Schrödinger Operators on Graphs and Geometry III . General Vertex Conditions and Counterexamples

Schrodinger operators on metric graphs with general vertex conditions are studied. Explicit spectral asymptotics is derived in terms of the spectrum of reference Laplacians. A geometric version of ...


I. INTRODUCTION
Quantum graphs are an important area of mathematical analysis providing an excellent setting for experiments in the spectral theory of differential operators.Under quantum graphs we understand differential operators on metric graphs -such operators possess properties of both ordinary and partial differential equations.The classical spectral theory of the one-dimensional Schrödinger equation play a very important role in spectral theory of quantum graphs, since quantum graphs can be seen as coupled systems of one-dimensional equations.Direct spectral problems study properties of the spectrum such as asymptotics or how potentials may perturb the eigenvalues.The corresponding inverse problem study how to reconstruct the metric graph, vertex couplings and potentials on the edges from a set of spectral data.More generally, one studies how topological and geometrical properties of graphs are reflected by the spectrum.
In 20 it was proven that the Euler characteristic of the graph can be calculated from the spectrum of the Schrödinger equation with so-called standard vertex conditions.The proof of this result uses a certain trace formula for the Laplace operator combined with the spectral asymptotics for the Schrödinger equation.Roughly speaking the eigenvalues are always situated close to the eigenvalues of the Laplacian.This is an easy fact in the case of essentially bounded potentials, and was considered in 20 .In 6 spectral asymptotics in the case of absolutely integrable potential and standard vertex conditions were proven.
From this it was concluded that the same formula for the Euler characteristic holds for L 1 -potentials, as well as a geometric version of the celebrated Ambartsumian theorem.The original theorem 1 states that the spectrum of the Schrödinger equation on an interval with Neumann conditions at the end points is equal to the spectrum of the Laplacian if and only if the potential is identically equal to zero.This theorem was the starting point for the theory of inverse problems in one dimension.In 6 it was proved that the spectrum of a Schrödinger equation on a metric graph is equal to the spectrum of the Laplacian on an interval if and only if the graph is equivalent to the interval and the potential is identically zero, under the assumptions that the vertex conditions are standard and potential is absolutely integrable.This type of result we call a geometric Ambartsumian's Theorem.The aim of the current paper is to generalise the obtained results to the case of general vertex conditions.
It appears that similar estimates for the spectrum hold, but with different values of the constants involved.This similarity of the obtained formulas may be misleading, since an important difference is that the spectrum of a Schrödinger operator on a graph Γ is proven to be close to the spectrum of a certain reference Laplace operator (see Definition 4) on what may be a different graph Γ ∞ obtained from Γ by dividing some of its vertices to reflect the connectivity of the new vertex conditions introduced there.The new vertex conditions are determined by the asymptotics of the vertex scattering matrix for Γ.If the large energy asymptotics of the vertex scattering matrix for one of the vertices is given by a block-diagonal matrix, then the corresponding vertex should be divided and thus Γ ∞ is not the same metric graph as Γ.It appears that the geometric Ambartsumian theorem cannot be generlised to quantum graphs with arbitrary vertex conditions without making additional assumptions: explicit counterexamples are constructed in the paper.Therefore it is possible to prove only that isospectrality of the Schrödinger equation to the Laplacian on an interval implies that Γ ∞ is equivalent to the interval.It is proven under the condition that the vertex conditions are asymptotically standard, which means that for large energies such conditions approach standard conditions (see Definition 3).

Outline of the paper
The following two sections are devoted to definitions and recollection of necessary facts from the theory of quantum graphs.In particular we define what is meant by a Schrödinger operator with an L 1 -potential acting on a metric graph Γ and how general vertex conditions can be parameterised using the vertex scattering matrix.We explain how to obtain the graph Γ ∞ from Γ using the vertex conditions.Section IV is devoted to the proof of the spectral estimates.The geometric version of the Ambartsumian theorem is proven in Section V, while Section VI is entirely devoted to the counterexamples showing that the geometric Ambartsumian theorem is not always valid for quantum graphs, even if one restricts the analysis to just Laplace operators.Finally we treat the formula for the Euler characteristic, again under the assumption that the vertex conditions are asymptotically standard.

