Appendix A: An Additional Approach for Deriving the Mean Robin–Neumann Gap

In this appendix, we present an alternative derivation for the mean value of the Robin–Neumann gap. This is done by considering a so-called local average of the RNG with respect to the wave number k (instead of averaging with respect to n). This approach is not as rigorous as the proof of Theorem 1.3. Nevertheless, it is advantageous in that it provides not only the limiting mean value of the RNG, but also the running mean as it depends on k, see Fig. 2.

We begin by considering the situation where the Robin condition is imposed at a single vertex, and later generalize to multiple vertices. In this case, we know that the eigenvalues interlace (see [8, thm. 3.1.8]); if \(\sigma <\sigma '\), then for all \(n\in \mathbb \)

$$\begin k_\left( \sigma \right) \le k_\left( \sigma '\right) <k_\left( \sigma \right) . \end$$

(A.1)

This can be rewritten in terms of the number counting function \(\mathcal \left( k,\sigma \right) :=\left| \left\:k_\le k\right\} \right| \). Mainly, this means that the spectral shift, which is the difference between the number counting functions at fixed k, may only take the values zero and one,

$$\begin \Delta ^\mathcal \left( k\right) :=\mathcal \left( k,0\right) -\mathcal \left( k,\sigma \right) \in \left\ . \end$$

(A.2)

We denote the length of the intervals where the spectral shift is equal to one by

$$\begin \delta _\left( \sigma \right) :=k_\left( \sigma \right) -k_\left( 0\right) . \end$$

(A.3)

Note that these intervals are defined similarly as the RNG, but for the difference between the k values rather than the eigenvalues.

By the Weyl law for metric graphs (see [8, 15]) for a fixed value of \(\sigma \), the mean distance between consecutive values of \(\left\(\sigma )\right\} \) is

$$\begin&\langle \Delta k\rangle :=\pi /\left| \Gamma \right| . \end$$

(A.4)

Hence, for large \(K>0\), the interval \(\left[ k-K/2,k+K/2\right] \) contains on average \(N:=K/\langle \Delta k\rangle \) values from \(\left\(\sigma )\right\} \). Thus, defining a local k-average, the spectral shift in k is equal to:

$$\begin \overline\mathcal }\left( k\right)= & \frac\int _^\Delta \mathcal ^\left( k'\right) dk'\approx \frac\cdot \frac\sum _+1}^+N}\delta _\left( \sigma \right) =\frac\left( k\right) },\nonumber \\ \end$$

(A.5)

where \(\+1,\ldots ,N_+N\}\) are the indices of the \(\left\(\sigma )\right\} \) values which are contained in the interval \(\left[ k-K/2,k+K/2\right] \) (on average). Hence, \(\delta ^\left( k\right) :=\frac\sum _+1}^+N}\delta _\left( \sigma \right) \) is the mean spectral shift around k. The expression above holds up to an error of order \(N^\) due to the limits of the integration interval.

To evaluate the mean spectral shift above, we use the trace formula for the counting function (as derived in [16, 17]):

$$\begin \mathcal ^\left( k,\sigma \right) =\frac+\frac\cdot Im\sum _^\fracU^\left( k,\sigma \right) }. \end$$

(A.6)

Here, \(U\left( k,\sigma \right) :=S^e^\) is the unitary scattering matrix (as in Theorem 2.4) and \(\Theta \left( k,\sigma \right) :=\log \left( \det \left( U\left( k,\sigma \right) \right) \right) \) is known as the total phase of \(U\left( k,\sigma \right) \). Under the assumption that K is large enough (mainly, that \(K\gg \pi /\ell _\), see [16, 17]), the contribution of the oscillatory term in (A.6) is suppressed by the averaging, and in leading order we have that

$$\begin \overline\mathcal }\left( k\right) =\frac-\overline}. \end$$

(A.7)

The total phase was evaluated in [17] as

$$\begin \Theta \left( k,\sigma \right) =2k\left| \Gamma \right| -2\sum _}} \arctan \left( \frac\right) . \end$$

(A.8)

Plugging this into (A.7) and then using (A.5) gives that for a single Robin vertex

$$\begin \delta ^\left( k\right)= & \frac\arctan \left( \frac\right) . \end$$

(A.9)

