Quadratic Forms and Spectral Estimates

Abstract

The goal of this Chapter is to provide a systematic study of quadratic forms associated with Schrödinger operators on metric graphs.

The goal of this Chapter is to provide a systematic study of quadratic forms associated with Schrödinger operators on metric graphs. These forms will be used to prove spectral estimates in terms of a certain reference Laplace operator. The spectrum of a Laplacian is easier to calculate, but this is not the only reason to obtain spectral estimates. It turns out that the reference Laplacian does not necessarily correspond to the same metric graph $\Gamma$ as the original operator. The corresponding reference metric graph may have different topological structure from $\Gamma$. Hence the spectral estimates obtained here will imply that in certain cases the topological structure of the graph $\Gamma$ cannot always be deduced from the spectrum of the corresponding quantum graph, despite that the vertex conditions are properly connecting (as always). Moreover, we are going to use spectral estimates to prove several generalisations of the celebrated Ambartsumian theorem.

The main mathematical tool we are going to use is the one-to-one correspondence between semi-bounded self-adjoint operators and closed semi-bounded quadratic forms [90, 442]. Working with the quadratic forms directly allows one to obtain effective spectral estimates much faster and use the full power of perturbation theory.

We are going to consider the case of zero magnetic potential in order to simplify formulas, but it is not a restriction, since arbitrary vertex conditions will be treated and a nontrivial magnetic potential is equivalent to introducing certain phases in the vertex conditions (see Chap. 16).

11.1 Quadratic Forms (Integrable Potentials)

Quadratic forms associated with quantum graphs have already been considered in Sect. 3.4, where parametrisation of vertex conditions via Hermitian matrices was considered. Our focus here will be on determining the quadratic form domain in the case of absolutely integrable potentials. The assumption that the potential is just summable and not necessarily uniformly bounded forces us to be more careful.

11.1.1 Explicit Expression

Let us denote by $Q_{L_q^{\mathbf {S}}} (u,v)$ the quadratic (more precisely sesquilinear) form associated with the operator $L_q^{\mathbf {S}}$ (see Definition 4.1 with $a(x) \equiv 0$). The quadratic form is first defined on the domain of the operator $u, v \in \mathrm {Dom}\, (L_q^{\mathbf {S}}) :$

$$\displaystyle \begin{aligned} Q_{L_q^{\mathbf{S}}} (u,v) = \left\langle u, L_q^{\mathbf{S}} v \right\rangle_{L_2(\Gamma)}.\end{aligned}$$

Hence the functions satisfy on every edge

$$\displaystyle \begin{aligned} u, v \in W_2^1 (E_n), \; \; - u'' + q u, - v'' + q v \in L_2 (E_n), \quad n= 1,2, \dots, N.\end{aligned}$$

As we have shown in Sect. 4.1, these conditions imply that the functions are not only continuous, but have continuous first derivatives on every edge and one may impose vertex conditions. Moreover this implies that one may integrate by parts in the expression for the quadratic form:

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \displaystyle Q_{L_q^{\mathbf{S}}} (u,v) & = & \displaystyle \left\langle u, L_q^{\mathbf{S}} v \right\rangle \\ & = & \displaystyle \sum_{n=1}^N \int_{E_n} \overline{u(x)} \Big( - v'' (x) + q(x) v(x) \Big) dx \\ & \!=\! & \displaystyle \sum_{n=1}^N \int_{E_n} \left( \overline{u'(x)} v'(x) \!+\! q(x) \overline{u(x)} v(x) \right) dx \!+\! \sum_{m=1}^M \langle \vec{u} (V^m), \partial \vec{v} (V^m) \rangle_{\mathbb C^{d_m}}, \end{array} \end{aligned} $$

(11.1)

where the vectors $\vec {u} (V^m), \partial \vec {u}(V^m)$ of boundary values at the vertex $V^m$ were introduced in formula (3.2).

The expression for the quadratic form may be simplified further if one takes into account that the vertex values of functions and their derivatives are not independent, but satisfy vertex conditions (4.8)¹

$$\displaystyle \begin{aligned} {} i (S^m - I) \vec{u} (V^m) = (S^m+I) \partial \vec{u}(V^m), \end{aligned} $$

(11.2)

where $S^m$ is an irreducible unitary $d_m \times d_m$ matrix. Consider the eigensubspace for $S^m$ associated with the eigenvalue $-1$ and its orthogonal complement in $\mathbb C^{d_m}$. Let us denote the corresponding orthogonal projectors by $P_{-1}^m$ and $P_{-1}^{m\perp }$ respectively.

Applying $P_{-1}^m$ to both sides of Eq. (11.2) we get

$$\displaystyle \begin{aligned} P_{-1}^m i (S^m-I) \vec{u} (V^m) = P_{-1}^m (S^m+I) \partial \vec{u} (V^m) \end{aligned}$$

and therefore

$$\displaystyle \begin{aligned} {} P_{-1}^m \vec{u} (V^m) = 0, \end{aligned} $$

(11.3)

where we used that $P_{-1}^m$ is an eigenprojector for $S^m$ and therefore commutes with it. It follows that the function $u$ satisfies certain generalised Dirichlet conditions at $V^m$—not all boundary values of $u$ at $V^m$ are equal to zero, but a certain combination of them is zero (more precisely the projection on the eigensubspace $P_{-1}^m \mathbb C^{d_m}$ is zero).

Let us apply now the projector $P_{-1}^{m\perp } = I- P_{-1}^m$ to both sides of (11.2) and use again that the projector and the matrix $S^m$ commute:

$$\displaystyle \begin{aligned} i P_{-1}^{m\perp}(S^m - I) P_{-1}^{m\perp}\vec{u} (V^m) = P_{-1}^{m\perp}(S^m+I) P_{-1}^{m\perp}\partial \vec{u} (V^m). \end{aligned} $$

(11.4)

Note that the matrix $P_{-1}^{m\perp }(S^m + I) P_{-1}^{m\perp }$ is invertible in the subspace $P_{-1}^{m\perp } \mathbb C^{d_m}$ and therefore the last condition can be written in the Robin form as follows:

$$\displaystyle \begin{aligned} {} P_{-1}^{m\perp}\partial \vec{u}(V^m) = P_{-1}^{m\perp} i \frac{S^m-I}{S^m+I} P_{-1}^{m\perp}\vec{u} (V^m). \end{aligned} $$

(11.5)

The matrix $P_{-1}^{m\perp } i \frac {S^m-I}{S^m+I} P_{-1}^{m\perp }$ is Hermitian in $P_{-1}^{m\perp } \mathbb C^{d_m}$ and was denoted by $A^m$ in (3.28)

$$\displaystyle \begin{aligned} {} A^m := P_{-1}^{m\perp} i \frac{S^m-I}{S^m+I} P_{-1}^{m\perp}. \end{aligned} $$

(11.6)

Hence every function from the domain of the operator satisfies the generalised Robin condition:

$$\displaystyle \begin{aligned} {} P_{-1}^{m\perp}\partial \vec{u}(V^m) = A^m P_{-1}^{m\perp}\vec{u} (V^m). \end{aligned} $$