II. DEFINITIONS
A quantum graph is identified with a self-adjoint operator acting in the Hilbert space L 2 (Γ), where Γ is a metric graph considered as a collection of edges E n = [x 2n−1 , x 2n ], n = 1, 2, . . ., N joined together at the vertices V m , m = 1, 2, . . ., M seen as equivalence classes in the set V = {v j } 2N j 1 of all end-points.The action of the operator is given by the differential expression − d 2 dx 2 + q(x) where q is a real-valued absolutely integrable potential q ∈ L 1 (Γ).To make the differential operator self-adjoint we impose vertex conditions connecting limiting values of functions and their first derivatives at each vertex separately.In the previous papers 6,20 only the case of so-called standard vertex conditions was studied.Our goal here is to consider the most general vertex conditions.

Vertex Conditions
The differential operator is defined on the functions u from the Sobolev space Therefore the first derivative u is an absolutely continuous function on each edge.
The vertex conditions connect together the limiting values of the functions and its first derivatives at each vertex separately (local vertex conditions).The limiting values at the end points of the edges are where the limit is taken over x inside the edge.To make our model independent of the parameterisation of the edges we introduce the normal derivatives Then the vertex conditions at any vertex V m = {x m 1 , . . ., x m dm } of degree d m can be written by relating the d m -dimensional vectors Here S m is an arbitrary unitary matrix 21,24 .We shall require that each S m is irreducible, which ensures that the conditions are properly connecting: the requirement means that the vertex cannot be divided into several vertices while still preserving locality.Such vertex conditions reflect the topology of the graph.We form the matrix S as the block-diagonal matrix with blocks equal to the S m (in the basis where all boundary values u(x j ), ∂ n u(x j ) are arranged in accordance to the vertices they belong to).

The operator
When all members of the triple (Γ, L q , S) are properly introduced we may give a formal definition of the Schrödinger operator on a metric graph.
Definition 1.Let Γ be a finite compact metric graph, q a real valued absolutely integrable potential on the graph q ∈ L 1 (Γ), and S m be d m × d m irreducible unitary matrices.Then the operator L S q (Γ) is defined on the functions from the Sobolev space u ∈ W 1 2 (Γ \ V) such that −u + qu ∈ L 2 (Γ) and satisfy vertex conditions (1).
The operators L S q (Γ) are self-adjoint and their most important spectral properties are described below.

