Let \(\mathcal {G} = ({\textsf{V}},{\textsf{E}})\) be a finite, connected, compact metric graph with vertex set \({\textsf{V}}\) and edge set \({\textsf{E}}\), see, for instance, [3, 15, 17, 18] for a more detailed introduction. The graph Hilbert space is

$$\begin{aligned} L^2(\mathcal {G})=\bigoplus _{{\textsf{e}}\in {\textsf{E}}} L^2(0,\ell _{\textsf{e}}), \end{aligned}$$

where \(\ell _{\textsf{e}}\in (0,\infty )\) denotes the edge length of the edge \({\textsf{e}}\in {\textsf{E}}\). On \(L^2(\mathcal {G})\), assuming—as in [4]—that

$$\begin{aligned} q =(q_{\textsf{e}})_{{\textsf{e}}\in {\textsf{E}}} \in L^\infty ({\mathcal {G}}) \cap \bigoplus _{{\textsf{e}}\in {\textsf{E}}} C^\infty (0,\ell _{\textsf{e}}), \end{aligned}$$

and \(\sigma =(\sigma _{\textsf{v}})_{{\textsf{v}}\in {\textsf{V}}} \in \mathbb {R}^{|{\textsf{V}}|}\) (for instance, one could think of potentials \(q \in \bigoplus _{{\textsf{e}}\in {\textsf{E}}}C^{\infty }[0,\ell _{\textsf{e}}]\)), we introduce the self-adjoint Schrödinger operator

$$\begin{aligned} H_\mathcal {G}^{q,\sigma }=-\frac{\textrm{d}^2}{\textrm{d}x^2}+q(x) \end{aligned}$$

associated with the quadratic form,

$$\begin{aligned} h_\mathcal {G}^{q,\sigma }(f) = \mathop {\int }\limits _\mathcal {G} \vert f^{\prime } \vert ^2\ \textrm{d}x + \mathop {\int }\limits _\mathcal {G} q \vert f \vert ^2\ \textrm{d}x+\sum _{{\textsf{v}}\in {\textsf{V}}}\sigma _{\textsf{v}}|f({\textsf{v}})|^2 \end{aligned}$$

with form domain

$$\begin{aligned} H^1(\mathcal {G}):= \bigg (\bigoplus _{{\textsf{e}}\in {\textsf{E}}} H^1(0,\ell _{\textsf{e}})\bigg ) \cap C(\mathcal {G}); \end{aligned}$$

here \(C(\mathcal {G})\) denotes the space of continuous functions on \(\mathcal {G}\). Note that functions \(f=(f_{{\textsf{e}}})_{{\textsf{e}}\in {\textsf{E}}} \in C(\mathcal {G})\) are, in particular, continuous across vertices, that is, \(f_{\textsf{e}}({\textsf{v}}) = f_{{\textsf{e}}'}({\textsf{v}})\) for all \({\textsf{v}}\in {\textsf{V}}\) and any edges \({\textsf{e}},{\textsf{e}}' \in {\textsf{E}}\) that are incident to \({\textsf{v}}\). Although this is mostly not relevant for us in this paper, we shall recall that each \(\sigma _{\textsf{v}}\) determines the so-called \(\delta \)-coupling conditions in a given vertex \({\textsf{v}}\in {\textsf{V}}\) of the graph: More explicitly, the operator domain of the self-adjoint operator \(H^{q,\sigma }_{\mathcal {G}}\) is given by all functions \(f=(f_{{\textsf{e}}})_{{\textsf{e}}\in {\textsf{E}}} \in H^1(\mathcal {G})\) for which \(f_{{\textsf{e}}} \in H^2(0,\ell _{\textsf {e}})\) for every \({\textsf{e}}\in {\textsf{E}}\) and

$$\begin{aligned} \forall {\textsf{v}}\in {\textsf{V}}: \quad \sum _{{\textsf{e}}\sim {\textsf{v}}}(\partial _{n}f_{{\textsf{e}}})({\textsf{v}})=\sigma _{{\textsf{v}}}f({\textsf{v}}). \end{aligned}$$
(1)

Hence, the sum of all (normal) derivatives of functions (meaning the derivative pointing into the adjacent edges) in the operator domain equals \(\sigma _{\textsf{v}}\) times the value of the functions in the vertex \({\textsf{v}}\in {\textsf{V}}\) which is well-defined due to continuity of the functions across vertices. Whenever \(\sigma _{\textsf{v}}=0\), one speaks of standard vertex conditions.