(11.7)

Using this notation the expression for the sesquilinear form can be written as follows:

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle Q_{L_q^{\mathbf{S}}} (u,v) & = & \displaystyle \sum_{n=1}^N \int_{E_n} \overline{u'(x)} v'(x) dx + \sum_{n=1}^N \int_{E_n} q(x) \overline{u(x)} v(x) dx \\[3mm] & & \displaystyle + \sum_{m=1}^M \langle P_{-1}^{m\perp} \vec{u} (V^m), A^m P_{-1}^{m\perp} \vec{v} (V^m) \rangle_{\mathbb C^{d_m}}. \end{array} \end{aligned} $$

(11.8)

Here we split the integral term, since we know that the functions $u, v$ are continuous and $q \in L_1.$ The corresponding quadratic form is

$$\displaystyle \begin{aligned} {} \begin{array}{ccl} \displaystyle Q_{L_q^{\mathbf{S}}} (u,u) & = & \displaystyle \sum_{n=1}^N \int_{E_n} \vert u'(x) \vert^2 dx + \sum_{n=1}^N \int_{E_n} q(x) \vert u(x) \vert^2 dx \\[3mm] & & \displaystyle + \sum_{m=1}^M \langle P_{-1}^{m\perp} \vec{u} (V^m), A^m P_{-1}^{m\perp} \vec{u} (V^m) \rangle_{\mathbb C^{d_m}}. \end{array} \end{aligned} $$

(11.9)

It is common for unbounded operators that the quadratic form is defined on a domain which is larger than the domain of the operator. If the operator is strictly positive, then the domain of the quadratic form is obtained by closing the operator domain with respect to the norm given by the quadratic form.

The quadratic form we got is not necessarily positive, since there is no reason to assume that the Hermitian matrices $A_m$ are positive, the potential $q$ can also be negative. In order to proceed we need to show that the quadratic form is semi-bounded, i.e. there exists a constant $K$ such that

$$\displaystyle \begin{aligned} {} \| u \|{}^2_{Q_{L_q^{\mathbf{S}}}} := Q_{L_q^{\mathbf{S}}} (u,u) + K \| u \|{}^2_{L_2 (\Gamma)}, \end{aligned} $$

(11.10)

is positive definite.

11.1.2 An Elementary Sobolev Estimate

In order to proceed we need the following elementary Sobolev estimate (a special case of the Gagliardo-Nirenberg estimate), showing that every function from $W_2^1$ on a compact interval is essentially bounded.

Lemma 11.1

Assume that $u \in W_2^1 [0, \ell ],$ then it holds

$$\displaystyle \begin{aligned} {} \| u \|{}^2_{L_\infty [0,\ell] } \leq \epsilon \| u' \|{}^2_{L_2 [0,\ell]} + \frac{2}{\epsilon} \| u \|{}^2_{L_2 [0,\ell]}, \end{aligned} $$

(11.11)

where $\epsilon > 0$ can be chosen arbitrarily, provided it is sufficiently small: $\epsilon \leq \ell .$

Proof

We prove first the estimate for continuous, piecewise continuously differentiable functions. Let us denote by $x_{\mathrm {min}}$ one of the the points at which $|u|$ attains the minimum, then it holds

and hence

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle \vert u(x) \vert^2 & \leq & \displaystyle \| u \|{}^2 / \ell + 2 \int_{x_{\mathrm{min}}}^x \vert u(y) \vert \; \vert u'(y) \vert dy \\ & \leq & \displaystyle \| u \|{}^2 / \ell + \epsilon \int_0^\ell \vert u'(y) \vert^2 dy + \frac{1}{\epsilon} \int_0^\ell \vert u (y) \vert^2 dy \\ & \leq & \displaystyle \epsilon \| u '\|{}^2_{L_2[0,\ell]} + \left(\frac{1}{\epsilon} + \frac{1}{\ell} \right) \| u \|{}^2_{L_2[0,\ell]}. \end{array}\end{aligned}$$

Taking into account that $\epsilon$ is positive and less than $\ell$ we obtain (11.11).

It remains to note that continuous piecewise continuously differentiable functions form a dense subset in $W_2^1 [0,\ell ]$ and hence estimate (11.11) holds for any function from $W_2^1 [0,\ell ] .$ □

The obtained estimate will be used for sufficiently small values of $\epsilon$, more precisely estimates with $\epsilon$ tending to zero will be interesting for us. Therefore the restriction $\epsilon \leq \ell$ is not essential. It is important to remember that, taking smaller and smaller values of $\epsilon$, the coefficient in front of $\| u \|{ }^2$ increases.

The estimate (11.11) can be generalised for the case of metric graphs as follows

$$\displaystyle \begin{aligned} {} \| u \|{}^2_{L_\infty (\Gamma) } \leq \epsilon \| u' \|{}^2_{L_2 (\Gamma)} + \frac{2}{\epsilon} \| u \|{}^2_{L_2 (\Gamma)}, \end{aligned} $$

(11.12)

provided $u \in W_2^1 (\Gamma )$ without any required vertex conditions and $\epsilon \leq \ell _{\mathrm {min}}$, where $\ell _{\mathrm {min}}$ denotes the length of the shortest edge

$$\displaystyle \begin{aligned} \ell_{\mathrm{min}} = \min_{n=1,2, \dots, N} \ell_n. \end{aligned} $$

(11.13)

The obtained estimate may be improved taking into account topological properties of the metric graph and vertex conditions. For example, in the case of standard vertex conditions the estimate holds for $\epsilon$ less than the diameter of the metric graph. The diameter is the longest distance between any two points on the metric graph.

Problem 43

Prove the Sobolev estimate (11.12) for $\epsilon$ less than the diameter of the metric graph $\Gamma$, provided the function $u \in W_2^1 (\Gamma \setminus \mathbf V)$ is in addition continuous at the vertices.

11.1.3 The Perturbation Term Is Form-Bounded

Our immediate goal is to estimate the second and third terms in the quadratic form expression (11.9) in order to show that the form is semibounded.