III. ON VERTEX CONDITION
The commonly used standard vertex conditions (also known as Kirchoff-, Neumann-and natural conditions) are given by imposing the relations In other words, u is required to be continuous in each vertex V m and the sum of normal derivatives should vanish there.In this case S m is given by: The corresponding self-adjoint operator will be denoted by L st q (Γ).We note that this operator is uniquely defined by the metric graph Γ and the potential q.The unitary matrix determining standard vertex conditions at all vertices in Γ will be denoted by S st (Γ).In the basis where all boundary values are arranged in accordance to the vertices, S st (Γ) has a block-diagonal form with the blocks given by S st m .It is clear that vertex conditions at different vertices can be treated separately.Pick a vertex of Γ and consider a star-graph with the same vertex conditions (as given in (1)).
Then the corresponding vertex scattering matrix S m,v (k) is given by where k 2 = E is the energy of the waves, see 16,17,24 .In particular (4) implies that S m is just the vertex scattering matrix for k = 1, i.e. S m = S m,v (1).
S m,v (k) is unitary and may be written using the spectral representation of S m , with eigenvalues e iθn and eigenvectors e n , as 14,15,24 S v (k) = ( From this it is seen that S v (k) has the same eigenvectors as S but the corresponding eigenvalues are in general k-dependent.The eigenvalues ±1 are invariant, and all other eigenvalues tend to 1 as k → ∞.As a consequence, if S is Hermitian -so that e iθn = ±1 for all n -S v (k) does not depend on k.We call such conditions non-Robin, and all other conditions resonant.We note that standard conditions (3) are non-Robin.If S m is not Hermitian we introduce the high energy limit of S m,v (k): where P (−1) is the orthogonal projection onto the eigenspace associated with −1.The highenergy limit of the block-diagonal matrix S v (k) is defined in an analogous way leading to the matrix S v (∞), which block structure may be finer that those in the parameter matrix S.
We let L Sv(∞) 0 denote the operator obtained from L S 0 by letting the vertex conditions be given by S v (∞) instead of S, i.e. we substitute S (= S v (1)) with S v (∞).The operator L Sv(∞) 0 (Γ) is not necessarily equal to L st 0 (Γ), even though the only possible eigenvalues of S v (∞) are 1 and −1.In fact the vertex conditions given by S v (∞) need not even be properly connecting -the blocks S m,v (∞) might be reducible and the operator therefore appropriate to a graph Γ ∞ obtained from Γ by dividing some vertices in Γ into several vertices.For later use we introduce the following terminology: Definition 2. We say that vertex conditions on Γ given by S are asymptotically properly connecting if the high energy limits of all vertex scattering matrices S m,v (∞) are irreducible.
If vertex conditions are asymptotically properly connecting, then Γ ∞ = Γ.Definition 3. We say that vertex conditions on Γ given by S are asymptotically standard This means that in a certain basis S v (∞) has a block-diagonal form with the blocks given by unitary matrices corresponding to standard vertex conditions.Note that this block structure may be incompatible with Γ, but it corresponds to Γ ∞ , which might be different from Γ.
Of course any non-Robin conditions are asymptotically properly connecting, similarly standard vertex conditions are asymptotically standard.Among all the non-Robin vertex conditions, only the standard conditions are asymptotically standard.
Every non-Dirichlet condition at a vertex of degree one is asymptotically standard, i.e.
Neumann.As noted, for vertices of degree d m ≥ 2, standard conditions are given by d m × d m blocks S m of the form (3). S m is Hermitian and unitary, and so has eigenvalues λ = ±1.For d m ≥ 2 it is easily verified that λ = −1 is of multiplicity 1 and the corresponding eigenspace is spanned by the vector (1, . . ., 1), and the eigenspace corresponding to λ = 1 is then the orthogonal compliment of (1, . . ., 1) in C dm .As we observed, the eigensubspaces of S v (k) are invariant under changes of k, the eigenvalues ±1 are stable and all other eigenvalues of S v (k) tend to 1 as k → ∞ 14,15 .Since S v (k) is unitary its set of eigenvectors is a basis, this implies Lemma 1.A necessary and sufficient condition for S to be asymptotically standard and asymptotically properly connecting is that for each m with d m ≥ 2, the multiplicity of −1 as an eigenvalue for S m is 1 with eigenspace spanned by the vector (1, . . .1), and for all m with d m = 1, the vertex condition is not Dirichlet.
The Laplace operator L Sv(∞) 0 (Γ ∞ ) plays a very important role in the studies of spectral asymptotics.Definition 4. Let L S q (Γ) be a Schrödinger operator on a metric graph Γ with vertex conditions determined by the matrix S. Let S v (∞) be the high-energy limit of the vertex scattering matrix and Γ ∞ -the corresponding metric graph obtained by chopping (if necessary) certain vertices in Γ so that the vertex conditions determined by S v (∞) are properly connecting for Note that the reference Laplacian is determined by the original graph Γ and the vertex conditions, but independent of the potential q.If the vertex conditions are non-Robin, then the reference Laplacian is just L S 0 (Γ).
The spectrum.
We denote the spectrum of an operator L by σ(L).Any self-adjoint operator L S q (Γ) is a finite rank perturbation in the resolvent sense of the operator L D q (Γ) with Dirichlet conditions imposed at the endpoints of all the edges.Since the spectrum of L D q (Γ) is discrete (Γ is finite and compact) and satisfies Weyl asymptotics this therefore also holds for the spectrum of L S q (Γ) (see 4,21 or 6 for details), i.e. λ n (L S 0 ) ∼ (πn/L) 2 , as n → ∞.Here L = N n=1 n is the total length of the graph.We will denote the eigenvalues, counting multiplicities, of L S q (Γ) by λ n (L S q ), with n = 0, 1, 2, . . .When we write σ(L S 1 q 1 (Γ 1 )) = σ(L S 2 q 2 (Γ 2 )) we mean not only equality as sets, but also that the multiplicities of all eigenvalues are equal.
The spectrum of the Laplace operator is much easier to determine, compared to the spectrum of the Schrödinger operator, since the eigenfunction equation on each edge possesses an explicit solution in terms of exponentials.More precisely the spectrum σ(L S 0 (Γ)) \ {0} is determined by the secular equation 12,18,19,23 det(S v (k)S e (k) − I) where S e (k) is the edge-scattering matrix given by 2×2 blocks in the representation, where the end-points are arranged in accordance to the edges.For non-Robin vertex conditions (i.e.Hermitian S) this is a generalised trigonometric polynomial, since S v (k) does not depend on k in that case S v (k) ≡ S (this follows from ( 4)).
Proposition 1. [After Theorem 5.1 6 ] Let the vertex conditions on Γ be non-Robin, i.e. defined by a matrix S which is not only unitary, but also Hermitian: Then the non-zero eigenvalues of L S 0 (Γ) are given as zeroes of a generalised trigonometric polynomial: • The function p(k) defined by ( 7) can be written in the form: with k-independent coefficients a j ∈ C and ω j ∈ R.
• The multiplicity of every eigenvalue λ n (L S 0 (Γ)) = k 2 n coincides with the order of the corresponding zero of the function p.
Proof.The proof of the first two statements is straightforward, since the matrix SS e (k) − 1 has entries that are generalised trigonometric polynomials and its determinant is also a generalised trigonometric polynomial.The last statement is proven in 21 and 32 .
For general vertex conditions the function p(k) is given by an expression similar to (8) but with the difference that the coefficients a j are polynomials in k.If p(k) is just a generalised trigonometric polynomial, then in special cases its zeroes are uniquely determined by their asymptotics (see Proposition 4).This fact that was used in the proof of Proposition 3, we are going to use it to prove Theorem 5.
We note also the following estimate for later use: Lemma 2. Let Γ be a finite compact graph of total length L with N edges, then for n > 2N the eigenvalues of L S 0 (Γ) satisfy the following estimates π L The proof is similar to the proof of formula (2.2) in 6 but with different constants that are obtained via the following observation: if we let L0 denote the closure of the Laplacian with domain C ∞ 0 (Γ \ V), then both L D 0 and L S 0 are self-adjoint extensions of L0 .This operator has deficiency indices (2N, 2N ), so L S 0 is at most a rank 2N perturbation of L D 0 in the resolvent sense.This Theorem implies in particular that the eigenvalues of L S 0 (Γ) cannot have multiplicities greater than 5N .This bound is far from optimal, but it is sufficient for our studies in Section IV.

