A Gelfand-Levitan trace formula for generic quantum graphs
aa r X i v : . [ m a t h - ph ] J a n A GELFAND-LEVITAN TRACE FORMULA FOR GENERIC QUANTUMGRAPHS
PEDRO FREITAS AND JIˇR´I LIPOVSK ´Y
Abstract.
We formulate and prove a Gelfand-Levitan trace formula for general quantum graphswith arbitrary edge lengths and coupling conditions which cover all self-adjoint operators onquantum graphs, except for a set of measure zero. The formula is reminiscent of the originalGelfand-Levitan result on the segment with Neumann boundary conditions. Introduction
Given a Schr¨odinger operator with a potential q on a line segment of length π with Neumannboundary conditions, let us denote the corresponding eigenvalues by λ n ( q ). In [GL53] Gelfand andLevitan found and proved a formula for the sum of the differences between λ n ( q ) and the eigenvaluesof the null potential λ n (0), namely, ∞ X n =1 (cid:20) λ n ( q ) − λ n (0) − π Z π q ( x ) d x (cid:21) = 14 [ q ( π ) + q (0)] − π Z π q ( x ) d x , (1)under certain regularity conditions – see also [Dik53, HK60]. Since then, regularised trace formulasof this type have been present in the literature more or less continuously and were extended tomany different settings and forms, including more general operators and potentials – see [SP06] fora review of the topic, including some historical notes. Also, and as was pointed out in [Bar74],there is a relation between the trace formula (1) and the short-time asymptotic expansion of thetrace of the heat kernel.Of interest to us here are the extensions to the case of quantum graphs, where this type ofresult may be traced back to a paper by Roth in 1983 [Rot83], with further developments in severaldirections such as those in [BER15, BK13, FK16, Nic87]. Quantum graphs have also received muchattention in the literature within the past 30 years and, in particular, there have been severalattempts at generalizing Gelfand and Levitan’s result to this setting. So far, the results obtainedhave been restricted to specific graphs and include, for instance, the case of equilateral graphs forwhich Carlson proved a formula involving integrals of the potential and the eigenfunctions [Car12],and the work of C–F. Yang and J.-X. Yang for equilateral star graphs with different boundaryconditions and coupling at the central node, which are closer in form to (1) [YY07, Yan13].A main difficulty with extending (1) to graphs with a general topology and arbitrary edge lengthsis that there will then exist eigenvalue sequences with different asymptotic behaviours, making theregularisation of the trace by associating the different eigenvalues of the problem with a potentialto those with the null potential a delicate issue. The purpose of the present paper is to provide ananswer to this question in this general setting. We thus consider graphs with arbitrary edge lengthsand topology, while the coupling is generic in the following sense. The whole class of couplingconditions defining a self-adjoint operator is allowed, with the exception of a set of measure zero corresponding to a particular eigenvalue in the coupling matrix. This exception leaves out someimportant coupling conditions such as Dirichlet, standard or δ -coupling, but it does include Robin,Neumann or δ ′ -coupling (the last one with the exception of the case when the coupling parameteris zero). On the other hand, notice also that the trace formula is expected to be slightly differentin the cases which are left out by the present approach, as this is already the case for one singleinterval.A key point in our approach is that, unlike in [YY07, Yan13], for instance, we do not subtractfrom the eigenvalue of the Hamiltonian with the potential a particular known value – in (1), λ n (0)is in fact k –, but for elegance of the result, we find the formula for the difference between theeigenvalues of the Hamiltonian with the potential and those for when this potential is zero. This is,in fact, what allows us to assign a correspondence between the different eigenvalues in such a way asto make the involved series convergent, while not making the corresponding formula cumbersome.If, for instance, one considers the first terms in the asymptotics of the different sequences, then theformula will include terms related to the coupling matrix, for instance.The discussion above also implies that a second ingredient which is necessary to obtain fora formula of this type to work is the asymptotic behaviour of the different sequences up to anorder such that we can both group eigenvalues according to their asymptotic behaviour and ensureconvergence of the series involved. As far as we are aware, previous results along these lines forgeneral graphs only considered remainders of order zero [Nic87], while for our purposes we need togo up to the term with remainder of order n − .The separation of the full spectrum into different sequences may be done in several differentways, so we now briefly explain our procedure. It is clear that, with the end in view, the basis forthis separation has to be the asymptotic behaviour of the spectrum. We first note that the leadingterm of the secular equation is Q di =1 ( − k sin ( kℓ i )), where d is the number of the edges of the graph, k is the square root of the energy and ℓ i are the edge lengths. In Section 4 we prove that thesquare roots of eigenvalues are close to the zeros of the given product and that they can be groupedin sets of at most d eigenvalues and d zeros. We thus partition the spectrum into d subsequencesof eigenvalues in the following way. Denote the sequence of all eigenvalues in increasing order by { λ n } ∞ n =1 and let the sequence { µ n } ∞ n =1 correspond to the non-negative zeros of the above product,also arranged in increasing order, with the first d entries being 0. We now pair λ n with µ n anddefine the subsequences { λ in } ∞ n =0 as subsequences of { λ n } ∞ n =1 which are paired with those zeros of Q di =1 ( − k sin ( kℓ i )) which are zeros of sin ( kℓ i ) for a given i (the first entry of this sequence λ i ispaired with 0).We may now formulate the main result of the paper. Theorem A.