The second term can be estimated as follows. Inequality (11.11) implies that

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle \left\vert \int_{E_n} q(x) \vert u(x) \vert^2 dx \right\vert & \leq & \displaystyle \| q \|{}_{L_1 (E_n)} \max_{x \in E_n} \vert u(x) \vert^2 \\[3mm] & \leq & \displaystyle \| q \|{}_{L_1 (E_n)} \left( \epsilon \| u'\|{}^2_{L_2 (E_n)} + \frac{2}{\epsilon} \| u \|{}^2_{L_2(E_n)} \right). \end{array}\end{aligned}$$

Hence we have

$$\displaystyle \begin{aligned} {} \left\vert \sum_{n=1}^N \int_{E_n} q(x) \vert u(x) \vert^2 dx \right\vert \leq \| q \|{}_{L_1 (\Gamma)} \left( \epsilon \| u'\|{}^2_{L_2 (\Gamma)} + \frac{2}{\epsilon} \| u \|{}^2_{L_2(\Gamma)} \right), \end{aligned} $$

(11.14)

of course provided $\epsilon < \ell _{\mathrm {min}}.$

Under the same assumption on $\epsilon$ the third term satisfies

$$\displaystyle \begin{aligned} {} \begin{array}{cl} & \displaystyle \left\vert \sum_{m=1}^M \langle P_{-1}^{m\perp} \vec{u} (V^m), A^m P_{-1}^{m\perp} \vec{u} (V^m) \rangle_{\mathbb C^{d_m}} \right\vert \\ \leq & \displaystyle \sum_{m=1}^M d_m \| A^m \| \; \| u \|{}_{L_\infty}^2 \\ \leq & \displaystyle \left( \sum_{m=1}^M d_m \| A^m \| \right) \; \left( \epsilon \| u' \|{}^2_{L_2(\Gamma)} + \frac{2}{\epsilon} \| u \|{}^2_{L_2(\Gamma)} \right) , \end{array} \end{aligned} $$

(11.15)

where $d_m$ is the degree of the vertex $V^m.$ Note that the obtained estimates (11.14) and (11.15) are far from being optimal.

Let us consider the quadratic form $Q_{L_q^{\mathbf {S}}} (u,u)$ as a perturbation of the Dirichlet form $\| u' \|{ }^2_{L_2(\Gamma )}$:

$$\displaystyle \begin{aligned} Q_{L_q^{\mathbf{S}}} (u,u) = \| u' \|{}^2_{L_2(\Gamma)} + B_{L_q^{\mathbf{S}}} (u,u). \end{aligned} $$

(11.16)

We have proven that the perturbation term

$$\displaystyle \begin{aligned} B_{L_q^{\mathbf{S}}} (u,u) : = \displaystyle \int_{\Gamma} q(x) \vert u(x) \vert^2 dx + \sum_{m=1}^M \langle P_{-1}^{m\perp} \vec{u} (V^m), A^m P_{-1}^{m\perp} \vec{u} (V^m) \rangle_{\mathbb C^{d_m}} \end{aligned} $$

(11.17)

possesses the estimate

$$\displaystyle \begin{aligned} {} \vert B_{L_q^{\mathbf{S}}} (u,u) \vert \leq \left( \sum_{m=1}^M d_m \| A^m \| + \| q \|{}_{L_1(\Gamma)} \right) \left( \epsilon \| u' \|{}^2_{L_2(\Gamma)} + \frac{2}{\epsilon} \| u \|{}^2_{L_2(\Gamma)} \right) . \end{aligned} $$

(11.18)

This inequality has two important implications:

• 1. The perturbation term is infinitesimally form bounded with respect to the Dirichlet form [442].

• 2. The constant $K$ in (11.10) can be chosen equal to

  $$\displaystyle \begin{aligned} {} K = \left( \sum_{m=1}^M d_m \| A^m \| + \| q \|{}_{L_1(\Gamma)} \right) \frac{2}{\epsilon} + 1, \end{aligned} $$

  (11.19)

  where $\epsilon \leq \ell _{\mathrm {min}}$ satisfies in addition

  $$\displaystyle \begin{aligned} \epsilon \leq \frac{1}{2} \left( \sum_{m=1}^M d_m \| A^m \| + \| q \|{}_{L_1(\Gamma)} \right)^{-1}.\end{aligned}$$

With such $\epsilon$ and $K$ we have the two-sided estimate:

$$\displaystyle \begin{aligned} {} \frac{1}{2} \| u' \|{}^2_{L_2 (\Gamma)} + \| u \|{}^2_{L_2(\Gamma)} \leq Q_{L_q^{\mathbf{S}}} (u,u) + K \| u \|{}^2_{L_2(\Gamma)} \leq \frac{3}{2} \| u' \|{}^2_{L_2 (\Gamma)} + 2K \| u \|{}^2_{L_2(\Gamma)}. \end{aligned} $$

(11.20)

In other words, expression (11.10) determines a norm, which is equivalent to the Sobolev $W_2^1$-norm:

$$\displaystyle \begin{aligned} \| u \|{}^2_{W_2^1 (\Gamma)} = \| u' \|{}^2_{L_2(\Gamma)} + \| u \|{}^2_{L_2(\Gamma)}.\end{aligned}$$

Our next goal is to determine the domain of the quadratic form—the closure of $\mathrm {Dom}\, (L_q^{\mathbf S})$ with respect to the quadratic form norm introduced above. It is natural to study first the closure of the Dirichlet form.

11.1.4 The Reference Laplacian

Consider the Dirichlet form

$$\displaystyle \begin{aligned} {} \| u' \|{}^2_{L_2(\Gamma)} = \int_\Gamma \vert u' (x) \vert^2 dx \end{aligned} $$

(11.21)

defined on the functions $u \in \mathrm {Dom}\, (L_q^{\mathbf S} (\Gamma )).$ The quadratic form is not closed on this domain and our aim is to determine its closure. We may restrict the quadratic form further by considering just smooth functions $u \in C^\infty (E_n)$ satisfying the given vertex condition, which we write as a combination of generalised Dirichlet and Robin conditions (see formulas (11.3) and (11.7)).

Consider any Cauchy sequence $u_j \in C^\infty (\Gamma \setminus \mathbf V )$ such that $\| u_j - u_i \|{ }_{W_2^1(\Gamma \setminus \mathbf V)} \rightarrow 0.$ It follows that $\| u_j - u_i \|{ }_{L_2 (\Gamma )} \rightarrow 0$ and $\| u^{\prime }_j - u^{\prime }_i \|{ }_{L_2 (\Gamma )} \rightarrow 0$ and therefore the limiting function as well as its first derivative are square integrable. In other words the limit function belongs to $W_2^1 (\Gamma \setminus \mathbf V).$ Every such function is continuous on each edge, therefore the generalised Dirichlet conditions (11.3) are preserved. The generalised Robin conditions (11.5) disappear, since the functions from $W_2^1$ are not necessarily continuously differentiable. Summing up, the closure of the positive Dirichlet form is defined by the same expression (11.21) on the domain of functions from $W_2^1 (\Gamma \setminus \mathbf V)$ satisfying just the generalised Dirichlet conditions (11.3).