The quadratic form
We calculate the quadratic -or rather sesquilinear -form associated with L S q (Γ), which we denote by We rewrite the vertex conditions in (1) to eliminate the occurrence of normal derivatives commutes with S m , applying it to both sides of equation ( 1) we get which may be seen as a generalized Dirichlet condition that u satisfies at each V m .Note that this does not imply that all boundary values of u at V m are zero, only that the projection of the boundary value vector onto N m is zero.If we apply P to equation (1) we get instead is invertible in N ⊥ m , so we obtain the generalised Robin condition The matrix P u(V m ) = 0, With this notation the expression for the quadratic form (10) may be rewritten as where the scalar product in N ⊥ m is inherited from C dm .It remains to determine the domain of the quadratic form, which of course containes the domain of the operator.We start with the reference Laplace operator L Sv(∞) 0 . Substituting (6) in the definition of A m , we see from (13) at every vertex V m , i.e. a generalized Dirichlet condition, and a generalized Neumann condition instead of the generalized Robin condition in (13).A comparison with (13)   and (14) shows that the quadratic form just coincides with the Dirichlet form u 2 and its domain is the closure of Dom(L ) with respect to the positive Dirichlet norm Γ) .To determine this closure, let u j be a Cauchy sequence with respect to u W 1 2 (Γ\V) in Dom(L S 0 ), so that u j ∈ W 1 2 (Γ) and satisfy (15).We then have that so the limit function as well as its first derivative will be square integrable, i.e. the limit belongs to W 1 2 (Γ\V) and satisfy the generalized Dirichlet condition (11).The limit function might not be continuously differentiable, hence ∂ n u(V m ) might not exists, so the generalised Neumann condition in ( 15) is not preserved.
We have proven that the domain of the quadratic form consists of all functions from the Sobolev space W 1 2 (Γ \ V) satisfying generalised Dirichlet conditions ( 11) at all vertices.The quadratic forms associated with the operators L S 0 (Γ) and L S q (Γ) can be considered as infinitesimally bounded perturbations of Q L Sv(∞)

0
. For this we note the following Sobolev or Gagliardo-Nirenberg inequality for arbitrary satisfying 0 < ≤ .
For a proof see for instance elementary proof in 6 .The estimate holds for finite compact metric graphs as well, provided ≤ min = min n=1,...,n n and the norms are replaced with the Lebesgue norm of the graph.
Applying Lemma 3 the potential and vertex terms in ( 14) may now be estimated as and M m=1 where A m is the operator norm of the matrix A m and d m is the degree of the vertex.Thus both perturbation terms are infinitesimally bounded with respect to Q L Sv(∞) 0 so the domains of all quadratic forms coincide and the operators are semibounded.