We assume a quantum graph with d edges with arbitrary lengths ℓ i , i = 1 , . . . , d , andassociated coupling matrix U not having − in its spectrum. Then, denoting the eigenvalues of theHamiltonian with a potential q and with the zero potential by λ in ( q ) and λ in (0) , respectively, in theway described above, and the component of the potential on the i -th edge by q i ∈ W , ((0 , ℓ i )) , thefollowing trace formula holds d X i =1 ∞ X n =0 " λ in ( q ) − λ in (0) − ℓ i Z ℓ i q i ( x ) d x = d X i =1 (
14 [ q i ( ℓ i ) + q i (0)] − ℓ i Z ℓ i q i ( x ) d x ) . The paper is structured as follows. In the next section we describe the model of quantum graphs,and in Section 3 the secular equation is found and some preparatory calculations for Section 4 are
ELFAND-LEVITAN TRACE FORMULA ON QUANTUM GRAPHS 3 performed. In Section 4 we give the proofs of the main results leading to the proof of Theorem A.Several technical results used throughout the paper are given in the Appendix A.2.
Description of the model
We briefly introduce the model of quantum graphs; for more details we refer the reader to [BK13].Let us consider a metric graph Γ consisting of the set of vertices v ∈ V , which are connected by theset of d finite edges e j ∈ E . The number d is finite and the lengths of the edges are ℓ j ∈ (0 , ∞ ). Weequip the graph Γ with a self-adjoint operator H = − d d x + q j ( x ) , x ∈ e j with the real potentials q j ∈ W , ( e j ). The domain of H consists of functions with the edgecomponents in the Sobolev spaces W , ( e j ) and satisfying the coupling conditions( U v − I )Ψ v + i ( U v + I )Ψ ′ v = 0at the vertices. Here U v is a d v × d v unitary matrix ( d v is the degree of the vertex v ), Ψ v is thevector of the limiting values of functions at the vertex v from its incident edge and, similarly, Ψ ′ v is the vector of the derivatives outgoing from v ; I is the d v × d v identity matrix. Throughout thepaper, we will assume that − σ ( U v ).With the use of the flower-like model (see [Kuc08, EL10]), where all the vertices are joined intoone and the topology of the graph is described by the larger 2 d × d coupling matrix U , one maywrite the coupling condition as ( U − I )Ψ + i ( U + I )Ψ ′ = 0 . Here, with a small abuse of notation, I refers now to the 2 d × d identity matrix, Ψ is the vectorwith the limiting values of functions defined on each edge, as the vertex is approached from eitherend of the edge, and Ψ ′ is the vector of limits of the corresponding outgoing derivatives. We assumethat the first entry of the vector Ψ is the functional value at the beginning of the first edge, thesecond entry is the functional value at the end of the first edge, the third entry is the functionalvalue at the beginning of the second edge, and so on, and similarly for Ψ ′ . Using the fact that − σ ( U ), we may write H Ψ + Ψ ′ = 0 , (2)where H = − i ( U + I ) − ( U − I ) is a Hermitian 2 d × d matrix. We denote the entries of the matrix H in the following way H i − , i − =: H i , H i − , i =: H i , H i, i =: H i ,H i − , j − =: H ij , H i − , j =: H ij , (3) H i, j − =: H ij , H i, j =: H ij , i < j . The secular equation
In this section, we will construct the secular equation. On each edge, we assume two independentsolutions of the eigenvalue problem with the eigenvalue k c j ( x, k ) = cos ( kx ) + Z x sin ( k ( x − t )) k q j ( t ) c j ( t, k ) d t ,s j ( x, k ) = sin ( kx ) k + Z x sin ( k ( x − t )) k q j ( t ) s j ( t, k ) d t . PEDRO FREITAS AND JIˇR´I LIPOVSK´Y
One can easily check that both functions satisfy H c j ( x, k ) = k c j ( x, k ) and H s j ( x, k ) = k s j ( x, k ).The following lemma (the asymptotic expansion follows the idea of [Yur00]) is proven in Appen-dix A. Lemma 3.1.