In the next step we calculate the self-adjoint operator associated with the closure of the Dirichlet form. Consider the sesquilinear form

$$\displaystyle \begin{aligned} \langle u ', v' \rangle\end{aligned}$$

assuming that $u$ and $v$ are in the domain of the quadratic form. The domain of the corresponding operator is given by all $v$ such that the sesquilinear form determines a bounded linear functional with respect to $u$ in the Hilbert space norm:

$$\displaystyle \begin{aligned} {} \langle u ', v' \rangle \leq C_v \| u \|{}_{L_2(\Gamma)}. \end{aligned} $$

(11.22)

Taking first $u \in C_0^\infty (\Gamma \setminus \mathbf V)$ we see that (11.22) holds only if the generalised derivative of $v'$ lies in $L_2 (\Gamma )$, i.e. if $v \in W_2^2 (\Gamma \setminus \mathbf V).$ Consider now any $u \in W_2^1 (\Gamma \setminus \mathbf V)$ satisfying the generalised Dirichlet condition (11.3); one may integrate by parts in the sesquilinear form to get

The integral term gives a bounded functional with respect to $u$. Hence the sesquilinear form determines a bounded functional if and only if

$$\displaystyle \begin{aligned} \langle P_{-1}^{m \perp} \vec{u}^m , P_{-1}^{m \perp} \partial \vec{v}^m \rangle_{\mathbb C^{d_m}}\end{aligned}$$

are bounded functionals with respect to the $L_2$-norm. The functionals $u \mapsto \vec {u}^m$ are not bounded, since square integrable functions are not necessarily bounded (see estimate (11.11), where $\epsilon$ could be taken arbitrarily small but not equal to zero). Therefore we get a bounded functional only if $v$ satisfies the generalised Neumann condition

$$\displaystyle \begin{aligned} {} P_{-1}^{m \perp} \partial \vec{v}^m = 0. \end{aligned} $$

(11.23)

Summing up, the self-adjoint operator corresponding to the closure of the Dirichlet form is the Laplace operator defined on the functions from $W_2^2 (\Gamma \setminus \mathbf V)$ satisfying generalised Dirichlet (11.3) and generalised Neumann (11.23) conditions at the vertices. These vertex conditions are scaling-invariant and sometimes are called non-Robin. This operator will play the role of a reference operator, when spectral estimates for quantum graphs will be derived. Using our notations, the reference operator can be written as $L^{\mathbf S_{\mathbf {v}}(\infty )}$, where following (3.31) we introduced the high energy limit of the vertex scattering matrix

$$\displaystyle \begin{aligned} \mathbf S_{\mathbf{v}}(\infty) = \lim_{k \rightarrow \infty} \mathbf S_{\mathbf{v}} (k) = I - 2 P_{-1} ,\end{aligned}$$

where $P_{-1} = \sum _{m=1}^M P^m_{-1}$. The matrices $S_{\mathbf {v}}^m(\infty )$ have eigenvalues $-1$ and $1$ and the corresponding eigensubspaces are $P_{-1}^m \mathbb C^{d_m}$ and $P_{-1}^{m \perp } \mathbb C^{d_m} .$ Hence the functions from the domain of $L^{\mathbf S_{\mathbf {v}}(\infty )}$ satisfy precisely the Dirichlet and Neumann conditions (see (11.3) and (11.23).

Note that the operator $L^{\mathbf S_{\mathbf {v}}(\infty )}$ coincides with $L^{\mathbf S}$ only if $\mathbf S$ is Hermitian. In general, substituting $\mathbf S$ with $\mathbf S_{\mathbf {v}}(\infty )$ may change the topology of the described system—the metric graph corresponding to $L^{\mathbf S_{\mathbf {v}}(\infty ) }$ may be slightly different from the original graph $\Gamma .$ The original matrix $\mathbf S$ has block structure, each block associated with a vertex in the graph $\Gamma .$ Therefore the limiting matrix $\mathbf S_{\mathbf {v}}(\infty )$ also has block structure, but the blocks in it could be finer than those in $\mathbf S$. We have already met this phenomenon in Sect. 9.3.3.

Example 11.2

Consider the $2 \times 2$ unitary matrix

$$\displaystyle \begin{aligned} \mathbf S = \frac{1}{2} \left( \begin{array}{cc} 1 + i & 1-i \\ 1-i & 1+ i \end{array} \right).\end{aligned}$$

The eigenvalues are $1$ and $i$

$$\displaystyle \begin{aligned} \mathbf S = 1\; P_{(1,1)} + i \;P_{(1,-1)},\end{aligned}$$

where $P_{(1,1)}$ and $P_{(1,-1)}$ are the orthogonal projectors on the corresponding eigenvectors $(1,1)$ and $(1,-1)$ respectively. It follows, that $\mathbf S_{\mathbf {v}}(\infty )$ is the unitary matrix with $1$ as the double eigenvalue. Really, formula (3.20), or more explicitly (3.30) implies that

$$\displaystyle \begin{aligned} \mathbf S_{\mathbf{v}} (k) = 1 P_{(1,1)} + \frac{k (i+1) + (i-1)}{k (i+1) - (i-1)} P_{(1,-1)} \rightarrow_{k \rightarrow \infty} 1\; P_{(1,1)} + 1\; P_{(1,-1)} = \mathbb I.\end{aligned}$$

The limit matrix $\mathbf S_{\mathbf {v}}(\infty )$ describes not a single vertex of degree $2$, but two independent degree one vertices with Neumann conditions.

Let us denote by $\Gamma ^\infty$ the metric graph corresponding to the new vertex scattering matrix $\mathbf S_{\mathbf {v}}(\infty ).$ The graph $\Gamma ^\infty$ is obtained from the original metric graph $\Gamma$ by chopping all vertices, for which $S_{\mathbf {v}}^m (\infty ) = \lim _{k \rightarrow \infty } S^m_{\mathbf {v}} (k)$ has block structure. For future use we formulate the following definition, which involves the Schrödinger operators with maybe non-zero potentials.

Definition 11.3

Let $L_q^{\mathbf S} (\Gamma )$ be a Schrödinger operator on a metric graph $\Gamma$ with vertex conditions determined by the matrix $\mathbf S.$ Let $\mathbf S_{\mathbf {v}} (\infty )$ be the high-energy limit of the vertex scattering matrix and $\Gamma ^\infty$ - the corresponding metric graph obtained by chopping (if necessary) certain vertices in $\Gamma$ so that the vertex conditions determined by $\mathbf S_{\mathbf {v}} (\infty )$ are properly connecting for $\Gamma ^\infty .$ Then the non-Robin Laplace operator $L^{\mathbf S_{\mathbf {v}} (\infty )} (\Gamma ^\infty ) = L_0^{\mathbf S_{\mathbf {v}} (\infty )} (\Gamma ^\infty )$ is called the reference Laplacian for the Schrödinger operator $L_q^{\mathbf S} (\Gamma ).$

Note that the Hilbert spaces $L_2(\Gamma )$ and $L_2(\Gamma ^\infty )$ can always be identified, since these metric graphs have the same set of edges. The reference operator will play a very important role in spectral estimates, where the spectrum of the Schrödinger operator will be compared to the spectrum of just the reference Laplacian.

It will be convenient to distinguish vertex conditions that do not lead to a different reference graph:

Definition 11.4

Vertex conditions are called asymptotically properly connecting if the high energy limits of all vertex scattering matrices $S_{\mathbf {v}}^m (\infty )$ are irreducible, i.e. if $\Gamma ^\infty = \Gamma .$

Definition 11.5

Vertex conditions are called asymptotically standard if and only if $\mathbf S_{\mathbf {v}}(\infty ) = \mathbf S^{\mathrm {st}} (\Gamma ^\infty )$, where $\mathbf S^{\mathrm {st}} (\Gamma ^\infty )$ is the unitary matrix determining standard vertex conditions on $\Gamma ^\infty .$

Note that we do not require that $\mathbf S_{\mathbf {v}} (\infty ) = \mathbf S^{\mathrm {st}} (\Gamma )$ for vertex conditions to be asymptotically standard. If this is the case, then the vertex conditions are called asymptotically properly connecting and standard.