IV. SPECTRAL ESTIMATES FOR NON-ROBIN CONDITIONS
We start with a definition, which we found very useful in our studies.
Definition 5. Two unbounded, semi-bounded self-adjoint operators A and B with discrete spectra {λ n (A)} ∞ n=1 and {λ n (B)} ∞ n=1 , respectively are called asymptotically isospectral if If the spectra satisfy Weyl's asymptotics, then to ensure that the operators are asymptotically isospectral it is enough to require that for a certain C > 0 or at least where > 0.
For example if the operator B is a bounded perturbation of A, then (20) trivially holds.Hence the standard Schrödinger operator L st q (Γ) and the standard Laplacian L st 0 (Γ) are asymptotically isospectral, provided q ∈ L ∞ (Γ).In 6 it was proven that the standard Schrödnger and Laplace operators are asymptotically isospectral even if the potential is absolutely integrable q ∈ L 1 (Γ) (Theorem 4.3 from 6 ).We are going to generalise this theorem in order to include arbitrary vertex conditions.Repeating precisely the same arguments one may prove that L S q (Γ) and L S 0 (Γ) are asymptotically isospectral.But our goal is slightly more ambitious: we are going to prove that the Schrödinger operator L S q (Γ) is asymptotically isospectral to the reference Laplacian L Sv(∞) 0 (Γ ∞ ).This is useful primarily because the vertex conditions determined by S v (∞) are non-Robin and therefore the spectrum of the reference Laplacian is given by zeroes of generalised trigonometric polynomials.The price we need to pay is that the graph Γ ∞ may be different from Γ as described above.
Theorem 4. Let Γ be a finite compact metric graph, q ∈ L 1 (Γ) and S be a unitary matrix parameterising properly connecting vertex conditions.Then the Schrödinger operator L S q (Γ) and its reference Laplacian L Sv(∞) 0 (Γ ∞ ) (see Definition 4) are asymptotically isospectral, moreover the difference between their eigenvalues is uniformly bounded where C = C(Γ, q L 1 (Γ) , S) is independent of n.
Proof.The proof follows the same lines as one of Theorem 4.3 in 6 , but extra constants that are due to the more general vertex conditions are accommodated.We are going to use the min-max and max-min principles, see e.g. 30 .It is necessary to take into account that the eigenvalues are enumerated starting from λ 0 (not λ 1 ).We note that for a concrete choice of an n-dimensional subspace V 0 n the Rayleigh quotient will give an upper estimate when the min-max Theorem is used We choose V 0 n as the linear span of the eigenfunctions corresponding to the n lowest eigenvalues of L We will split the quadratic form and then estimate the terms separately; for this we introduce some notation.The Hermitian matrices A m in ( 14) can be written as m are defined using the spectral representation Here λ j (A m ) and e j denote the eigenvalues and eigenvectors of A m .
Writing q = q + − q − in terms of positive and negative parts, we may then estimate Thus using ( 24) and ( 22) we get an upper estimate for λ n−1 (L S q ) as The first quotient is equal to ) and the maximum is attained on For the second quotient we may use that To estimate the L ∞ -norm of the function u we are going to use the following: Proposition 2 (formula (2.4) from 6 ).Let Γ be a compact, finite metric graph.Then there exists a constant c = c(Γ) such that for any eigenfunction ψ of L S 0 (Γ) Hence we obtain the following naive estimate which we will use later, but does not in itself yield an estimate of λ n−1 (L S q ) in terms of ) that is uniform in n.
We will now consider the upper and lower estimates separately.We shall also assume that n is sufficiently large, since the precise value of a finite number of eigenvalues may affect the value of C, but not the existence of the estimate (21).

Upper estimate.
We will use the estimate (25).We divide functions u Here p is a natural number independent of n, but dependent on Γ and B + L S q , to be fixed later.Therefore as n increases the first function u 1 will contain an increasing number of terms, while the second function will always be given by a sum of p terms.

