction ϕ ∈ T and split the expression in the following way u [ ˆ ϕ t ] = 2 (cid:88) j ≤ J (cid:60) ( ˆ ϕ t ( k j )) (cid:124) (cid:123)(cid:122) (cid:125) S J ( t ) + 2 (cid:88) j>J (cid:60) ( ˆ ϕ t ( k j )) (cid:124) (cid:123)(cid:122) (cid:125) R J ( t ) . (11)We call S J ( t ) the truncated sum and R J ( t ) the remainder or tail . We have | S J ( t ) − χ | ≤ | u [ ˆ ϕ t ] − χ | + | R J ( t ) | . (12)Let ϕ ∈ T such that T = 1 in Condition (ii) be fixed. Because of Proposition 2.6 when t ≥ /(cid:96) min the term | u [ ˆ ϕ t ] − χ | is zero, hence | S J ( t ) − χ | ≤ | R J ( t ) | . | R J ( t ) | ≤ (cid:88) j>J |(cid:60) ˆ ϕ ( k j /t ) | (13) ≤ (cid:88) j>J sup k ≥ k j |(cid:60) ˆ ϕ ( k/t ) | = 2 (cid:88) j>J F ϕ ( k j /t ) , (14)6here F ϕ , is defined in Condition (iv) of T . In particular, F ϕ is positive decreasing to zero .At this point we employ the classical eigenvalue estimate k j ≥ ( j − M ) π/ L (see for instance[21]) and we bound the summation over the integers of a monotone function by its integral. | R J ( t ) | ≤ (cid:88) j>J − M F ϕ (cid:18) π L jt (cid:19) (15) ≤ (cid:90) ∞ J − M F ϕ (cid:16) π L xt (cid:17) dx (16)= 2 L tπ (cid:90) ∞ π L t ( J − M ) F ϕ (x) d x =: E ϕ ( J − M, L t ) . (17)The function E ϕ : R → R + is continuous, positive, and monotone with respect to both ar-guments. In particular E ϕ ( J − M, · ) is strictly monotone increasing, while E ϕ ( · , L t ) is strictlymonotone decreasing to zero. Since χ is an integer number (generally negative) it is enough toensure that | R J ( t ) | < / χ . Therefore we define J min ( M, L t ) := min { J ∈ N : E ϕ ( J − M, L t ) < / } , this is the minimum number of eigenvalues needed for computing χ .This can be further extended to the case where only partial information on Γ is known. Theresult is summarised in Theorem 3.1.
Let Γ be a graph which • has at most M vertices, i.e. M ≤ M ; • has total length at most L , i.e. L ≤ L ; • has its shortest orbits at least (cid:96) min long, i.e. (cid:96) min ≥ (cid:96) min .Let ϕ ∈ T with T = 1 and let t = 1 /(cid:96) min . Then J ≥ J min = J min ( M , L t ) eigenvalues of thestandard Laplacian on Γ suffice for recovering the Euler characteristic of Γ via the formula χ (Γ) = nint [ S J ( t )] , (18) where nint( x ) denotes the nearest integer function.Proof. Because t = 1 /(cid:96) min ≥ /(cid:96) min then Proposition 2.6 implies | χ − u ( ˆ ϕ t ) | = 0, hence (12)reads | χ − S J ( t ) | ≤ | R J ( t ) | ≤ E ϕ ( J − M, L t ). Because of the monotonicity of E ϕ , we have E ϕ ( J − M, L t ) ≤ E ϕ ( J − M , L t ). Hence for J ≥ J min it follows that | χ − S J ( t ) | < /
2, and thuswe obtain (18).Notice that the cases S J ( t ) = χ ± / · ] over the half integers.We notice that the hypothesis of the above theorem cannot be relaxed too much. In partic-ular, the condition (cid:96) min ≥ (cid:96) min is crucial, because adding small loops to any graph perturbs itssmallest eigenvalues only a little, but at the same time it does dramatically affect its topologicalstructure, decreasing χ by 1.We also notice that the quantity L can be obtained from other information on the graphwhen available. For instance if the number of edges of the graph is at most N then L ≤ min ≤ j ≤ J ( j + N ) π/k j . Remark . The above Theorem 3.1 can be reproduced after replacing F ϕ by any other ma-jorant function monotone decreasing. The J min computed will be different. We will use thisremark in Section 4. This follows immediately from the fact that (cid:60) ˆ ϕ has only isolated zeros. Indeed, it is a real-analytic functionwhich is not identically zero. .2 Calculation with approximate spectrum Since we have in mind to apply our method in an experimental context