Problem 44

Consider a degree three vertex. Provide examples of properly connecting vertex conditions, such that in $\Gamma ^\infty$ the vertex splits into:

• three degree one vertices;

• one degree one and one degree two vertices.

Give an example of asymptotically properly connecting vertex conditions.

Problem 45

How to describe all asymptotically standard vertex conditions?

11.1.5 Closure of the Perturbed Quadratic Form

We return now to the operator $L_q^{\mathbf S} (\Gamma )$ and the corresponding quadratic form norm given by (11.10), where the constant $K$ is chosen as in (11.19). For simplicity, let us consider this norm not on the domain of $L_q^{\mathbf S} (\Gamma )$, but on the space of smooth functions $C^\infty (\Gamma \setminus \mathbf V)$ satisfying vertex conditions with the parameter matrix $\mathbf S$. These conditions can be written as a combination of generalised Dirichlet and Robin conditions: (11.3) and (11.5).

The estimate (11.20) implies that the closure of this domain with respect to the quadratic form norm of $L_q^{\mathbf S}$ and with respect to the Dirichlet form just coincide. Hence the quadratic form (11.9) is closed on the set of $W_2^1 (\Gamma \setminus \mathbf V)$-functions satisfying just generalised Dirichlet conditions (11.3). As before, the Robin part of vertex conditions disappears. The difference is that the Robin part can be reconstructed taking into account the terms involving the matrices $A_m.$

Let us summarise our studies:

Theorem 11.6

Let$\Gamma$be a finite compact metric graph and let$q \in L_1(\Gamma ) .$Then the quadratic form of the operator$L_q^{\mathbf {S}}$is defined on the domain$\mathrm {Dom}\, (Q_{L_q^{\mathbf {S}}})$of functions from$W_2^1 (\Gamma \setminus \mathbf V)$satisfying the generalised Dirichlet conditions (11.3) at the vertices and is given by the following expression

$$\displaystyle \begin{aligned} {} Q_{L_q^{\mathbf{S}}} (u,u) &= \int_{\Gamma} \left(\vert u' (x) \vert^2 + q(x) \vert u(x) \vert ^2 \right) dx \\ & \quad + \sum_{m=1}^M \langle P_{-1}^{m\perp} \vec{u} (V^m), A^m P_{-1}^{m\perp} \vec{u} (V^m) \rangle_{\mathbb C^{d^m}}. \end{aligned} $$

(11.24)

We have already mentioned that there is a one-to-one correspondence between semibounded quadratic forms and self-adjoint operators. Let us see how the unique self-adjoint operator $L_q^{\mathbf {S}}$ can be reconstructed from its quadratic form $Q_{L_q^{\mathbf {S}}}.$ Assume that the quadratic form is known and therefore is given by the expression (11.24) on the domain of functions from $W_2^1 (\Gamma \setminus \mathbf V)$ satisfying the generalised Dirichlet conditions (11.3). Assume that $u, v \in \mathrm {Dom}\, (Q_{L_q^{\mathbf {S}}})$, then the corresponding sesquilinear form is

$$\displaystyle \begin{aligned} {} Q_{L_q^{\mathbf{S}}} (u,v) &= \int_{\Gamma} \left( \overline{u' (x) } v'(x) + q(x) \overline{u(x)} v(x) \right) dx \\ & \quad + \sum_{m=1}^M \langle P_{-1}^{m\perp} \vec{u} (V^m), A^m P_{-1}^{m\perp} \vec{v} (V^m) \rangle_{\mathbb C^{d_m}}. \end{aligned} $$

(11.25)

The domain of the operator consists of all functions $v$ such that (11.25) defines a bounded linear functional with respect to $u$ i.e. an estimate similar to (11.22) holds. Consider first $u \in C_0^\infty (\Gamma \setminus \mathbf V),$ then the form is equal to

$$\displaystyle \begin{aligned} Q_{L_q^{\mathbf{S}}} (u,v) = \int_{\Gamma} \overline{u' (x) } v'(x) dx + \int_{\Gamma} q(x) \overline{u(x)} v(x) dx. \end{aligned}$$

Using generalised derivatives, one may write that this sesquilinear form determines a bounded functional only if

$$\displaystyle \begin{aligned} {} - u'' + q u \in L_2 (\Gamma) \end{aligned} $$

(11.26)

holds.

Assume now that $u \in C^\infty (\Gamma \setminus \mathbf V)$, i.e. $u$ is not necessarily equal to zero in a neighbourhood of the vertices. We already know that (11.26) holds and hence $v$ and its first derivative are continuous on each edge, this allows us to integrate by parts:

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle Q_{L_q^{\mathbf{S}}} (u,v) & = & \displaystyle \int_{\Gamma} \overline{u(x)} \left( - v'' (x) + q(x) v(x) \right) dx - \sum_{m=1}^M \langle \vec{u} (V^m), \partial \vec{v} (V^m) \rangle_{\mathbb C^{d_m}} \\ && \displaystyle + \sum_{m=1}^M \langle P_{-1}^{m\perp} \vec{u} (V^m), A^m P_{-1}^{m\perp} \vec{v} (V^m) \rangle_{\mathbb C^{d_m}}. \end{array}\end{aligned}$$

We already know that the integral term is a bounded functional. Taking into account that $u$ satisfies the generalised Dirichlet conditions (11.3) the vertex terms can be written as

$$\displaystyle \begin{aligned} {} \sum_{m=1}^M \langle P_{-1}^{m\perp} \vec{u} (V^m), A^m P_{-1}^{m\perp} \vec{v} (V^m) - P_{-1}^{m\perp} \partial \vec{v} (V^m) \rangle_{\mathbb C^{d_m}} .\end{aligned} $$

(11.27)

Since the functionals $u \mapsto u (x_j)$ are not bounded with respect to the $L_2$-norm of $u$ and the vectors $P_{-1}^{m\perp } \vec {u} (V^m)$ are arbitrary, (11.27) determines bounded functionals if and only if

$$\displaystyle \begin{aligned} A^m P_{-1}^{m\perp} \vec{v} (V^m) - P_{-1}^{m\perp} \partial \vec{v} (V^m) = 0 ,\end{aligned}$$

which is equivalent to the generalised Robin condition (11.7). Hence the domain of the operator is given by the set of functions from $W_2^1 (\Gamma \setminus \mathbf V)$ satisfying (11.26) and vertex conditions (11.3) and (11.7) as expected. Provided $v$ belongs to this domain, the quadratic form is given by

which shows that the action of the operator corresponding to the quadratic form is given by the differential expression $\tau _q$ as indicated above.