Finally, we can define the k-averaged Robin–Neumann gap by

$$\begin \left\langle d\right\rangle _\left( \sigma \right) :=\left( k+\delta ^\left( k\right) \right) ^-k^, \end$$

(A.10)

which under the assumption \(\delta ^\left( k\right) \ll k\), and together with (A.9), gives the following:

$$\begin \left\langle d\right\rangle _\left( \sigma \right) \approx 2k\delta ^\left( k\right) =\frac\arctan \left( \frac\right) . \end$$

(A.11)

For the more general case where the Robin condition is imposed at several vertices, we can repeat the same proof (applying the additivity of Eq. (A.8)) to obtain

$$\begin \left\langle d\right\rangle _\left( \sigma \right) =\frac \sum __}}\arctan \left( \frac\right) . \end$$

(A.12)

Note that for \(k\rightarrow \infty \), one can recover the rigorously obtained expression from Theorem 1.3 by first-order approximation of (A.12). On the other hand, for \(k\rightarrow 0\) the average gap approaches zero. The average sensitivity of the gaps with respect to a change of the Robin parameter is obtained by differentiating (A.12) with respect to \(\sigma \):

$$\begin&\left\langle \frac\right\rangle _=\frac\sum _}} \frac+\lambda \deg \left( v\right) ^} \end$$

(A.13)

$$\begin&\approx _\frac\sum _}}\frac. \end$$

(A.14)

A crucial assumption in the derivations above was that the averaging interval K contains many eigenvalues, which is required to neglect the oscillating terms in (A.6). At the same time, K must be small enough so that the value of (A.9) does not change by a large amount inside the given interval. Otherwise, the definition of a local average of the gap is not meaningful. The two conditions can only be met for graphs with a large metric length \(\left| \Gamma \right| \rightarrow \infty \). Alternatively, one may employ an ensemble average over graphs where the topology is fixed and the edge lengths are varied. Having written the above, we refer to Figs. 2, 12, and 6, which demonstrate how close is (A.12) to a running mean value obtained by averaging over 21 adjacent eigenvalues.

Appendix B: Omitting the Assumption of Independence Over \(\mathbb \)

Recall that in order to apply the ergodic theorem (Theorem 2.7), we added the assumption that the entries of the vector of edge lengths \(\vec \) are linearly independent over \(\mathbb \). We now show that the results of Theorem 1.4 in fact hold without this assumption.

Proposition B.1

Assumption 4.1 can be omitted in Theorem 1.4.

Proof

We give the proof for expression (1.8), where the proof of (1.10), (1.11) is similar. This is a simple denseness argument. Fix a discrete graph \(G=\left( \mathcal ,\mathcal \right) \) and \(v\in \mathcal \). Denote:

$$\begin \mathbb _^=\left\\in \mathbb ^:x_>0,\,\,\forall i\in \left\ \right\} . \end$$

(B.1)

For \(\vec \in \mathbb _^\), denote by \(\Gamma _}\) the metric graph obtained by assigning the vector of edge lengths \(\vec \) to the fixed combinatorial graph G.

Define the following function:

$$\begin&\phi _:\mathbb _^\rightarrow \mathbb ^}, \end$$

(B.2)

$$\begin&\phi _\left( \vec \right) =\left( \left| f_^}\left( v\right) \right| ^, \left| f_^}\left( v\right) \right| ^,...\right) , \end$$

(B.3)

where \(f_^}\) is an \(n^}\) \(L^\) normalized eigenfunction for the metric graph \(\Gamma _}\) as in Theorem 1.4.

Denote by P the subset of \(\mathbb _^\) of vectors whose coordinates are rationally independent. This is a dense subset of \(\mathbb _^\). Denote the set of Cesàro summable sequences by \(\mathcal \). Define the following functions:

$$\begin&\phi _:\mathcal \rightarrow \mathbb , \end$$

(B.4)

$$\begin&\phi _\left( \left( c_\right) _^\right) =\lim _\frac\sum _^c_\end$$

(B.5)

$$\begin&\phi :P\rightarrow \mathbb ,\end$$

(B.6)

$$\begin&\phi =\phi _\circ \left( \phi _|_\right) . \end$$

(B.7)

By the version we proved for Theorem 1.4, \(\phi \) is a well-defined function on P. For all \(\vec \in \mathbb _^\), there exists a neighborhood \(U\subset \mathbb _^\) of \(\vec \), such that \(\phi \) is uniformly continuous on \(U\cap P,\) since it is simply given by the expression:

$$\begin \phi \left( \vec \right) =\frac^\ell _}. \end$$

(B.8)

Since P is dense in \(\mathbb _^\) and \(\phi \) is locally uniformly continuous (in the sense mentioned above), it can be extended into a continuous function \(}\) on \(\mathbb _^\).