we need to consider thecase where the eigenvalues are known only up to some approximation. We denote by { (cid:101) k j } Jj =1 the approximate spectrum where each eigenfrequency is measured with precision δ , meaningthat (cid:12)(cid:12)(cid:12)(cid:101) k j − k j (cid:12)(cid:12)(cid:12) ≤ δ for all 1 ≤ j ≤ J . Hence we define (cid:101) S J ( t ) = 2 (cid:80) j ≤ J (cid:60) ˆ ϕ t (˜ k j ) the truncatedsum evaluated at the approximated eigenvalues, hence our new aim is to approximate χ from (cid:101) S J ( t ). Therefore to the error contribution E ϕ ( J − M, L t ) we need to add a new term (cid:12)(cid:12)(cid:12) (cid:101) S J ( t ) − χ (cid:12)(cid:12)(cid:12) ≤ E ϕ ( J − M, L t ) + (cid:12)(cid:12)(cid:12) (cid:101) S J ( t ) − S J ( t ) (cid:12)(cid:12)(cid:12) . (19)This second term can be bounded linearly in δ in the following way (cid:12)(cid:12)(cid:12) (cid:101) S J ( t ) − S J ( t ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j ≤ J (cid:60) ( ˆ ϕ ( (cid:101) k j /t ) − ˆ ϕ ( k j /t )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (20) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j ≤ J (cid:90) (cid:16) cos (cid:101) k j (cid:96)/t − cos k j (cid:96)/t (cid:17) ϕ ( (cid:96) ) d(cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (21) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j ≤ J (cid:90) − (cid:101) k j − k j ) (cid:96) t sin ( (cid:101) k j + k j ) (cid:96) t ϕ ( (cid:96) ) d(cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (22) ≤ δt (cid:88) j ≤ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) sin ( (cid:101) k j + k j ) (cid:96) t ϕ ( (cid:96) ) d(cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (23) ≤ δJt . (24)Therefore we define (cid:101) E ϕ ( J − M, L , t ) := E ϕ ( J − M, L t ) + 2 δJt . (25)For this new function there does not necessarily always exists J such that (cid:101) E ϕ ( J − M, L , t ) < / • E ϕ ( J − M, L t ) < / • δJ/t < / • J ≥ (cid:101) J min ( M, L t ) := min { J ∈ N : J > M, E ϕ ( J − M, L t ) < / } ; (26) • δ ≤ (cid:101) δ max ( t, J ) := t/ (8 J ) . (27)If the first J eigenvalues are known such that both the above conditions are satisfied then we areable to compute the Euler characteristic via the approximated spectrum. The above analysiscan be summarised in the following Theorem 3.3.
Let Γ , ϕ and t be as in the hypotheses of Theorem 3.1, and moreover let thefirst J eigenvalues of the standard Laplacian on Γ be known up to error δ . Assume that J and δ satisfy both conditions (26) and (27) with M, L replaced by M , L . Then the Euler characteristicof Γ can be recovered via the following formula χ (Γ) = nint (cid:104) (cid:101) S J ( t ) (cid:105) . (28)8 Example of a family of test functions
We now apply the analysis in Section 3 to the test functions ϕ d introduced at the end of Section2. In the rest of the section, we assume that t = 1 /(cid:96) min . We have (cid:60) ( ˆ ϕ d ( k )) = ( − d ( d !) sin( k )2 π Π dj = − d ( k/ (2 π ) + j ) , so that, when k > πd , F ϕ ( k ) ≤ ( d !) π ( k/ (2 π ) − d ) d +1 . Let us write R d,J ( t ) for the remainder denoted in Section 3 by R J ( t ), in the case where ˆ ϕ = ˆ ϕ d .According to Remark 3.2, we find, for J > t L d + M , | R d,J ( t ) | ≤ t L π (cid:90) + ∞ πt L ( J − M ) ( d !) π ( k/ (2 π ) − d ) d +1 dk = ( d !) (2 t L ) d +1 πd ( J − M − t L d ) d . Using the inequality d ! ≤ √ πdd d e − d +1 / (see for instance Formula (3.9) in [22]), we find that | R d,J ( t ) | ≤ ε ( M, ρ, d, J ) , (29)with ρ := 2 t L and ε ( µ, γ, α, β ) := α α e − α +1 / γ α +1 ( β − µ − γα ) α . (30)Let us note that we have replaced the discrete (integer) values d and J by the continuousparameters α and β . Theorem 4.1.