The functions c j and s j defined above satisfy c j ( x, k ) = cos ( kx ) + sin ( kx ) k Z x q j ( t ) d t ++ cos ( kx ) k "
14 ( q j ( x ) − q j (0)) − (cid:18)Z x q j ( t ) d t (cid:19) + o (cid:18) e | Im k | x k (cid:19) ,c ′ j ( x, k ) = − k sin ( kx ) + cos ( kx ) 12 Z x q j ( t ) d t ++ sin ( kx ) k "
14 ( q j ( x ) + q j (0)) + 18 (cid:18)Z x q j ( t ) d t (cid:19) + o (cid:18) e | Im k | x k (cid:19) ,s j ( x, k ) = sin ( kx ) k − cos ( kx ) k Z x q j ( t ) d t ++ sin ( kx ) k "
14 ( q j ( x ) + q j (0)) − (cid:18)Z x q j ( t ) d t (cid:19) + o (cid:18) e | Im k | x k (cid:19) ,s ′ j ( x, k ) = cos ( kx ) + sin ( kx ) k Z x q j ( t ) d t −− cos ( kx ) k "
14 ( q j ( x ) − q j (0)) + 18 (cid:18)Z x q j ( t ) d t (cid:19) + o (cid:18) e | Im k | x k (cid:19) . We will now transform equation (2) into a form that is more appropriate for our purposes.Writing a j := 12 Z ℓ j q j ( t ) d t , b j := q j ( ℓ j ) + q j (0)4 + 18 Z ℓ j q j ( t ) d t ! , and using the expression of the components of the eigenfunction as the linear combination f j ( x ) = A j c j ( x, k ) + B j s j ( x, k ) , and the corresponding conditions c j (0 , k ) = 1, c ′ j (0 , k ) = 0, s j (0 , k ) = 0, s ′ j (0 , k ) = 1, it is possibleto rewrite equation (2) as[ HM ( k ) + M ( k )]( A , B , A , B , . . . , A d , B d ) T = 0 . ELFAND-LEVITAN TRACE FORMULA ON QUANTUM GRAPHS 5
Here the matrices M and M are given by M ( k ) = . . .c ( ℓ , k ) s ( ℓ , k ) 0 0 . . . . . . c ( ℓ , k ) s ( ℓ , k ) . . . ... ... ... ... . . . = . . . cos ( kℓ ) + a sin ( kℓ ) k sin ( kℓ ) k − a cos ( kℓ ) k . . . . . . kℓ ) + a sin ( kℓ ) k sin ( kℓ ) k − a (cos kℓ ) k . . . ... ... ... ... .. . + +o e | Im k | max ℓ j k ! , and M ( k ) = . . . − c ′ ( ℓ , k ) − s ′ ( ℓ , k ) 0 0 . . . . . . − c ′ ( ℓ , k ) − s ′ ( ℓ , k ) . . . ... ... ... ... . . . = . . .k sin ( kℓ ) − a cos ( kℓ ) − b sin ( kℓ ) k − cos ( kℓ ) − a sin ( kℓ ) k . . . . . . k sin ( kℓ ) − a cos ( kℓ ) − b sin ( kℓ ) k − cos ( kℓ ) − a sin ( kℓ ) k . . . ... ... ... ... ... + +o e | Im k | max ℓ j k ! , PEDRO FREITAS AND JIˇR´I LIPOVSK´Y
Hence the secular equation can be obtained as det [ HM ( k ) + M ( k )] = 0. From this we obtainafter a straightforward, but a bit tedious computation0 = ϕ ( k ) = d Y i =1 ( − k sin ( kℓ i )) + d X i =1 d Y j =1 j = i ( − k sin ( kℓ j )) [cos ( kℓ i )( a i − Tr H i ) − H i ]++ d X i,j =1 i Using the Taylor expansion for the logarithm around one we obtain(4) ln ϕ ( k ) Q di =1 ( − k sin ( kℓ i )) = 1 k d X i =1 (cid:20) cot ( kℓ i )(Tr H i − a i ) + 2Re H i sin ( kℓ i ) (cid:21) +1 k d X i,j =1 i 12 (Tr H i − a i ) (cid:21) + ( a i Tr H i − b i − det H i )+ cot ( kℓ i )sin ( kℓ i ) [ − H i − a i )Re H i ] − H i ) sin ( kℓ i ) (cid:27) + o (cid:18) k (cid:19) . Writing ϕ ( k ) for the function in the secular equation when q j ( x ) = 0 , j = 1 , . . . , d we obtain ina similar way(5) ln ϕ ( k ) ϕ ( k ) = − k d X i =1 cot ( kℓ i ) a i + 1 k d X i =1 (cid:20) ( kℓ i ) a i Tr H i − b i − 12 cot ( kℓ i ) a i + cot ( kℓ i )sin ( kℓ i ) 2 a i Re H i (cid:21) + o (cid:18) k (cid:19) . Proof of the main result Let us define the counter-clockwise contour Γ N as a square with vertices N − iN , N + iN , − N + iN , − N − iN . Then, using the symmetric version of Rouch´e’s theorem, we can prove thefollowing theorem relating the number of zeros of Q di =1 ( − k sin ( kℓ i )) and zeros of ϕ ( k ) (the proofis given in the Appendix). Theorem 4.1. For all ε > there exists K > so that for all N > K and N 6∈ ∪ di =1 ∪ n ∈ N (cid:16) nπℓ i − εℓ i , nπℓ i + εℓ i (cid:17) the functions Q di =1 ( − k sin ( kℓ i )) and ϕ ( k ) have the same number of zeros insidethe contour Γ N . Let us denote the sequence of all eigenvalues of the operator H arranged by ascending orderby { λ n } ∞ n =1 . We denote by { µ n } ∞ n =1 the sequence in which the first d elements are 0 and allsubsequent elements are positive zeros of Q di =1 sin ( kℓ i ) arranged in increasing order. We pair λ n with µ n . In view of Theorem 4.1, k n := √ λ n with Re k n ≥ µ n , as we will see in thefollowing lemma. We will denote the sequence of eigenvalues corresponding to the zeros of sin ( kℓ i )by { λ in } ∞ n =0 , where λ i corresponds to 0 and the remaining values to positive zeros of sin ( kℓ i ). PEDRO FREITAS AND JIˇR´I LIPOVSK´Y Lemma 4.2. It is possible to choose ε > and K > such that there exists a strictly increasingsequence { N p } ∞ p =1 with K < N and satisfying N p 6∈ ∪ di =1 ∪ n ∈ N (cid:18) nπℓ i − εℓ i , nπℓ i + εℓ i (cid:19) and lim p →∞ N p = + ∞ . and there are at most d eigenvalues λ = k of H with N p ≤ k ≤ N p +1 , for all p ∈ N . Fur-thermore, all these eigenvalues belong to different sequences λ in and there are at most d zeros µ of Q di =1 sin ( kℓ i ) with N p ≤ µ ≤ N p +1 , ∀ p ∈ N . The number of eigenvalues and zeros with thisproperty is the same.Proof. We choose ε < π j ℓ j d X i =1 ℓ i . The width of each interval (cid:16) nπℓ i − εℓ i , nπℓ i + εℓ i (cid:17) is 2 εℓ i and so the sum of the lengths of these“forbidden” intervals for all sequences is 2 ε d X i =1 ℓ i . We choose ε sufficiently small to ensure thatthis expression is smaller than π j ℓ j (half of the smallest distance between two neighbouringzeros of the sine function from the given sequence). Hence the “forbidden intervals” do not coverthe whole interval between two neighbouring zeros of a given sine function, and it is possible tochoose a contour in Theorem 4.1 between them and obtain that the number of zeros of the sine andthe eigenvalues in that contour is the same. (cid:3) Now we choose for contours C p the rectangles with vertices N p +1 − iN p +1 , N p +1 + iN p +1 , N p + iN p +1 and N p − iN p +1 , traversed counter-clockwise. Inside the contour there is the same numberof square roots of eigenvalues of H and zeros of Q di =1 k sin ( kℓ i ) and this number is at most d . Letus first consider the case when there is only one square root of eigenvalue and one zero inside C p . ELFAND-LEVITAN TRACE FORMULA ON QUANTUM GRAPHS 9 Theorem 4.3. Let us assume that inside the contour C p there are the points nπℓ i and k in = √ λ in for a given i . Then λ in = k in behaves asymptotically as λ in = (cid:18) nπℓ i (cid:19) + 2 ℓ i [ a i − Tr H i − ( − n H i ]++ 2 nπ d X j =1 j = i (cid:20) cot nπℓ j ℓ i ( | H ij | + | H ij | + | H ij | + | H ij | )++ 1sin nπℓ j ℓ i H ij ¯ H ij + H ij ¯ H ij )+( − n sin nπℓ j ℓ i H ij ¯ H ij + H ij ¯ H ij )+( − n cot nπℓ j ℓ i H ij ¯ H ij + H ij ¯ H ij ) (cid:21) + O (cid:18) n (cid:19) . Proof. We use the integral k in − (cid:18) nπℓ i (cid:19) = − πi I C p ln ϕ ( k ) Q dj =1 ( − k sin ( kℓ j )) 2 k d k . A straightforward computation using equation (4) and Lemma A.3 leads to the result. (cid:3) When the number of square roots of the eigenvalues (and zeros of the product) is larger thanone, we sum over the eigenvalues. Theorem 4.4. Let us assume that inside the contour C p there are the points n i πℓ i and k in = p λ in i for i from the index set I . Then P i ∈ I λ in i behaves asymptotically as X i ∈ I λ in i = X i ∈ I (cid:18) n i πℓ i (cid:19) + X i ∈ I ℓ i [ a i − Tr H i − ( − n i H i ]++ X i ∈ I n i π d X j =1 n i ℓ j = n j ℓ i (cid:20) cot n i πℓ j ℓ i ( | H ij | + | H ij | + | H ij | + | H ij | )++ 1sin n i πℓ j ℓ i H ij ¯ H ij + H ij ¯ H ij )+( − n i sin n i πℓ j ℓ i H ij ¯ H ij + H ij ¯ H ij )+( − n i cot n i πℓ j ℓ i H ij ¯ H ij + H ij ¯ H ij ) (cid:21) + O (cid:18) max i ∈ I n i (cid:19) . Proof. Again, we obtain a similar integral as in the previous lemma X i ∈ I " k in − (cid:18) nπℓ i (cid:19) = − πi I C p ln ϕ ( k ) Q dj =1 ( − k sin ( kℓ j )) 2 k d k . If there are no common zeros of the different sine functions, we may apply the same argument asin the previous lemma and obtain the sum of the right-hand side of the previous theorem. If thereis a multiple zero of a sine function (i.e. n i ℓ j = n j ℓ i for any i, j so that n i πℓ i lies inside the contour C p ), we may apply Lemma A.3 g), i), and k) to show that the contribution of this zero to the thirdterm on the rhs is of order O (cid:16) n i (cid:17) . (cid:3) Combining the previous two theorems together yields the following corollary Corollary 4.5. The sum d X i =1 ∞ X n =0 (cid:20) λ in ( q ) − λ in (0) − a i ℓ i (cid:21) is absolutely convergent, where λ in ( q ) and and λ in (0) denote the eigenvalues for the potential q andfor the null potential, respectively.Proof. Subtracting the right-hand side of the formulæ in Theorems 4.3 and 4.4 one obtains the termsof the sum. (Note that the term by n depends only on the matrix H and not on the potential.)Hence the sum P i ∈ I λ in i ( q ) − λ in i (0) − a i ℓ i is of order O (cid:16) max i ∈ I n i (cid:17) and the sum of these sumsis absolutely convergent. (cid:3) Finally, we can prove the main result. Proof of Theorem A. We integrate around the contours Γ N in the “allowed regions” with N goingto infinity. For sufficiently large N , there are d + P di =1 (cid:4) Nℓ i π (cid:5) eigenvalues of H with square rootssmaller than N (here ⌊·⌋ denotes the floor function, that is, the largest integer not larger than itsargument). The number of k n with the same property in the k -plane is double. We obtain2 d X i =1 ⌊ Nℓiπ ⌋ X n =0 [ λ in ( q ) − λ in (0)] = − πi I Γ N ln ϕ ( k ) ϕ ( k ) 2 k d k . (6)We can evaluate the integral with the use of equation (5) and Lemma A.3, we find after dividingthe equation by 2 d X i =1 ⌊ Nℓiπ ⌋ X n =0 [ λ in ( q ) − λ in (0)] = d X i =1 a i ℓ i (cid:18) (cid:22) N ℓ i π (cid:23)(cid:19) + d X i =1 (cid:18) b i − a i (cid:19) + O (cid:18) N (cid:19) . (7)We have used the sums M X n =1 n = π (cid:18) M (cid:19) , M X n =1 ( − n n = − π 12 + O (cid:18) M (cid:19) . Subtracting P di =1 2 a i ℓ i (cid:0) (cid:4) Nℓ i π (cid:5)(cid:1) from both sides of (7), using b i − a i = 14 [ q i ( ℓ i ) + q i (0)] , and sending N to infinity we find the sought result. The contribution of the term o (cid:16) k (cid:17) in (6)resulting from the logarithm expansion (5) goes to zero as N → ∞ , because the length of thecontour is of order N and the value of the function on it is o (cid:16) N (cid:17) × N . (cid:3) ELFAND-LEVITAN TRACE FORMULA ON QUANTUM GRAPHS 11 Appendix A. Some auxiliary results First, we prove Lemma 3.1 – this is already partially given in [Yur00]. Proof of Lemma 3.1. For the sake of simplicity we omit the subscript j . Repeatedly substituting c j into its defining formula we get c ( x, k ) = cos ( kx ) + Z x sin ( k ( x − t )) k cos ( kt ) q ( t ) d t ++ Z x sin ( k ( x − t )) k q ( t ) Z t sin ( k ( t − s )) k q ( s ) cos ( ks ) d s d t + o (cid:18) e | Im k | x k (cid:19) . Then we use the formula sin ( α − β ) cos β = 12 [sin α + sin ( α − β )]to obtain c ( x, k ) = cos ( kx ) + 12 k Z x [sin ( kx ) + sin ( k ( x − t ))] q ( t ) d t ++ Z x Z t sin ( k ( x − t )) k q ( t ) 12 k [sin ( kt ) + sin ( k ( t − s ))] q ( s ) d s d t + o (cid:18) e | Im k | x k (cid:19) . Finally, using integration by parts we have Z x sin ( k ( x − t )) q ( t ) d t = Z x q ( t ) ∂∂t cos ( k ( x − t ))2 k d t == 12 k [ q ( x ) − q (0)] cos ( kx ) − k Z x cos ( k ( x − t )) ∂q ( t ) ∂t d t == 12 k [ q ( x ) − q (0)] cos ( kx ) + o (cid:18) e | Im k | x k (cid:19) , where we have used the fact that q ∈ W , ( e ). Using this we can write c ( x, k ) = cos ( kx ) + sin ( kx ) k Z x q ( t ) d t ++ 14 k [ q ( x ) − q (0)] cos ( kx ) + 14 k Z x Z t q ( t ) q ( s )[cos ( k ( x − t )) − cos ( kx )] d s d t ++ 12 k Z x q ( t ) sin ( k ( x − t )) Z t sin ( k ( t − s )) q ( s ) d s d t + o (cid:18) e | Im k | x k (cid:19) . By similar arguments as before (with the use of integration by parts) the term in the last line andthe term k R x R t q ( t ) q ( s )[cos ( k ( x − t )) d s d t are of order o (cid:16) e | Im k | x k (cid:17) . Finally, sincecos ( kx )4 k Z x q ( t ) Z t q ( s ) d s d t = cos ( kx )8 k Z x Z x q ( t ) q ( s ) d s d t = cos ( kx )8 k (cid:18)Z x q ( t ) d t (cid:19) , we obtain the formula for c ( x, k ). The formulæ for the function s ( x, k ) and the corresponding derivatives can be derived in a similarway. For c ′ we have c ′ ( x, k ) = − k sin ( kx ) + Z x cos ( k ( x − t )) q ( t ) cos ( kt ) d t ++ Z x cos ( k ( x − t )) q ( t ) Z t sin ( k ( t − s )) k q ( s ) cos ( ks ) d s d t + o (cid:18) e | Im k | x k (cid:19) == − k sin ( kx ) + Z x q ( t ) 12 [cos ( kx ) + cos ( k ( x − t ))] d t ++ Z x cos ( k ( x − t )) q ( t ) Z t k q ( s )[sin ( kt ) + sin ( k ( t − s ))] d s d t + o (cid:18) e | Im k | x k (cid:19) For the different particular terms we get12 Z x q ( t ) cos ( k ( x − t )) d t = 12 Z x q ( t ) ∂ sin ( k ( x − t )) ∂t (cid:18) − k (cid:19) d t == 14 k Z x ∂q ( t ) ∂t sin ( k ( x − t )) d t + 14 k [ q ( x ) + q (0)] sin ( kx ) == 14 k [ q ( x ) + q (0)] sin ( kx ) + o (cid:18) e | Im k | x k (cid:19) . Z x cos ( k ( x − t )) q ( t ) Z t k q ( s ) sin ( kt ) d s d t == 14 k Z x q ( t )[sin ( kx ) − sin ( k ( x − t ))] Z t q ( s ) d t == sin ( kx )8 k (cid:18)Z x q ( t ) d t (cid:19) + o (cid:18) e | Im k | x k (cid:19) . Z x cos ( k ( x − t )) q ( t ) Z t k q ( s ) sin ( k ( t − s )) d s d t = o (cid:18) e | Im k | x k (cid:19) . We also briefly show the derivation of formulæ for s and s ′ . s ( x, k ) = sin ( kx ) k + Z x sin ( k ( x − t )) k q ( t ) sin ( kt ) d t ++ Z x sin ( k ( x − t )) k q ( t ) Z t sin ( k ( t − s )) q ( s ) sin ( ks ) d s d t + o (cid:18) e | Im k | x k (cid:19) == sin ( kx ) k − cos ( kx ) k Z x q ( t ) d t + 14 k sin ( kx )[ q ( x ) + q (0)] −− k Z x [sin ( kx ) + sin ( k ( x − t ))] q ( t ) Z t q ( s ) d s d t ++ 12 k Z x sin ( k ( x − t )) q ( t ) Z t cos ( k ( t − s )) q ( s ) d s d t + o (cid:18) e | Im k | x k (cid:19) . ELFAND-LEVITAN TRACE FORMULA ON QUANTUM GRAPHS 13 s ′ ( x, k ) = cos ( kx ) + 1 k Z x cos ( k ( x − t )) sin ( kt ) q ( t ) d t ++ 1 k Z x cos ( k ( x − t )) q ( t ) Z t sin ( k ( t − s )) sin ( ks ) q ( s ) d s d t + o (cid:18) e | Im k | x k (cid:19) == cos ( kx ) + 12 k Z x [sin ( kx ) − sin ( k ( x − t ))] q ( t ) d t ++ 12 k Z x cos ( k ( x − t )) q ( t ) Z t [cos ( k ( t − s )) − cos ( kt )] q ( s ) d s d t =+ o (cid:18) e | Im k | x k (cid:19) = cos ( kx ) + sin ( kx ) k Z x q ( t ) d t − cos ( kx )4 k [ q ( x ) − q (0)] −− k Z x Z t cos ( k ( x − t )) q ( t ) q ( s ) d s d t −− k Z x cos ( kx ) q ( t ) Z t q ( s )d s d t + o (cid:18) e | Im k | x k (cid:19) . (cid:3) Lemma A.1. On the contour Γ N defined in Section 4 with large enough N satisfying N 6∈ ∪ n ∈ Z (cid:18) nπℓ j − εℓ j , nπℓ j + εℓ j (cid:19) , it holds e | Im k | ℓ j | sin ( kℓ j ) | ≤ K ε , where the constant K ε depends only on ε .Proof. The proof will be similar to the proof of [Yan14, Lemma 2.4]. We will first prove theinequality for the right edge of the square Γ N , i.e. for k = N + iτ , τ ∈ ( − N, N ). We know thatthere exist such C ε > | sin ( kℓ j ) | > C ε . We have | sin ( kℓ j ) | = | sin ( N ℓ j ) cos ( iτ ℓ j ) + cos ( N ℓ j ) sin ( iτ ℓ j ) | == 12 q | sin ( N ℓ j )(e − τℓ j + e τℓ j ) | + | cos ( N ℓ j )(e − τℓ j − e τℓ j ) | ≥≥ | sin ( N ℓ j ) || e − τℓ j + e τℓ j | > C ε e | τ | ℓ j and hence e | Im k | ℓ j | sin ( kℓ j ) | ≤ | Im k | ℓ j C ε e | Im k | ℓ j = 2 C ε . For the upper edge of the square k = σ + iN , σ ∈ ( − N, N ) we have for sufficiently large N | sin ( kℓ j ) | = 12 | e − Nℓ j + iσℓ j − e Nℓ j − iσℓ j | ≥ 12 (e Nℓ j − e − Nℓ j )and hence for N large enough e | Im k | ℓ j | sin ( kℓ j ) | ≤ Nℓ j e Nℓ j − e − Nℓ j ≤ . We have chosen N such that e − Nℓ j < . The proof for the other edges of the square Γ N issimilar. (cid:3) For the sake of completeness we present the symmetric version of Rouch´e’s theorem (for theproof see e.g. [Est62, p. 156] or [Bur79, p. 265]). Theorem A.2. Let f and g be holomorphic functions in the bounded subset V of C and continuousat its closure ¯ V . Let us assume that on the boundary ∂V of V the following relation holds | f − g | < | f | + | g | . Then functions f and g have the same (finite) number of zeros in V . Now we can proceed with the proof of Theorem 4.1. Proof of Theorem 4.1. Since we assume that N 6∈ ∪ n ∈ N (cid:16) nπℓ i − εℓ i , nπℓ i + εℓ i (cid:17) for each i , we have | sin ( N ℓ i ) | > C ε > C ε depending only on ε . We use the Rouch´e’s theorem with f = ϕ ( k )and g = Q di =1 ( − k sin ( kℓ i )).On the contour Γ N we have | f | + | g | = 2 | k | d d Y i =1 | sin ( kℓ i ) | + O( | k | d − e | Im k | P di =1 ℓ i ) , | f − g | ≤ O( | k | d − e | Im k | P di =1 ℓ i ) . Using Lemma A.1 we obtain | f | + | g | − | f − g | = | k | d d Y i =1 | sin ( kℓ i ) | | k | O e | Im k | P dj =1 ℓ j Q do =1 | sin ( kℓ o ) | !! > | k | d C dε > | k | large enough and hence the inequality in Theorem A.2 is satisfied which completes theproof. (cid:3) Lemma A.3. Let us assume a counterclockwise contour γ n which encircles nπℓ i once and does notencircle any other zeros of sin ( kℓ i ) . Then a) 12 πi I γ n cot ( kℓ i ) d k = 1 ℓ i , n ∈ Z , b) 12 πi I γ n kℓ i ) d k = ( − n ℓ i , n ∈ Z , c) 12 πi I γ n k cot ( kℓ i ) d k = 1 nπ , n ∈ Z \{ } , d) 12 πi I γ n k cot ( kℓ i ) d k = 0 , n = 0 , e) 12 πi I γ n k sin ( kℓ i ) d k = ( − n nπ , n ∈ Z \{ } , f) 12 πi I γ n k sin ( kℓ i ) d k = 0 , n = 0 , ELFAND-LEVITAN TRACE FORMULA ON QUANTUM GRAPHS 15 g) 12 πi I γ n k cot ( kℓ i ) d k = − n π , n ∈ Z \{ } , h) 12 πi I γ n k cot ( kℓ i ) d k = − , n = 0 , i) 12 πi I γ n cot ( kℓ i ) k sin ( kℓ i ) d k = − ( − n n π , n ∈ Z \{ } , j) 12 πi I γ n cot ( kℓ i ) k sin ( kℓ i ) d k = − , n = 0 , k) 12 πi I γ n k sin ( kℓ i ) d k = − n π , n ∈ Z \{ } , l) 12 πi I γ n k sin ( kℓ i ) d k = 13 , n = 0 , m) 12 πi I γ n k d k = 1 , n = 0 .Proof. The lemma can be proven by standard techniques of complex analysis, i.e. the