From the inequality |u
and the fact that q + and A (+) m are nonnegative we have That u 1 , u 2 are orthogonal is clear, and from this the orthogonality of u 1 and u 2 also follows: Taking this into account we arrive at To estimate the first form we use ( 26) and the Sobolev estimate ( 16) to estimate u(x) L∞ .
We get where we now introduced the constant we get The key point is that and p can be chosen in such a way that ) holds (see (34) below).
On the other hand, our naive approach ( 27) can be applied to the second form in (29)   with the only difference being that the number of eigenfunctions involved is p, not n Adding the estimates for Q(u 1 , u 1 ) and Q(u 2 , u 2 ) and using that u 2 2 we get We would get the desired estimate with C = 2c 2 Ip if we manage to prove that for a certain that may depend on n and p.We use the estimate for the eigenvalues of L Sv(∞) 0 given in Theorem 2: substituting ) with the lower bound and with the upper and setting = 1/n, we get the following inequality for the left-hand side of ( 34) We see that for any integer p > 5N − 1 + (1 + 2(L/π) 2 )I (where we remind the reader that I was defined in ( 30)) the expression is positive for sufficiently large n, and the difference between the eigenvalues possesses a uniform upper estimate: The constant C is at least as large as 2c 2 pI, but our proof holds only for sufficiently large n.Hence in order to obtain a uniform estimate one might need to increase C further.
Lower estimate.
To obtain a lower estimate we are going to use the max-min principle, and the first inequality in ( 24) Using the subspace V 0 n−1 defined in (23) we get Since u is orthogonal to V 0 n−1 it may be written as As before let us split the function u = u 1 + u 2 Note two important differences: • the function u 1 is given by the sum of p terms, where the number p is independent of n and will be chosen later, • the function u 2 is given by an infinite series, not by an increasing number of terms as the function u 1 in the proof of upper estimate.
The functions u 1 and u 2 thus exchange roles compared with the proof of the upper estimate.
Using the fact that A (−) m and q − are nonnegative we may split the quadratic form (compare (29)) Now the function u 1 is given by a finite number of terms and we may similarly to (32)   estimate where again To estimate the second form we use (26) and the Sobolev estimate (16) for max |u(x)| 2 .We get Taking into account we arrive at Summing the estimates (39) and ( 41) and taking into account that u 2 2 we get As before, to prove the desired uniform estimate it is sufficient to show that for large enough n the following expression can be made positive by choosing an appropriate : Again we use Theorem 2: we substitute λ n+p−1 (L Sv(∞) 0 ) with the lower bound and λ n−1 (L Sv(∞) 0 ) with the upper.As before we choose = 1/n, so the left-hand side of (42) becomes )I, then the expression is positive for sufficiently large n, hence a lower estimate holds Remark.A general inequality of the type |λ n (L S 1 q 1 (Γ))−λ n (L S 2 q 2 (Γ))| < C for two Schrödinger operators on the same metric graph Γ does not hold.The proven asymptotic isospectrality implies that the spectra of Schrödinger operators on Γ are close to the spectra of the reference Proof.The Schrödinger operator L S q (Γ) and its reference Laplacian L Sv(∞) 0 (Γ ∞ ) are asymptotically isospectral in accordance with Theorem 4. Hence the reference Laplacian, which is is asymptotically isospectral to the standard Laplacian L st 0 (I) on the interval.A standard Laplacian is asymptotically isospectral to the standard Laplacian on the interval only if it is isospectral, since the spectrum is given as zeroes of a generalised trigonometric polynomial and the following statement holds 6 : Proposition 4 (Theorem 5.2 6 ).Let f be the trigonometric polynomial The spectrum of L st 0 (I) is given by equidistant points k n , hence a scaling argument implies that the standard Laplacians on Γ ∞ and I are isospectral.The graphs Γ ∞ and I have the same total length, which can be seen from Weyl's asymptotics.Hence Γ ∞ is a metric graph having the same spectral gap -the difference between the two lowest eigenvalues -as the interval.It is well-known that the interval is a unique metric graph (up to removal of all degree two vertices) minimising the spectral gap among all graphs of the same total length.This result was proved independently and by different methods in 26 , 11 and 22 (See e.g.Theorem 3 from 22 ).Therefore the graph Γ ∞ is equivalent to the interval I.
Note that this theorem does not imply that the graph Γ ∞ coincides with the interval, it implies that Γ ∞ is equivalent to the interval, i.e. one gets the interval by removing all vertices of degree two in Γ ∞ .The standard conditions imply that the function and its derivative are continuous at such vertices.The points on the two metric graphs can be identified pairwise making the corresponding quantum graphs indistinguishable.Such graphs are considered equivalent.
Therefore comparing spectra of two standard Laplacians one may only hope to prove that the underlying metric graphs are equivalent.Degree two vertices is not the only problem when a metric graph is reconstructed from the spectrum of the standard Laplacian: explicit examples provided e.g. in 7 show that there exist isospectral not equivalent graphs.In the special case of an interval, however, the spectrum does determine the graph.
Moreover, the theorem does not imply that Γ is equivalent to the interval, a counterexample is provided at the end of Section VII.One of the graphs presented there is a cycle, but its spectrum is close to the spectrum of an interval.In order to guarantee that Γ coincides with I one has to assume that the vertex conditions are not only asymptotically standard, but are asymptotically properly connecting: Theorem 6.Let Γ be a connected, finite compact metric graph, with asymptotically standard and asymptotically properly connecting (see Definitions 2 and 3) vertex conditions given by the unitary matrix S, and let q ∈ L 1 (Γ).Suppose that the Schrödinger operator L S q (Γ) is asymptotically isospectral to the standard Laplacian on the interval I. Then Γ is equivalent to I, i.e. the interval I can be obtained from Γ by removing all vertices of degree two.
The proof is straightforward, since asymptotically properly connecting vertex conditions imply that Γ ∞ = Γ.Note that the class of asymptotically standard and properly connecting vertex conditions is characterised by Lemma 1. Theorem 6 does not imply that Γ = I.