For fixed M and ρ positive and ε ∈ (0 , , there exists J ∗ , the smallest amongthe integers J such that ε ( M, ρ, d, J ) ≤ ε is satisfied for some d . There exists then d ∗ , thesmallest positive integer d such that ε ( M, ρ, d, J ∗ ) ≤ ε . Furthermore, these integers are givenby the following formulas J ∗ = min {(cid:100) β ( ε, M, ρ, d ) (cid:101) : d ∈ {(cid:98) α ∗ ( ε, ρ ) (cid:99) , (cid:100) α ∗ ( ε, ρ ) (cid:101)}} (31) and d ∗ = min { d : d ∈ { , . . . , (cid:100) α ∗ ( ε, ρ ) (cid:101)} and (cid:100) β ( ε, M, ρ, d ) (cid:101) = J ∗ } , (32) where β ( ε, M, ρ, d ) = M + ρd e (cid:32) e / ρε (cid:33) / (2 d ) and α ∗ ( ε, ρ ) = 12(1 + W (1)) ln (cid:32) e / ρε (cid:33) . emark . We denote by W (1) the unique positive number satisfying W (1) e W (1) = 1. Thenotation W (1) is standard and denotes the value of the Lambert W-function at 1 (see [23, Eq.4.13.1]). Remark . Existence of the integers J ∗ and d ∗ follows immediately from the fact that β (cid:55)→ ε ( M, ρ, d, β ) decreases to 0 for any positive integer M and d and any ρ >
0. Our actual goal isthus to prove Equations (31) and (32), which we use to compute them.
Proof of Theorem 4.1.
It follows from Equation (30) that the unique β ∈ ( M + ρα, + ∞ ) whichsolves the equation ε ( M, ρ, α, β ) = ε is β ( ε, M, ρ, α ) = M + ρα e (cid:32) e / ρε (cid:33) / (2 α ) . (33)This implies that, for any positive integer d , the smallest integer J such that ε ( M, ρ, d, J ) ≤ ε is equal to (cid:100) β ( ε, M, ρ, d ) (cid:101) . Differentiating Equation (33) with respect to α , we find ddα β ( ε, M, ρ, α ) = ρ e (cid:32) e / ρε (cid:33) / (2 α ) (cid:32) − α ln (cid:32) e / ρε (cid:33)(cid:33) . Using the substitution s := 12 α ln (cid:32) e / ρε (cid:33) − , we see that this derivative vanishes if, and only if, se s = 1. This equation has a uniquepositive solution which is, by definition, s = W (1). Therefore, the derivative vanishes only at α = α ∗ ( ε, ρ ), with α ∗ ( ε, ρ ) = 12(1 + W (1)) ln (cid:32) e / ρε (cid:33) . We can easily check that the derivative is negative below that value and positive above, sothat α (cid:55)→ β ( ε, M, ρ, α ) is first decreasing and then increasing. The function d (cid:55)→ β ( ε, M, ρ, d )takes its smallest value among all positive integers for the largest integer below α ∗ ( ε, ρ ) or thesmallest integer above. Equations (31) and (32) then follow easily. We have numerically tested our method on some graphs for which we have computed the exactspectrum up to machine error precision and then added an error term uniformly distributedwith prescribed variance. In this section we give an overview of the implemented algorithm andshow a few examples of its application. We recall the quantities that appear in the numericalanalysis: • t is the rescaling parameter of the test function; • d, ( d ∗ ) is the order of the test function ϕ d , (respectively the minimal order according toequation (32)); 10 J, ( J ∗ ) is the number of the first smallest eigenfrequencies used to compute S J ( t ), (respec-tively the minimal number according to Equation (31)); • S J ( t ) is the truncated sum defined in (11) computed on the first J eigenfrequencies of theexact spectrum where the test function is rescaled by t ; • (cid:101) S J ( t ) is the same truncated sum as S J ( t ), but computed on a sample of the spectrumaffected by a certain error term.In view of the result of Proposition 