Since we only look at finite, connected, compact metric graphs, the operator \(H_\mathcal {G}^{q,\sigma }\) has compact resolvent and therefore purely discrete spectrum; we denote its eigenvalues by

$$\begin{aligned} \lambda _0^{q,\sigma } < \lambda _1^{q,\sigma } \le \lambda _2^{q,\sigma } \le \dots \rightarrow +\infty , \end{aligned}$$

taking into account that the ground state is unique [13]. As a main result of [4], the authors established the relation

$$\begin{aligned} \lim _{N \rightarrow \infty }\frac{1}{N}\sum _{n=1}^{N}(\lambda _n^{q,\sigma }-\lambda _n^{0})=\frac{1}{\mathcal {L}}\bigg (\mathop {\int }\limits _{\mathcal {G}}q\ \textrm{d}x+2\sum _{{\textsf{v}}\in {\textsf{V}}}\frac{\sigma _{\textsf{v}}}{\text {deg}({\textsf{v}})} \bigg ), \end{aligned}$$
(2)

where \(\mathcal {L}=\sum _{{\textsf{e}}\in {\textsf{E}}} \ell _{\textsf{e}}\) denotes the total length of the graph \(\mathcal {G}\) and \(\text {deg}({\textsf{v}})\) is the degree of a vertex \({\textsf{v}}\in {\textsf{V}}\). Here, \((\lambda _n^0)_{n \in \mathbb {N}_0}\) shall denote the eigenvalues of the operator \(H^{q=0,\sigma =0}_\mathcal {G}=:H^{0}_\mathcal {G}\).

Using (2) and based on the well-known papers [1, 6, 10, 11, 16], the classical question we would like to address in this note is the following: Assuming we know that the eigenvalues of a Schrödinger operator \(H^{q,\sigma }_\mathcal {G}\) are identical to those of \(H^{0}_\mathcal {G}\), is it possible to conclude that \(q=0\) as well as \(\sigma =0\)? Or more generally, is it possible to reconstruct the potential and the vertex conditions in the vertices of the graph by knowing the spectrum of \(H_\mathcal {G}^{q,\sigma }\)? The second question was already answered in the negative by Borg in [6] where he looked at Laplacians on intervals subject to Robin boundary conditions, see also [16]. However, as already noted by Ambarzumian in [1], assuming the eigenvalues of \(H^{q,\sigma =0}_\mathcal {G}\) on an interval are exactly those of the Neumann Laplacian on this interval, then one necessarily has that \(q=0\). As we shall see in the sequel, a key role (as already noted by Borg) is played by the fact that the lowest Neumann eigenvalue—in our case of a graph with \(\sigma =0\)—is zero (and therefore non-negative).

For us, an important paper which provided a generalization of Ambarzumian’s classical theorem to the context of quantum graphs is the one by Davies [8]. In this paper, Davies compared two self-adjoint operators with purely discrete spectrum: a non-negative self-adjoint operator \(H_0\) with eigenvalues \(({\tilde{\lambda }}_n)_{n \in \mathbb {N}_0}\) and the self-adjoint operator \(H=H_0+V\), where V is real-valued and bounded, and with eigenvalues \((\mu _n)_{n \in \mathbb {N}_0}\), both defined on a compact metric measure space. In [8, Theorem 3.3], Davies then proved that, under certain conditions which are satisfied in the special case of finite, compact, connected quantum graphs with \(\sigma =0\), that if the first eigenvalue of H is non-negative and \(\limsup _{n \rightarrow \infty }(\mu _n-{\tilde{\lambda }}_n) \le 0\), then \(V=0\)! Note that Davies’ result contains the classical theorem of Ambarzumian for the Neumann Laplacian on an interval as a special case.

Before we proceed, we shall also refer the reader to the papers [5, 7, 9, 12, 19, 20] and references therein. In particular, we encourage the reader to study [15, Chapter 14] which provides an extensive discussion of Ambarzumian-type theorems in the context of quantum graphs. For example, in our paper, we always assume that the underlying metric graph is fixed and we try to reconstruct the potential as well as the vertex conditions in the vertices. However, as discussed in [15, Theorem 14.11] based on [5], one may also be able to reconstruct the graph!

We start by formulating a direct but relevant consequence of the results obtained in [4], see (2).