VI. COUNTEREXAMPLES: GRAPHS ISOSPECTRAL TO THE INTERVAL [0, π].
Theorem 6 requires that the vertex conditions are asymptotically standard and asymptotically properly connecting.In this section we show that this requirement cannot always be relaxed by constructing a family of quantum graphs with non-Robin conditions that are all isospectral to the standard Laplacian L st 0 ([0, π]).The operator we consider is the Laplace operator, but the vertex conditions are not standard.

Elementary counterexample 1
Consider any unimodular function Θ(x) constant on each of the two edges, say Then the operators and L st 0 ([0, π]) are unitary equivalent.The operator A θ is the Laplace operator defined on the functions satisfying Neumann conditions at V 1 and V 3 and vertex conditions with The operators A θ and L st 0 are not only unitary equivalent, but the eigenfunction satisfy i.e. the probability densities for the eigenfunctions coincide.The eigenfunctions have support on one of the half-intervals, hence there is no chance that a version of formula (45) holds.On the other hand, this example is not very interesting, since the graph corresponding to the operator B is not connected.

Nontrivial counterexample
Consider again the graph Γ and impose Neumann conditions at V 1 and V 3 .At V 2 we impose any properly connecting non-Robin conditions via a vertex scattering matrix S 2 As Neumann conditions at the endpoints x 1 and x 4 determine scattering that is just reflection with reflection coefficient 1, we get the total scattering matrix for Γ In this basis the edge scattering matrix S e is given by As noted in (7) Hence the operators L S ( a,θ) 0 (Γ) and L st 0 ([0, π]) are isospectral.We note that the eigenfunctions for the eigenvalues λ = (2m) 2 are of the form The explicit formulas for the eigenfunctions show that a formula similar to (45) does not hold unless a = 0. Note that both operators A θ and B are formally included in the family L S(a,θ) 0 We have thus shown that the insertion of an extra middle vertex with vertex conditions given by an arbitrary non-diagonal unitary and Hermitian matrix does not change the spectrum of L st 0 ([0, π]).These conditions correspond to all possible properly connecting For a proof see 12,19,20,23,31 .
In the proof of Proposition 5 it is shown that the distribution which is determined by the spectrum of L S 0 may be calculated via the formula where S(k) = S v S e (k), and D is the following diagonal matrix D = diag { 1 , 1 , 2 , . . ., N }.
The graph Γ above has no loops and therefore no closed paths of discrete length 1, and similarly there are no closed paths of odd discrete length.Thus all periodic paths have discrete length 4n or 2(2n + 1).In the case of Γ we have D = π/2I, with I the 4 × 4 identity matrix, and S v and S e given by ( 46) and (47) respectively.Thus

VII. THE EULER CHARACTERISTIC
For finite compact connected quantum graphs with standard vertex conditions, it was noted in 20,23 that the secular equation ( 7) in general does not give the correct multiplicity of the ground state λ = 0.The difference between the multiplicities is determined by the Euler characteristic of the metric graph.This observation together with the trace formula (49) leads to an explicit formula formula for the Euler characteristic through the spectrum of the standard Laplacian on a metric graph Γ.The same formula holds if the Laplacian eigenvalues are substituted with the Schrödinger eigenvalues under the assumption that the vertex conditions are standard.This was proven for L ∞ and L 1 in 6,20 using the fact that these eigenvalues are situated close to each other.We have proven that this property holds for arbitrary vertex conditions with the only difference that the reference operator is not necessarily the standard Laplacian on Γ, but the Laplace operator L Sv(∞) 0 instead.Let us assume that the vertex conditions are asymptotically standard and asymptotically properly connecting so that L Sv(∞) 0 (Γ ∞ ) = L st 0 (Γ).Then following the proof in 6 Section 7 we get Theorem 8. Let Γ be a finite connected compact graph, let the potential be absolutely integrable q ∈ L 1 (Γ) and the vertex conditions determined by a unitary matrix S be asymptotically standard and asymptotically properly connecting.Then it holds with the convention that λ n = 0 implies sin λ n (L st q )/2t λ n (L st q )/2t 1)e iθn + (k − 1) (k − 1)e iθn + (k + 1) e n , • C d e n = d n=1 k(e iθn + 1) + (e iθn − 1) k(e iθn + 1) − (e iθn − 1) e n , • C d e n = θn=π (−1) e n , • C d e n + θn =π k(e iθn + 1) + (e iθn − 1) k(e iθn + 1) − (e iθn − 1) e n , • C d e n .
the eigenspace associated with the eigenvalue −1 of S m , and denote by P (−1) m the orthogonal projection onto N m and P (−1)⊥ m the projection onto N ⊥ m .Noting that P (−1) m m =: A m is Hermitian in N ⊥ m .?Thus every function in the domain of L S 0 satisfies the generalised Dirichlet and Robin conditions  We start by providing two elementary examples.Let Γ be the graph formed by two edges of length π/2, e 1 = [x 1 , x 2 ] = [0, π/2] ande 2 = [x 3 , x 4 ] = [π/2, π] and vertices V 1 = {x 1 }, V 2 = {x 2 , x3 }, and V 3 = {x 4 }.The graph Γ can be seen as the interval [0, π] with certain conditions introduced at the middle point x = π/2.