To complete the proof, we should show that \(}=\phi _\circ \phi _\) and that \(}\) is given by expression (B.8). This is based on standard topological arguments which we merely sketch here, and refer the interested reader to [26, prop. 5.16] for further details. First, one shows that \(\phi _\) is continuous. From here follows \(\phi _\left( \mathbb _^\right) \subset \mathcal \), using that set of Cesàro summable sequences is closed. Now one gets that \(\phi _\circ \phi _\) is well defined and continuous and concludes \(}=\phi _\circ \phi _\) since those functions agree on the dense set P. Finally, the continuity of \(}\) implies that it is indeed given by (B.8) and completes the proof. \(\square \)

Remark B.2

It is worth noting that while the limiting mean values in Theorems 1.3, 1.4 are the same for the rationally dependent case, the behavior of the sequences themselves might be drastically different in that case. For instance, for the case of an equilateral star graph, one can show that the sequence of RNG accumulates around two values and does not get close to the mean value. Nevertheless, the Cesàro mean converges to its expected value as in Theorem 1.3. The above is nicely exemplified in Fig. 5.

Appendix C: Optimality of the Bounds on the RNG1.1 C.1: Optimality of the Lower Bound in Lemma 6.1 and Theorem 1.7

Under certain assumptions, the lower bound of zero in Lemma 6.1 and Theorem 1.7 is optimal. This happens when the corresponding graph allows for Robin eigenfunctions whose absolute values at the set \(\mathcal }\) are arbitrarily small. This results in very low sensitivity to the Robin condition (i.e., small value of \(\frac\lambda _}\sigma }\)), giving an arbitrarily small value to the RNG. We demonstrate this for two typical cases:

(i)

The graph contains a cycle.

(ii)

The graph is a tree, where at least two leaves (denoted by \(v_,v_\)) are not contained in \(\mathcal }\).

For the graphs above, eigenfunctions with low sensitivity to the Robin condition exist. In the case where the edge lengths of the graph are linearly dependent over \(\mathbb \) (rationally dependent), constructing such eigenfunctions is simple. In fact, one can construct eigenfunctions which vanish on the set \(\mathcal }\). By Lemma 2.1, these eigenfunctions have zero sensitivity to the Robin condition (i.e., \(\frac\lambda _}\sigma }=0\) for all \(\sigma \in \mathbb \)), and they thus give zero RNG—\(d_\left( \sigma \right) =0\) for all \(\sigma >0\). We now point out the existence of these eigenfunctions (see also Figs. 10 and 11).

Fig. 10

Two examples of states with zero sensitivity: Case (i) of a graph with a cycle, where the Robin vertices are placed at the four vertices of the inner square. Case (ii) of a star graph with Robin condition at the central vertex. In both cases the eigenfunction vanishes at the set \(\mathcal }\), resulting in zero sensitivity, i.e., \(\frac\lambda _}\sigma }\)=0 for all \(\sigma \in \mathbb \)

In case (i), one can select a cycle on the graph and choose the wave number k such that all edge lengths in the cycle are integer multiples of the wave length \(\Lambda :=2\pi /k\). Under these conditions, there exist eigenfunctions which vanish on the entire graph apart from the given cycle. In particular, those eigenfunctions vanish at all vertices on the given cycle. In case (ii), consider the (unique) path connecting the vertices \(v_,v_\). Then similar to before, one can choose edge lengths and k such that all edges in the path are integer multiples of \(\Lambda \), with the exception of the edges adjacent to \(v_,v_\), to which an additional \(\Lambda /4\) is added. Under these conditions, there exist eigenfunctions which vanish on all of the graph apart from the given path. In particular, those eigenfunctions vanish at all interior vertices along the given path, and their derivative at \(v_,v_\) vanishes. All eigenfunctions described above (for both cases (i) and (ii)) vanish at \(\mathcal }\), as required.