2.6 one might expect to obtain a good approximation of χ by computing the limit as t → ∞ of the right hand side of (7). Unfortunately this works outonly if all eigenvalues are taken into account in the computation. The code consists of three distinct parts each of them developed to run on the software Math-ematica [24].1. The first part is a GUI for building a graph with a drawing, drag-and-drop method andinserting the lengths of the edge.2. The second part contains an algorithm which computes the eigenvalues of the standardLaplacian of the graph in input. This is based on the von Below formula [25] which holdsfor equilateral metric graphs, cf. [12, 10]. This code shares parts with an other study in[26] on how this method can be extended to approximate eigenvalues of non equilateralmetric graphs. Several samples of approximate spectra { (cid:101) k j } j ∈ N are realized by summinga pseudo-random number uniformly distributed over a prescribed interval to the exactspectrum: (cid:101) k j = k j + e j , e j ∼ U ([ − δ, + δ ]).3. The last part computes the parameters d ∗ , J ∗ , and Formula (18), plots the maps t, J (cid:55)→ S J ( t ) and several instances of the the maps t, J (cid:55)→ (cid:101) S J ( t ) over different samples of spectrawith error terms, and compares the error bound ε ( M, t L , d, J )+2 δJ/t with the numericalerrors | (cid:101) S J ( t ) − χ | over different values of t and J . Figure 1: On the left the Lasso graph, χ = 0, the loop has length (cid:96) = 1 and the pendant edgehas length (cid:96) = 5, hence (cid:96) min = 1, ρ = 12 and d ∗ = 1; on the right the graph K , χ = −
5, eachedge has assigned length one, hence (cid:96) min = 2, ρ = 10 and d ∗ = 1. The lengths of the edges arenot to scale. 11ere we present some computations of S J ( t ) and (cid:101) S J ( t ) under different values of J and t compared with χ performed for the graphs Lasso and K (see Figure 1). We used the testfunction of optimal order d ∗ , equal to 1 in both cases. - - - - - Figure 2: Plots for the Lasso graph: In the first three plots J = J ∗ = 48 is fixed and the value t = 1 /(cid:96) min = 1 is shown with the dashed vertical line. In the bottom right plot t = 1 /(cid:96) min = 1is fixed. The samples of approximate spectra are computed with δ ≈ . · − .Let us explain how the numerical results are displayed in Figures 2 and 3. Top left : the function t (cid:55)→ S J ( t ) (in black) is compared withseveral instances of t (cid:55)→ (cid:101) S J ( t ), corresponding to different samples of the approximatespectrum. The curves are not visibly distinguishable despite the different error terms. Top right : the function t (cid:55)→ S J ( t ) is compared for two different test functions, ϕ d (black solidcurve) and the triangular function ψ (blue dashed curve). Bottom left :several instances of the error t (cid:55)→ | χ − (cid:101) S J ( t ) | (purple solid curves) are compared with thetheoretical bounds t (cid:55)→ ε ( M, ρ, d, J ) + 2 δJ/t (the red thick dashed curve, with the thincurves representing the two addends) We recall that we have assumed t ≥ /(cid:96) min when wecomputed the theoretical bounds. Bottom right :several instances of the error J (cid:55)→ | χ − (cid:101) S J ( t ) | (purple dots) are compared with the theo-retical bound J (cid:55)→ ε ( M, ρ, d, J ) + 2 δJ/t (the red thick dashed curve, with the thin curvesrepresenting the two addends).