11.2 Spectral Estimates (Standard Vertex Conditions)

The spectrum of the Schrödinger operator with $L_1$-potential on a finite compact metric graph is purely discrete, since every such operator can be considered as a finite rank perturbation (in the resolvent sense) of the Schrödinger operator on the separate edges with Dirichlet conditions at all endpoints (so-called Dirichlet operator, see Definition 4.3 ). The aim of this section is to prove spectral estimates comparing the spectra of Schrödinger operators and the reference Laplacians, two differential operators acting essentially on the same metric graph.² The main reason for such studies is that the spectrum of a non-Robin Laplacian is much easier to calculate. Moreover as a by-product of our studies we shall prove a generalisation of the celebrated Ambartsumian theorem (see Chap. 14). It turns out that as in the case of a single interval the difference between Laplace and Schrödinger eigenvalues is uniformly bounded. The classical proof for this fact in the case of single interval relies heavily on the explicit formula for the resolvent kernel of the Laplacian [105, 381]. Since the corresponding formula in the case of metric graphs is not so explicit, we are going to use general perturbation theory. In particular the min-max and max-min principles giving explicit formulas for the eigenvalues will be exploited (see Proposition 4.19).

Our immediate goal is to use Proposition 4.19 to estimate the eigenvalues of the Schrödinger operator through the corresponding Laplacian eigenvalues. This is an easy exercise if we assume $q \in L_{\infty } (\Gamma ),$ but we want to cover the most general case of $q \in L_1 (\Gamma ).$

Problem 46

Assume that $q \in L_\infty (\Gamma ),$ show that the eigenvalue of the Laplace and Schrödinger operators satisfy the uniform in $n$ estimate

$$\displaystyle \begin{aligned} \vert \lambda_n (L^{\mathrm{st}}_q) - \lambda_n (L^{\mathrm{st}}) \vert \leq C, \end{aligned} $$

(11.28)

where $C$ is a certain constant independent of $n.$ Can you give an explicit formula for $C$?

If one takes into account that the eigenvalues satisfy Weyl’s law, one may prove that the square roots of the eigenvalues are asymptotically close:

(11.29)

This observation motivates the following definition.

Definition 11.7

Two unbounded, semi-bounded self-adjoint operators $A$ and $B$ with discrete spectra $\{ \lambda _n (A) \}_{n=1}^\infty$ and $\{ \lambda _n (B) \}_{n=1}^\infty$, respectively, are called asymptotically isospectral if

$$\displaystyle \begin{aligned} {} \lim_{n \rightarrow \infty} \left( \sqrt{\lambda_n (A)} - \sqrt{\lambda_n(B)} \right)= 0. \end{aligned} $$

(11.30)

This definition makes sense only if the operators are unbounded, otherwise the requirement in the definition is too weak. To guarantee asymptotic isospectrality of quantum graphs it is enough if the eigenvalues satisfy the estimate

$$\displaystyle \begin{aligned} {} \vert \lambda_n (L^{\mathbf{S}}_q) - \lambda_n (L^{\mathbf{S}}) \vert \leq C n^{1-\epsilon}, \quad \epsilon > 0. \end{aligned} $$

(11.31)

Our spectral estimates are stronger, but we prefer to use the notion of isospectrality, since precisely condition (11.30) will be used to prove that two generalised trigonometric polynomials have close zeroes if and only if the zeroes coincide.

In order to make the presentation more transparent, let us discuss the case of standard conditions first and return back to the case of general vertex conditions in the next section.

Theorem 11.8

Let $L^{\mathrm {st}}$ and $L_q^{\mathrm {st}}$ be the standard Laplace and standard Schrödinger operators on a compact finite metric graph $\Gamma$ and let the potential $q$ in the Schrödinger operator be absolutely integrable, $q \in L_1 (\Gamma ).$ Then the Laplace and Schrödinger operators, $L^{\mathrm {st}}$ and $L_q^{\mathrm {st}}$ , are asymptotically isospectral, moreover the difference between their eigenvalues is uniformly bounded

$$\displaystyle \begin{aligned} {} \vert \lambda_n (L^{\mathrm{st}}) - \lambda_n (L_q^{\mathrm{st}}) \vert \leq C (\Gamma, q), \end{aligned} $$

(11.32)

where the constant $C (\Gamma , q)$ depends on the graph $\Gamma$ and the potential $q$ , but is independent of $n.$

We postpone the proof of the theorem and try to derive an upper estimate for $\lambda _n (L_q^{\mathrm {st}})$ using a naive approach. In the case of standard conditions the quadratic form is

$$\displaystyle \begin{aligned} Q_{L_q^{\mathrm{st}}} (u, u) = \int_{\Gamma} \vert u ' \vert^2 dx + \int_{\Gamma} q(x) \vert u(x) \vert^2 dx,\end{aligned}$$

where $u$ is arbitrary $W_2^1 (\Gamma )$ function continuous at the vertices. The form can be estimated from above by

$$\displaystyle \begin{aligned} {} Q_{L_q^{\mathrm{st}}} (u, u) \leq Q_{L_{q_+}^{\mathrm{st}}} (u,u) = \int_{\Gamma} \vert u ' \vert^2 dx + \int_{\Gamma} q_+(x) \vert u(x) \vert^2 dx, \end{aligned} $$

(11.33)

where $q_+$ is the positive part of the potential $q$:

$$\displaystyle \begin{aligned} q(x) = q_+ (x) - q_- (x), \; q_\pm (x) \geq 0. \end{aligned} $$

(11.34)

This step cannot be improved much, since the new estimate coincides with the original one in the case $q$ is nonnegative.

The idea how to proceed is to choose a concrete $n$-dimensional subspace $\mathcal V^0_n$, then the Rayleigh quotient will give not an exact value for $\lambda _n (L^{\mathrm {st}}_q )$, but an upper estimate if formula (4.51) is used

$$\displaystyle \begin{aligned} \lambda_n (L^{\mathrm{st}}_q) = \min_{\mathcal V^n} \max_{u \in \mathcal V^n} \frac{Q_{L^{\mathrm{st}}_q} (u,u)}{\| u\|{}^2} \leq \max_{u \in \mathcal V^0_n} \frac{Q_{L^{\mathrm{st}}_q} (u,u)}{\| u\|{}^2}.\end{aligned}$$

The only candidate for $\mathcal V^0_n$ we have is the linear span of the Laplacian eigenfunctions $\psi _j^{L^{\mathrm {st}}}$ corresponding to the $n$ lowest eigenvalues

$$\displaystyle \begin{aligned} {} \mathcal V^0_n = \mathcal L \left\{ \psi_1^{L^{\mathrm{st}}} , \psi_2^{L^{\mathrm{st}}} , \dots, \psi_n^{L^{\mathrm{st}}} \right\}. \end{aligned} $$

(11.35)

If $q \equiv 0$ then this estimate gives the exact value for $\lambda _n.$ Therefore it is natural to split the quadratic form as follows:

$$\displaystyle \begin{aligned} \lambda_n (L_q^{\mathrm{st}}) \leq \max_{u \in \mathcal V^0_n} \frac{Q_{L_q^{\mathrm{st}}} (u,u)}{\| u\|{}^2} \leq \max_{u \in \mathcal V^0_n} \frac{\int_{\Gamma} \vert u' \vert^2 dx }{\| u\|{}^2} + \max_{u \in \mathcal V^0_n} \frac{\int_\Gamma q_+ (x) \vert u \vert^2 dx }{\| u\|{}^2} .\end{aligned}$$