FormalPara Theorem 1

Assume that \(H_\mathcal {G}^{q,\sigma }\) is a Schrödinger operator on a metric graph \(\mathcal {G}\) such that

$$\begin{aligned} \lambda _{n}^{q,\sigma } - \lambda _n^0 \rightarrow 0 \, \quad \text {as }n \rightarrow \infty , \end{aligned}$$

where \((\lambda _n^0)_{n \in \mathbb {N}_0}\) are the eigenvalues of \(H_\mathcal {G}^0\). Then, one has

$$\begin{aligned} \int \limits _{\mathcal {G}} q\ {\text {d}}x + 2 \sum _{{\textsf {v}}\in {\textsf {V}}} \frac{\sigma _{\textsf {v}}}{\textrm{deg}(\textsf {v})} = 0. \end{aligned}$$
(3)
FormalPara Remark 2

It is interesting to remark that, in the case of isospectrality, condition (3) can also be recovered from the the \(\sqrt{t}\)-coefficient of the heat-trace asymptotics as derived in [2, (8.14)].

In a next step, we formulate a statement that is somehow complementary to Theorem 1 in the sense that the assumed isospectrality of \(H_\mathcal {G}^{q,\sigma }\) and \(H_\mathcal {G}^0\) yields certain relations that have to be satisfied by q and \(\sigma \).

More explicitly, going back to what has been said above, Davies has focussed on the case where \(\sigma =0\). Therefore, we provide a generalization of [8, Theorem 3.1] to cover the case of non-vanishing \(\sigma \) (in other words, we allow for general \(\delta \)-coupling conditions rather than only standard vertex conditions in which case \(\sigma =0\)).

FormalPara Theorem 3

Assume that the lowest eigenvalue of \(H_\mathcal {G}^{q,\sigma }\) is non-negative and that

$$\begin{aligned} \mathop {\int }\limits _{\mathcal {G}}q\ \textrm{d}x + \sum _{{\textsf{v}}\in {\textsf{V}}}\sigma _{\textsf{v}}\le 0. \end{aligned}$$
(4)

Then \(q=0\) and \(\sigma =0\).

FormalPara Proof

One first observes that, inserting the function which is constant equal to one on \(\mathcal {G}\) into the quadratic form \(h_\mathcal {G}^{q,\sigma }(\cdot )\), the min-max principle yields the inequality

$$\begin{aligned} 0 \le \lambda _0^{q,\sigma }\le \frac{1}{\mathcal {L}}\left( \mathop {\int }\limits _{\mathcal {G}}q\ \textrm{d}x+\sum _{{\textsf{v}}\in {\textsf{V}}}\sigma _{\textsf{v}}\right) \le 0 \end{aligned}$$
(5)

which can be satisfied only whenever \(\int _{\mathcal {G}}q\ \textrm{d}x+\sum _{{\textsf{v}}\in {\textsf{V}}}\sigma _{\textsf{v}}=0\) as well as \(\lambda _0^{q,\sigma }=0\). In a next step, we employ the uniqueness of the ground state as established in [13]. More explicitly, (5) tells us that the function that is constant equal to one on \(\mathcal {G}\) is indeed a minimizer of the Rayleigh quotient

$$\begin{aligned} \frac{h_\mathcal {G}^{q,\sigma }(f)}{\Vert f\Vert ^2_{L^2(\mathcal {G})}},\quad f\in H^1(\mathcal {G}), \end{aligned}$$

and since the minimizer is unique, this constant function must be the ground state of the operator \(H^{q,\sigma }_{\mathcal {G}}\). Consequently, it would be an element of the operator domain which is, taking (1) into account, only possible whenever \(\sigma =0\). Hence we are left with \(\int _{\mathcal {G}}q \ \textrm{d}x = 0\) and we conclude \(q=0\) with [8, Theorem 3.1]. \(\square \)

FormalPara Remark 4

To put things into context, it is interesting to compare the statements of Theorem 1 and Theorem 3 with some results of [14]. More explicitly, it can be shown that the Schrödinger operator on the interval \(\mathcal {I}=[0,1/2]\) with potential \(q(x)=\frac{2}{(1+x)^2}\) and \(\sigma _0=-1\) and \(\sigma _1=\frac{2}{3}\) has the same spectrum as the Laplacian subject to Neumann boundary conditions. So, in general and as elaborated on in [14] (see also [15]), Ambarzumian-type theorems cannot be expected to hold in utmost generality. However, note that this counterexample does not contradict the above theorems (Theorem 1 and Theorem 3). Indeed, although one has

$$\begin{aligned} \mathop {\int }\limits _\mathcal {I} q\ \textrm{d}x + 2 \sum _{{\textsf{v}}\in {\textsf{V}}} \frac{\sigma _{\textsf{v}}}{\text {deg}({\textsf{v}})}= \mathop {\int }\limits _0^{1/2} \frac{2}{(1+x)^2}\ \textrm{d}x -\frac{2}{3} =0, \end{aligned}$$

one has at the same time that

$$\begin{aligned} \mathop {\int }\limits _\mathcal {I} q\ \textrm{d}x +\sum _{{\textsf{v}}\in {\textsf{V}}} \sigma _{\textsf{v}}= \mathop {\int }\limits _0^{1/2} \frac{2}{(1+x)^2}\ \textrm{d}x -\frac{1}{3}=\frac{1}{3} > 0. \end{aligned}$$
FormalPara Remark 5

In view of Remark 4, we note that Theorem 1 provides an answer to a question raised in [15, Problem 70].