Observation . We now gather some observations which hold for both Figures 2 and 3.12 .2 0.4 0.6 0.8 1.0 1.2 1.4 t - - -
22 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t - - -
41. 45. 50.50. 100. 500. 1000. J10 - Figure 3: Plots for the graph K : in the first three plots J = J ∗ = 41 is fixed and the value t = 1 /(cid:96) min = 0 . t = 1 /(cid:96) min = 0 . δ ≈ . · − .1. In the two plots at the top right we notice that the black curve approximates χ withhigher precision than the blue curve, i.e. it stays within distance 1 / χ for a certaininterval in t . This shows that the method with ϕ d is significantly more efficient comparedwith ψ in terms of number of eigenfrequencies sufficient in order to compute χ .2. The two plots at the bottom show that the theoretical bound exceeds the numerical errorby at least a factor 10 . This phenomenon can be explained by the several cancellationsoccurring inside the tail R d,J ( t ) which we did not take into account in (29).3. In the bottom left plots it can be observed that both the theoretical bound and thenumerical error grow with respect to the parameter t , indicating that it is important forthe effectiveness of the method to have a good lower bound of (cid:96) min .4. In the bottom right plot we observe that for J growing the numerical error stabilizes atsmall positive values different for different samples. This suggests that the precision doesnot improve by taking J (cid:29) J ∗ and that the error mainly depends on the contributionfrom the first terms in the summation (cid:101) S J ( t ).5. The theoretical bound shows to be even worse with respect to the numerical error for J (cid:29) J ∗ , this can be explained by the decreasing magnitude of the Fourier coefficients ofthe test function ϕ inside | (cid:101) S J ( t ) − S J ( t ) | , which we did not take into account in (20–24). We compare the numerical computation of χ for the following three equilateral graphs K , K with one edge made pendant, i.e. disconnected at one end-point, and K , . These graphshave Euler characteristic respectively χ = − , − −
3, they have all three been constructed13quilateral with each edge of length one. The first two graphs are chosen to have very closespectra. We can numerically observe that only 1 in 5 of their eigenfrequencies differ significantly(figure 4). - - - Figure 4: For each graph in figure we plot the functions t (cid:55)→ S J ( t ) with d = 1 and J = 30: K , (blue dotted curve), K with one endpoint disconnected (red dashed curve) and K (green solidcurve). From Theorem 4.1 we can notice that the values of d ∗ and J ∗ − M = (cid:100) β ( ε, M, ρ, d ) − M (cid:101) dependonly on the parameters ρ and ε . Following the idea described in Section 3.2, we fix ε = 1 / ρ , we compute the values of d ∗ and bounds of J ∗ − M . The resultsare shown in the following table. ρ d ∗ J ∗ − M ≤ ρ ≤ . ≤ J ∗ − M ≤ . ≤ ρ ≤ . ≤ J ∗ − M ≤ . ≤ ρ ≤
421 2 70 ≤ J ∗ − M ≤ ≤ ρ ≤
423 2 or 3 2912 ≤ J ∗ − M ≤ ≤ ρ ≤ ≤ J ∗ − M ≤ Cases for which d ∗ > ρ > . L > . · (cid:96) min . This depends on two factors: thedistribution of the length of the edges and the number of edges. We can identify two extremecases. • The distribution of the edge length has large variance, i.e. the length of the shortest edgeis very small compared to the total length. For instance, a Lasso graph with the loop atleast 8 times shorter than the pendant edge. • The distribution of the edge length has null, or very small variance, but the number ofedges is relatively large, at least 17 if the graph does not have loops. For instance, theequilateral K . A Proof of Proposition 2.6