Then the first quotient is equal to $\lambda _n (L^{\mathrm {st}})$ and the maximum is attained on

$$\displaystyle \begin{aligned} u = \psi_n ^{L^{\mathrm{st}}}.\end{aligned}$$

If nothing about $q$ is known, then to estimate the second quotient one may use

$$\displaystyle \begin{aligned} {} \int_\Gamma q_+ (x) \vert u \vert^2 dx \leq \| q_+ \|{}_{L_1(\Gamma)} \, \left( \max_{x \in \Gamma} \vert u(x) \vert \right)^2. \end{aligned} $$

(11.36)

We need to estimate $\vert u(x) \vert ^2$, provided $u = \sum _{j=1}^n \alpha _j \psi _j^{L^{\mathrm {st}}}.$ The Laplacian eigenfunctions $\psi _j ^{L^{\mathrm {st}}}$ possess the uniform upper bound

$$\displaystyle \begin{aligned} {} \vert \psi_j^{L^{\mathrm{st}}} (x) \vert \leq c \| \psi_j^{L^{\mathrm{st}}} \|{}_{L_2(\Gamma)} , \end{aligned} $$

(11.37)

where the constant $c$ depends on the graph $\Gamma$ only. To prove this one may use the fact that every Laplacian eigenfunction $\psi$ is a sine function on each edge and therefore admits

$$\displaystyle \begin{aligned} \| \psi \|{}^2_{L_2(\Gamma)} \geq \int_{E_i} \vert \psi (x) \vert^2 dx \geq \left( \max_{x \in E_i} \vert \psi(x) \vert \right)^2 \frac{1}{2} \left[ \frac{\ell_i k}{2 \pi} \right] \frac{2 \pi}{k}\end{aligned}$$

and take into account that the eigenvalues satisfy Weyl asymptotics implying that $\left [ \frac {\ell _i k}{2 \pi } \right ]$ may be equal to zero just for a finite number of eigenfunctions.

We get the following explicit estimate for the second quotient

$$\displaystyle \begin{aligned} \max_{u \in \mathcal V^0_n} \frac{\int_\Gamma q_+ (x) \vert u \vert^2 dx }{\| u\|{}^2} \leq \| q_+ \|{}_{L_1(\Gamma)} c^2 \max_{\alpha_j} \frac{\left\vert \alpha_1 + \alpha_2 + \dots + \alpha_n \right\vert^2}{\vert \alpha_1 \vert^2 + \vert \alpha_2 \vert^2 + \dots + \vert \alpha_n \vert^2}.\end{aligned}$$

The maximum is attained when all alpha are equal, for example if $\alpha _1 = \alpha _2 = \dots = 1 ,$ leading to

$$\displaystyle \begin{aligned} {} \max_{u \in \mathcal V^0_n} \frac{\int_\Gamma q_+ (x) \vert u \vert^2 dx }{\| u\|{}^2} \leq \| q_+ \|{}_{L_1(\Gamma)} c^2 n. \end{aligned} $$

(11.38)

It follows that

$$\displaystyle \begin{aligned} {} \lambda_n (L^{\mathrm{st}}_q) - \lambda_n (L^{\mathrm{st}}) \leq \| q_+ \|{}_{L_1(\Gamma)} c^2 n, \end{aligned} $$

(11.39)

i.e. we do not get an estimate uniform in $n$—the estimate grows linearly with $n.$ The reason is the splitting of the quadratic form of $L_q^{\mathrm {st}}$ into two parts. The maxima of the two parts are realized on intrinsically different vectors: the first term is maximized if $u = \psi _n^{L^{}},$ while since nothing is known about the potential all eigenfunctions may play the same role in the second estimate. This is the reason why the obtained estimate is not optimal. Let us prove the theorem getting a uniform estimate in $n$.

Proof of Theorem 11.8

It is enough to prove the theorem for any fixed total length $\mathcal L$, therefore we assume that $\mathcal L = \pi$ in order to simplify formulas. The proof is divided into two parts proving upper and lower estimates separately.

Upper Estimate As before we use the estimate

$$\displaystyle \begin{aligned} \lambda_n (L^{\mathrm{st}}_q) \leq \max_{u \in \mathcal V^0_n} \frac{\int_\Gamma \vert u' \vert^2 dx + \int_\Gamma q_+ \vert u \vert^2 dx }{\| u \|{}^2}, \end{aligned} $$

(11.40)

where $\mathcal V^0_n$ is defined by (11.35). Every function $u = \sum _{j=1}^n \alpha _j \psi _j^{L^{\mathrm {st}}}$ from $\mathcal V^0_n$ can be written as a sum $u = u_1 + u_2$, where

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle u_1 & := & \displaystyle \alpha_1 \psi_1^{L^{\mathrm{st}}} + \alpha_2 \psi_2^{L^{\mathrm{st}}} + \dots + \alpha_{n-p-1} \psi_{n-p}^{L^{\mathrm{st}}} , \\ \displaystyle u_2 & := & \displaystyle \alpha_{n-p} \psi_{n-p+1}^{L^{\mathrm{st}}} + \alpha_{n-p+1}\psi_{n-p+1}^{L^{\mathrm{st}}} + \dots + \alpha_{n} \psi_{n}^{L^{\mathrm{st}}} . \\ \end{array} \end{aligned} $$

(11.41)

Here $p$ is a certain natural number to be fixed later (depending on $\Gamma$ and $q$), 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.

Using the fact that $q_+$ is nonnegative we have

$$\displaystyle \begin{aligned} {} \int_\Gamma q_+ \vert u_1 + u_2 \vert^2 dx \leq 2 \int_\Gamma q_+ \vert u_1 \vert^2 dx + 2 \int_\Gamma q_+ \vert u_2 \vert^2 dx. \end{aligned} $$

(11.42)

Then taking into account that $u_1$ and $u_2$ as well as $u_1^{\prime }$ and $u_2^{\prime }$ are mutually orthogonal we arrive at

To estimate the first form we use the Sobolev estimate (11.12)

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle Q_{L^{\mathrm{st}}_{2q_+}} (u_1, u_1) & = & \displaystyle \int_\Gamma \vert u_1^{\prime}\vert^2 dx + 2 \int_\Gamma q_+ \vert u_1 \vert^2 dx \\ & \leq & \displaystyle (1+ 2 \epsilon \| q_+ \|{}_{L_1(\Gamma)} ) \int_\Gamma \vert u_1^{\prime}\vert^2 dx + \frac{4}{\epsilon} \| q_+ \|{}_{L_1(\Gamma)} \int_\Gamma \vert u_1 \vert^2 dx \\ & \leq & \displaystyle \left((1+ 2\epsilon \| q_+ \|{}_{L_1(\Gamma)} ) \lambda_{n-p} (L^{\mathrm{st}}) + \frac{4}{\epsilon} \| q_+ \|{}_{L_1(\Gamma)} \right) \| u_1 \|{}^2_{L_2(\Gamma)}. \end{array}\end{aligned}$$

The key point is that $\epsilon$ can be chosen in such a way that

$$\displaystyle \begin{aligned} (1+2 \epsilon \| q_+\|{}_{L_1(\Gamma)} ) \lambda_{n-p} + \frac{4}{\epsilon} \| q_+\|{}_{L_1(\Gamma)} < \lambda_n\end{aligned}$$

holds. This will be shown later.