FormalPara Example 6

The condition formulated in Theorem 3 gives a rather simple criterion for a Schrödinger operator \(H_\mathcal {G}^{q,\sigma }\) to have a spectrum different from that of the operator \(H^{0}_\mathcal {G}\). This is for instance the case for the operator \(H_\mathcal {G}^{0,\sigma }\) on a 3-star with a negative \(\delta \)-coupling condition at the central vertex and standard conditions at the outer vertices (see Fig. 1).

Fig. 1
figure 1

3-star with central vertex \({\textsf{v}}_c\) and boundary vertices \({\textsf{v}}_1,{\textsf{v}}_2,{\textsf{v}}_3\)

Full size image

As alluded to above, we list some consequences of Theorem 1 and Theorem 3, one being another proof of the classical Ambarzumian theorem, as discussed in [1, 6], regarding Schrödinger operators on intervals and subject to Neumann boundary conditions. The overall question here is: When does (3) imply (4)?

FormalPara Theorem 7

(Ambarzumian-type theorem for \(\delta \)-coupling conditions on graphs I). Assume that \(H_\mathcal {G}^{q,\sigma }\) is a Schrödinger operator on a metric graph \(\mathcal {G}\) such that

$$\begin{aligned} \lambda _n^{q,\sigma } =\lambda _n^0,\, \quad n=0,1,\dots , \end{aligned}$$

where \((\lambda _n^0)_{n \in \mathbb {N}_0}\) are the eigenvalues of \(H_\mathcal {G}^0\). Then, whenever one of the following assertions is satisfied,

  1. (i)

    \(\sum _{{\textsf{v}}\in {\textsf{V}}} \sigma _{\textsf{v}}\Big (1-\frac{2}{\deg ({\textsf{v}})} \Big ) \le 0\),

  2. (ii)

    \(\int _\mathcal {G} q \ \textrm{d}x \ge 0\) and \(\sigma \in [0,\infty )^{\vert {\textsf{V}}\vert }\),

  3. (iii)

    \(\Big ( \frac{\Delta _{\min }}{2}-1 \Big ) \int _\mathcal {G} q \ \textrm{d}x \ge 0\) and \(\sigma \in (-\infty , 0]^{\vert {\textsf{V}}\vert }\), where \(\Delta _{\min } := \min _{{\textsf{v}}\in {\textsf{V}}} \deg ({\textsf{v}})\) denotes the minimal degree of the underlying graph \(\mathcal {G}\),

one has that \(q = 0\) and \(\sigma =0\).

FormalPara Proof

The statement follows from a combination of Theorem 1 and Theorem 3.

If (i) holds, then (3) immediately gives (4) by direct calculation. Also, whenever (ii) holds, \(\int _\mathcal {G} q \ \textrm{d}x + 2\sum _{{\textsf{v}}\in {\textsf{V}}} \frac{\sigma _{\textsf{v}}}{\text {deg}({\textsf{v}})} = 0\) immediately gives \(\int _\mathcal {G} q \ \textrm{d}x = 0\) and \(\sum _{{\textsf{v}}\in {\textsf{V}}} \sigma _{\textsf{v}}=0\) and hence (4) is satisfied, too.

Now, assume (iii): Starting with \(\int _\mathcal {G} q \ \textrm{d}x + 2\sum _{{\textsf{v}}\in {\textsf{V}}} \frac{\sigma _{\textsf{v}}}{\text {deg}({\textsf{v}})} = 0\) and multiplying both sides with the minimal degree \(\Delta _{\min }\ge 1\), it follows that

$$\begin{aligned} \mathop {\int }\limits _\mathcal {G} q \ \textrm{d}x + \sum _{{\textsf{v}}\in {\textsf{V}}} \sigma _{\textsf{v}}\le \frac{\Delta _{\min }}{2} \mathop {\int }\limits _\mathcal {G} q \ \textrm{d}x + \sum _{{\textsf{v}}\in {\textsf{V}}} \frac{\Delta _{\min }}{\text {deg}({\textsf{v}})} \sigma _{\textsf{v}}= 0 \end{aligned}$$

and hence (4) is again satisfied. \(\square \)

FormalPara Remark 8

Condition Theorem 7(i) is remarkable since it does not contain the potential q! For example, for a given graph \(\mathcal {G}\), one could first choose \(\sigma \) such that condition Theorem 7(i) is satisfied. Then, one would conclude that the operator \(H^{q,\sigma }_{\mathcal {G}}\) is, for any choice of \(q\ne 0\), never isospectral to \(H^{0}_{\mathcal {G}}\).