Fig. 11

The eigenvalue curves for a tetrahedron graph with \(\mathcal }=\mathcal \). The horizontal red curve at \(k=2\pi \) corresponds to a state which is supported on a triangle subgraph and vanishes at all \(v\in \mathcal }\). This gives a state with zero sensitivity as in case (i), which results in \(d_\left( \sigma \right) =0\)

While it is impossible to construct eigenfunctions with zero sensitivity in the general case of rationally independent edge lengths, one can still use the construction above to find eigenfunctions with arbitrarily small sensitivity. This can be done by approximating the given edge lengths with rationally dependent edge lengths and then applying the method above. This will give a sequence of eigenfunctions whose value at \(\mathcal }\) is arbitrarily small, resulting in a subsequence of \(d_\left( \sigma \right) \) which tends to zero.

The frequency of such eigenfunctions with low sensitivity has been estimated in [24]. It depends on the number of edge lengths that determine the value of k in the construction above. For example, these eigenfunctions appear more frequently in Fig. 12a than in b. This is since in case (a) of a star graph the supporting path which determines k contains two edges, while in case (b) of a tetrahedron it contains three edges forming a cycle. The figures show that while the lower bound is not attained in these cases, it is still tight.

Fig. 12

Scatter plot of the eigenvalue sensitivities \(\frac}\) for a the star graph from Fig. 2 and b the tetrahedron graph from Fig. 6. The values are scaled so that the mean sensitivity is one. The light blue lines are running averages, and the blue lines on top of it are the analytic results from Eqs. (A.13), (A.14). The red lines are the upper bound of (6.2) with and without the second term, respectively

1.2 C.2 Optimality of the Upper Bound in Lemma 6.1

In order to construct eigenfunctions which attain the upper bound of Lemma 6.1, equality must hold in Eqs. (6.14), (6.18), (6.19), and (6.20). We provide two examples for specific graphs which satisfy this.

For the first example, consider an equilateral star graph with edges of length \(\ell \) and Robin condition at the central vertex \(v_\). On each edge, the \(L^\) normalized eigenfunctions can be written as

$$\begin f_\left( x_\right) =\frac)\ell }}\cos \left( kx_-\varphi _}\right) , \end$$

(C.1)

where the edges are parameterized so that the central vertex \(v_\) corresponds to \(x_=0\) for all e. Then by choosing \(k>0\) such that

$$\begin \tan k\ell =\frac)}, \end$$

(C.2)

and taking \(\varphi _}=k\ell \), we get a valid Robin eigenfunction. Moreover, this eigenfunction does not vanish at any vertex, so equality holds in (6.14). Now, choose a trivial star decomposition, where the points \(u_\) are located at the outer vertices. In this case, all terms in (6.14) which correspond to the outer vertices vanish. For the remaining vertex \(v_\), (6.18) is satisfied and equality holds in (6.19). This also gives equality in (6.20).

The assumption of equal edge lengths which was used for this construction can be relaxed. Similar to the previous subsection, choosing arbitrary edge lengths will lead to the existence of eigenfunctions with sensitivity arbitrarily close to the upper bound. However, since all edges of the graph are involved in the construction, the probability of such an eigenfunction is much lower than in the case of approaching the lower bound zero (where only a small subset of the edges were involved), see Fig. 12. Moreover, condition (C.2) which is used to determine k depends on \(\sigma \) and can be satisfied at most at isolated points along the integral in Eq. (1.14). Therefore, unlike the lower bound zero, the upper bound in Theorem 1.7 cannot be realized by this construction.

The second construction is similar, but concerns a regular graph with no neighboring Neumann vertices. For this construction, one should take the length of edges connecting a Robin vertex to a Neumann vertex to be \(\ell \) and the length of an edge connecting two Robin vertices to be \(2\ell \). Just as before, choose k according to (C.2) above and \(\varphi _=k\ell \) for all \(v\in \mathcal \). This will once again give a valid eigenfunction. Now, choose a star decomposition which is attained by splitting the edges which connect two Robin vertices in the middle, i.e., \(s_}=\ell \) for all Robin vertices and \(s_}=0\) for all Neumann vertices. Then just as in the example above, the Neumann vertices have zero contribution to (6.14), while for Robin vertices (6.18) hold and so there is equality in (6.19). Since all star graphs around Robin vertices are identical, equality holds in (6.20) as well.