Elementary counterexample 2
Consider the Laplace operator B defined on the functions satisfying Neumann conditions at the both end points of the left edge [x 1 , x 2 ] = [0, π/2] and Dirichlet and Neumann conditions at the end points of the right edge [x 3 , x 4 ] = [π/2, π].The conditions at the middle vertex of Γ are not properly connecting and the corresponding operator is an orthogonalsum of the operators on the two edges, hence the spectrum is given by the union of the two spectra 0, 2 2 , 4 2 , . . .and 1, 3 2 , 5 2 , . . .and hence coincides with the spectrum of L st 0 ([0, π]).
We only require that S 2 is unitary, Hermitian and non-diagonal (which ensures that S 2 is properly connecting), so that a, d ∈ R and c = b.From the normalisation of the columns we get that c = √ 1 − a 2 e −iθ for some θ ∈ [0, 2π) and so b = √ 1 − a 2 e iθ , and from the orthogonality we get that a √ 1 − a 2 e iθ + d √ 1 − a 2 e iθ = 0, and since |a| < 1 we have d = −a, so the nonzero spectrum σ(L S(a,θ) 0 (Γ))\{0} is determined by the secular function det(S(a, θ)S e (k) − I) = 0 ⇔ det(S(a, θ) − S e (k)) = 0.The secular equation is det(S(a, θ) − S e (k)) =  0, which is the same as the secular equation for an interval of length π with standard, i.e.Neumann, conditions at the endpoints.Hence the non-zero eigenvalues of L S(a,θ) 0 (Γ) and L st 0 ([0, π]) coincide and it remains to study λ = 0.The eigenvalue zero of L S(a,θ) 0 (Γ) has multiplicity one, and corresponding eigenfunction is given by

S 2 S 4 =
4e 2iπk .This shows that in general Tr S 2(2n+1) = 0, so that periodic paths with an odd number of loops in Γ (by which we mean a loop in the interval [0, π/2] or a loop in [π/2, π], from which all closed paths in Γ may be formed) do not contribute to the formula (49).This mirrors the situation for loops in the interval [0, π], where no such loops exist since there is no middle vertex, and therefore do not contribute to the trace formula either.The above calculation also imply that in general we have S 4 n = e 2iπkn I, and this contribution coincides with that for the interval [0, π] since S e ([0, π])S v ([0, π])

= 1 .Example 1 .
This theorem implies that any two isospectral connected quantum graphs with asymptotically standard and asymptotically properly connecting vertex conditions should have the same Euler characteristics, i.e. should have the same complexity.Note that in the counterexamples of isospectral graphs constructed in 2 the vertex conditions are not asymptotically standard (Dirichlet conditions are introduced as certain vertices), therefore their Euler characteristics are different.We now provide an explicit example showing that formula (52) does not in general give χ(Γ).Let Γ be a circle of circumference π with one vertex, i.e.Γ = [0, π] with V = {0, π}.Let the vertex condition at V be given by a unitary 2 × 2 matrix S with eigenvalues λ 1 , λ 2 = −1, e.g. S = noted in (6) we then haveS v (∞) = I − 2P (−1) = I. S v (∞) is therefore not properly connecting -the appropriate graph Γ ∞ is instead the interval [0, π]with Neumann-Neumann conditions, i.e. standard conditions at both vertices.Then Theorem 8 implies that the limit in (52) is equal to χ(Γ ∞ ) = 2 − 1 = 1 = χ(Γ) = 1 − 1 = 0. We mention that this example has already been used in33 as a model of topology change in a quantum system.