Another interesting aspect about the conditions formulated in Theorem 7 is that they do not (at least explicitly) depend on the lengths \((\ell _{{\textsf{e}}})_{{\textsf{e}}\in {\textsf{E}}}\) of the metric graph. Therefore, the combinatorial properties of the graph are seemingly more relevant in this context.

In the following and final statement, we call a graph a line graph whenever \(\text {deg}({\textsf{v}}) \in \{1,2\}\) for all \({\textsf{v}}\in {\textsf{V}}\). This is the case, for example, whenever the graph is actually an interval. Furthermore, we call a line graph a loop graph whenever \(\text {deg}({\textsf{v}})=2\) for all \({\textsf{v}}\in {\textsf{V}}\).

FormalPara Corollary 9

(Ambarzumian-type theorem for \(\delta \)-coupling conditions on graphs II). Assume that \(H_\mathcal {G}^{q,\sigma }\) is a Schrödinger operator defined over a line graph \(\mathcal {G}\) such that

$$\begin{aligned} \lambda _n^{q,\sigma } =\lambda _n^0,\, \quad n=0,1,\dots , \end{aligned}$$

where \((\lambda _n^0)_{n \in \mathbb {N}_0}\) are the eigenvalues of \(H_\mathcal {G}^0\). Then, whenever \(\mathcal {G}\) is a line graph with \(\sigma \ge 0\) or \(\mathcal {G}\) is a loop graph, one has that \(q = 0\) and \(\sigma =0\).

FormalPara Proof

Whenever \(\mathcal {G}\) is a line graph with \(\sigma \ge 0\), \(\big (1-\frac{2}{\text {deg}({\textsf{v}})} \big )\) is non-positive and hence the statement follows by Theorem 7(i).

Whenever \(\mathcal {G}\) is a loop graph, \(\big (1-\frac{2}{\text {deg}({\textsf{v}})}\big )=0\) and hence Theorem 7(i) is satisfied independent of the sign of \(\sigma \). \(\square \)

FormalPara Remark 10

As in Theorem 1, in Theorem 7 and Corollary 9 it is enough to assume \(\lambda _{n}^{q,\sigma } - \lambda _n^0 \rightarrow 0\) as \(n \rightarrow \infty \) provided the first eigenvalue of the Schrödinger operator is non-negative.

Let us finish this paper with some interpretation of Theorem 7 and Corollary 9. In Remark 4, we have encountered a Schrödinger operator \(H^{q,\sigma }_{\mathcal {G}}\) with non-vanishing potential on an interval that is isospectral to the Neumann Laplacian. More explicitly, in this example, one sees that condition Corollary 7(i) is not satisfied (as it should be) since

$$\begin{aligned} \sum _{{\textsf{v}}\in {\textsf{V}}} \sigma _{\textsf{v}}\Big (1-\frac{2}{\text {deg}({\textsf{v}})} \Big )=1-\frac{2}{3} > 0. \end{aligned}$$

In the classical setting of Ambarzumian as well as the setting of Davies, one has \(\sigma =0\) and hence condition Theorem 7(i) is always (and trivially) satisfied; in other words, the topology of the graph becomes “invisible”. On the other hand, for loop graphs, the analogue of the classical theorem by Ambarzumian (or Davies) holds in even greater generality since \(\delta \)-coupling conditions are allowed. Nevertheless, going back to condition Theorem 7(i), one could argue that loop graphs constitute a different extreme case in the sense that their geometrical features are exactly such that condition Theorem 7(i) is satisfied for all \(\sigma \). In the general setting, when studying isospectrality of \(H^{q,\sigma }_{\mathcal {G}}\) and \(H^{0}_{\mathcal {G}}\), Theorem 7 reveals some interesting interplay between the geometrical features of the graph \(\mathcal {G}\), the potential q, and the \(\delta \)-coupling conditions in the vertices. One might ask: Other than a loop graph, does there exist another graph for which the operator \(H^{q,\sigma }_{\mathcal {G}}\) is not isospectral to \(H^{0}_{\mathcal {G}}\) as soon as q and \(\sigma \) are not both identically zero?