A priori, we only know that Formula (5) holds for test functions ϕ in the Schwartz class. Inorder to extend it to functions in the set T , we use a standard argument of regularization byconvolution. Let us give more details. 14e choose a function θ : R → R of class C ∞ , non-negative, whose support is in [ − , θ is even and that (cid:82) + ∞−∞ θ ( (cid:96) ) d(cid:96) = 1. It follows that ˆ θ is real-valued and even,with ˆ θ (0) = 1 and | ˆ θ ( k ) | ≤ k ∈ R . For any positive integer n , we define θ n ( (cid:96) ) := nθ ( n(cid:96) ).We have ˆ θ n ( k ) = ˆ θ (cid:0) kn (cid:1) for all k ∈ R .Let us fix ϕ ∈ T and a positive integer n . The function ϕ n := θ n ∗ ϕ is of class C ∞ withsupport in (cid:2) − n , L + n (cid:3) and satisfies (cid:82) + ∞−∞ ϕ n ( (cid:96) ) d(cid:96) = 1. Formula (5) applies and gives us χ + 2 L ϕ n (0) + (cid:88) p ∈P (cid:96) (prim( p )) S V ( p ) ( ϕ n ( − (cid:96) ( p )) + ϕ n ( (cid:96) ( p )))= 2 ˆ ϕ (0) + 2 (cid:88) k j > ˆ θ (cid:18) k j n (cid:19) (cid:60) ( ˆ ϕ ( k j )) , (34)where the series in the left-hand side reduces to a finite sum.For all (cid:96) ∈ R , ϕ n ( (cid:96) ) → ϕ ( (cid:96) ) as n → ∞ , so the left-hand side of Equation (34) converges to χ + (cid:88) p ∈P (cid:96) (prim( p )) S V ( p ) ( ϕ ( − (cid:96) ( p )) + ϕ ( (cid:96) ( p ))) . Condition (iv) in the definition of the set T implies that the series (cid:88) k j > (cid:60) ( ˆ ϕ ( k j ))is absolutely convergent. On the other hand, (cid:12)(cid:12)(cid:12)(cid:12) ˆ θ (cid:18) k j n (cid:19) (cid:60) ( ˆ ϕ ( k j )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ |(cid:60) ( ˆ ϕ ( k j )) | , while ˆ θ (cid:16) k j n (cid:17) → ˆ θ (0) = 1 as n → ∞ . It follows that (cid:88) k j > ˆ θ (cid:18) k j n (cid:19) (cid:60) ( ˆ ϕ ( k j )) → (cid:88) k j > (cid:60) ( ˆ ϕ ( k j ))as n → ∞ , using for instance the Dominated Convergence Theorem for the counting measureassociated with the sequence ( k j ) j ≥ . Taking the limit n → ∞ in Equation (34) therefore yieldsthe desired result. B Fourier transform of the test functions
Let us recall that we are considering the family of test functions ϕ d ( (cid:96) ) = C d (1 − cos(2 π(cid:96) )) d [0 , ( (cid:96) ) , with d a positive integer. To apply Proposition 2.6, we need to compute the Fourier transformof ϕ d , and also to determine the normalisation constant C d . Let us set, for k ∈ R , I d ( k ) := (cid:90) (1 − cos(2 π(cid:96) )) d e ik(cid:96) d(cid:96). Standard trigonometric identities give us I d ( k ) := 2 d (cid:90) sin( π(cid:96) ) d e ik(cid:96) d(cid:96) = ( − d − d (cid:90) ( e iπ(cid:96) − e − iπ(cid:96) ) d e ik(cid:96) d(cid:96).
15e can evaluate quickly the integral on the right using complex integration, and assuming k > πd . Let us denote by γ the closed path in the complex plane C obtained by addingthe path γ , consisting of the segment [ − ,
1] traversed in the positive direction, and γ , thesemi-circle centered at 0 traversed in the anti-clockwise direction from 1 to −
1. Then I d ( k ) =( − d − d / ( iπ ) (cid:82) γ f ( z ) dz , with f ( z ) = 1 z (cid:18) z − z (cid:19) d z k/π = ( z − d z k/π − d − . Since (cid:82) γ f ( z ) dz = 0, we obtain I d ( k ) = − ( − d − d / ( iπ ) (cid:90) γ f ( z ) dz. We find I d ( k ) = ( − d +1 d iπ (cid:18)(cid:90) ( t − d t k/π − d − dt − e ik (cid:90) ( t − d t k/π − d − dt (cid:19) , the two terms on the right-hand side being the integral on the positive and negative part of γ ,respectively. Using the change of variable t = √ s , we transform the expression into I d ( k ) = ( − d +1 d +1 iπ (1 − e ik ) (cid:90) ( s − d s k/ (2 π ) − d − ds = ( − d d π e ik/ sin( k/ B (2 d + 1 , k/ (2 π ) − d ) , where B denotes the Euler beta function. We have B (2 d + 1 , k/ (2 π ) − d ) = Γ(2 d + 1)Γ( k/ (2 π ) − d )Γ( k/ (2 π ) + d + 1) = (2 d )!Π dj = − d ( k/ (2 π ) + j ) , where the last equality results from using, 2 d + 1 times, the identity Γ( x + 1) = x Γ( x ). Weobtain I d ( k ) = ( − d (2 d )!2 d π Π dj = − d ( k/ (2 π ) + j ) e ik/ sin( k/ . Both sides of the equation define a function of k which is holomorphic in C . The equality, whichwe have proved for k > πd , therefore holds for k ∈ C by the principle of isolated zeros. Inparticular, we find C − d = I d (0) = (2 d )!2 d ( d !) , and finallyˆ ϕ d ( k ) = ( − d ( d !) π Π dj = − d ( k/ (2 π ) + j ) e ik/ sin( k/ . Acknowledgement
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