residuetheorem, see e.g. [Bur79]. (cid:3) Acknowledgements P.F. was partially supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia, Portugal, throughproject PTDC/MAT-CAL/4334/2014. J.L. was supported by the project “International mobilitiesfor research activities of the University of Hradec Kr´alov´e” CZ.02.2.69/0.0/0.0/16 027/0008487.J.L. thanks the University of Lisbon for its hospitality during his stay in Lisbon. References [Bar74] Barcilon, V. A note on a formula of Gelfand and Levitan J. Math. Anal. Appl. (1974), 43–50.[BK13] Berkolaiko, G. and Kuchment, P. Introduction to Quantum Graphs . Mathematical Surveys and Mono-graphs 186. AMS, 2013, 270 pp.[BER15] Bolte, J., Endres, S. and Rueckriemen, R. Heat-kernel Asymptotics for Schr¨odinger Operators onGraphs. Applied Mathematics Research eXpress (2015), 129–165.[Bur79] Burckel, R. B. An Introduction to Classical Complex Analysis . Birkh¨auser, 1979, 558 pp.[Car12] Carlson, R. Eigenvalue cluster traces for quantum graphs with equal edge lengths Rocky Mountain J.Math. (2012), 467–490.[CW08] Currie, S. and Watson, B. A. Green’s functions and regularized traces of Sturm-Liouville operators ongraphs. Proc. Edinb. Math. Soc. Ser. 2 (2008), 315–335.[Dik53] Diki˘ı, L.A. On a formula of Gel ′ fand–Levitan (Russian), Uspehi Matem. Nauk (N.S.) (1953), 19–23.[Est62] Estermann, T. Complex Numbers and Functions . Athlone Press, London, 1962, 250 pp.[EL10] Exner, P. and Lipovsk´y, J. Resonances from perturbations of quantum graphs with rationally relatededges. J. Phys. A: Math. Theor. (2010), 105301.[FK16] Freitas, P. and Kennedy, J.B. Summation formula inequalities for eigenvalues of Schr¨odinger operatorsJ. Spectral Theory (2016), 483–503.[GL53] Gelfand, I. M. and Levitan, B. On a simple identity for the characteristic values of a differential operatorof the second order. (Russian). Doklady Akad. Nauk SSSR (1953), 593–596.[HK60] Halberg, C. J. A. and Kramer, V. A. A generalization of the trace concept. Duke Math. J. (1960),607–617.[Kuc08] Kuchment, P. Quantum graphs: an introduction and a brief survey. In Analysis on Graphs and itsApplications , Proc. Symp. Pure. Math. AMS. 2008, pp. 291–314. [Nic87] Nicaise, S. Spectre des r´eseaux topologiques finis Bull. Sc. math. (1987), 401–413.[Rot83] Roth, J.-P. Spectre du laplacien sur un graphe, C.R. Acad. Sci. Paris (1983), 793–795.[SP06] Sadovnichi˘ı, V.A. and Podol ′ ski˘ı, V.E. , Traces of operators, Uspekhi Mat. Nauk (2006), 89–156;translation in Russian Math. Surveys (2006), 885–953.[Yan13] Yang, C.-F. Regularized trace for Sturm-Liouville differential operator on a star-shaped graph. ComplexAnal. Oper. Theory (2013), 1185–1196.[Yan14] Yang, C.-F. Traces of Sturm-Liouville operators with discontinuities. Inverse Problems in Science andEngineering (2014), 803–813.[YY07] Yang, C.-F. and Yang, J.-X. Large eigenvalues and traces of Sturm-Liouville equations on star-shapedgraphs. Methods Appl. Anal. (2007), 179–196.[Yur00] Yurko, V. Integral transforms connected with discontinuous boundary value problems. Integral Transformsand Special Functions (2000), 141–164. Departamento de Matem´atica, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais1, P-1049-001 Lisboa, Portugal and Grupo de F´ısica M´atematica, Faculdade de Ciˆencias, Universidade deLisboa, Campo Grande, Edif´ıcio C6, P-1749-016 Lisboa, Portugal E-mail address : [email protected] Department of Physics, Faculty of Science, University of Hradec Kr´alov´e, Rokitansk´eho 62, 500 03Hradec Kr´alov´e, Czechia E-mail address ::