On the other hand, our naive approach (11.38) can be applied to the second form with the only difference that the number of eigenfunctions involved is $p$, not $n$

$$\displaystyle \begin{aligned} Q_{L^{\mathrm{st}}_{2q_+}} (u_2, u_2) \leq \left( \lambda_ n (L^{\mathrm{st}}) + 2 c^2 \| q_+ \|{}_{L_1 (\Gamma)} p \right) \| u_2 \|{}^2_{L_2(\Gamma) }.\end{aligned}$$

Putting together the obtained estimates we get

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle Q_{L^{\mathrm{st}}_q} (u,u) & \leq & \displaystyle \left((1+ 2 \epsilon \| q_+ \|{}_{L_1(\Gamma)} ) \lambda_{n-p} (L^{\mathrm{st}}) + \frac{4}{\epsilon} \| q_+ \|{}_{L_1(\Gamma)} \right) \| u_1 \|{}^2_{L_2(\Gamma)} \\ & & \displaystyle + \left( \lambda_ n (L^{\mathrm{st}}) + 2 c^2 \| q_+ \|{}_{L_1 (\Gamma)} p \right) \| u_2 \|{}^2_{L_2(\Gamma) } \\[5mm] & \leq & \displaystyle \lambda_n (L^{\mathrm{st}}) \| u \|{}^2_{L_2(\Gamma)} + 2 c^2 \| q_+ \|{}_{L_1(\Gamma)} p \| u \|{}^2_{L_2(\Gamma)} \\ && \displaystyle \!-\! \left( \lambda_n (L^{\mathrm{st}}) \!-\! (1\!+\! 2 \epsilon \| q_+ \|{}_{L_1(\Gamma)} ) \lambda_{n-p} (L^{\mathrm{st}}) \!-\! \frac{4}{\epsilon} \| q_+ \|{}_{L_1(\Gamma)} \right) \| u_1 \|{}^2_{L_2(\Gamma)} . \end{array} \end{aligned}$$

We would get the desired estimate

$$\displaystyle \begin{aligned} \lambda_n (L^{\mathrm{st}}_q) \leq \max_{u \in \mathcal V^0_n} \frac{Q_{L^{\mathrm{st}}_q} (u,u)}{\| u \|{}^2_{L_2 (\Gamma)}} \leq \lambda_n (L^{\mathrm{st}}) + C \| u \|{}^2_{L_2(\Gamma)} \end{aligned} $$

(11.43)

with $C = 2 c^2 \| q_+ \|{ }_{L_1(\Gamma )} p$ if we manage to prove that

$$\displaystyle \begin{aligned} {} \lambda_n (L^{\mathrm{st}}) - (1+ 2 \epsilon \| q_+ \|{}_{L_1(\Gamma)} ) \lambda_{n-p} (L^{\mathrm{st}}) - \frac{4}{\epsilon} \| q_+ \|{}_{L_1(\Gamma)} > 0 \end{aligned} $$

(11.44)

for a certain $\epsilon$ that may depend on $n$ and a certain $p$ independent of $n$. We use the following two-sided estimate for Laplacian eigenvalues proven in Sect. 4.6

$$\displaystyle \begin{aligned} {} (n-M)^2 \leq \lambda_n (L^{\mathrm{st}}) \leq (n+N)^2. \end{aligned} $$

(11.45)

Let us choose $\epsilon = 1/n$, this gives us

$$\displaystyle \begin{aligned} \begin{array}{cl} & \displaystyle \lambda_n (L^{\mathrm{st}}) - (1+ 2 \epsilon \| q_+ \|{}_{L_1(\Gamma)} ) \lambda_{n-p} (L^{\mathrm{st}}) - \frac{4}{\epsilon} \| q_+ \|{}_{L_1(\Gamma)} \\ = & \displaystyle \displaystyle \lambda_n (L^{\mathrm{st}}) - (1+ 2/n \| q_+ \|{}_{L_1(\Gamma)} ) \lambda_{n-p} (L^{\mathrm{st}}) - 4n \| q_+ \|{}_{L_1(\Gamma)} \\ \geq & \displaystyle (n-M)^2 - (1+2/n \| q_+ \|{}_{L_1(\Gamma)} ) (n-p+N)^2 - 4n \| q_+ \|{}_{L_1(\Gamma)} \\ = & \displaystyle 2 n \left( p-M-N-3 \| q_+ \|{}_{L_1(\Gamma)} \right) + \mathcal O(1). \end{array} \end{aligned}$$

We see that for any fixed integer $p > M + N + 3 \| q_+ \|{ }_{L_1(\Gamma )}$ the expression is positive for sufficiently large $n$ and the difference between the eigenvalues possesses the uniform upper estimate

$$\displaystyle \begin{aligned} {} \lambda_n (L^{\mathrm{st}}_q) - \lambda_n (L^{\mathrm{st}}) \leq C . \end{aligned} $$

(11.46)

If one is interested in the difference between the eigenvalues for large $n$ only, then the constant $C$ can be taken equal to

$$\displaystyle \begin{aligned} {} C =2 c^2 \| q \|{}_{L_1 (\Gamma)} (M+ N + 3 \| q \|{}_{L_1(\Gamma)} +1) , \end{aligned} $$

(11.47)

but this value of $C$ may be too small in order to ensure that (11.46) holds for all $n$, since proving (11.44) we assumed that $n$ is sufficiently large. The later assumption does not affect the final result, since for a finite number of eigenvalues estimate (11.46) is always satisfied, but the exact value of the constant $C$ may be affected.

Lower Estimate To get a lower estimate we are going to use the maxmin principle (4.52). The first step is to notice that

$$\displaystyle \begin{aligned} {} Q_{L^{\mathrm{st}}_q} (u,u) \geq \int_\Gamma \vert u' (x) \vert^2 dx - \int_\Gamma q_-(x) \vert u(x) \vert^2 dx. \end{aligned} $$

(11.48)

Using the same subspace $\mathcal V^0_{n-1}$ we get

$$\displaystyle \begin{aligned} \lambda_n (L^{\mathrm{st}}_q) \geq \min_{u \perp \mathcal V^0_{n-1}} \frac{Q_{L^{\mathrm{st}}_q} (u,u)}{\| u \|{}^2_{L_2(\Gamma)} }.\end{aligned}$$

Since $u$ is orthogonal to $\mathcal V^0_{n-1}$ it possesses the representation

$$\displaystyle \begin{aligned} u = \sum_{j=n}^\infty \alpha_j \psi_j^{L^{\mathrm{st}}}.\end{aligned}$$

As before let us split the function $u = u_1 + u_2$

$$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle u_1 & := & \displaystyle \alpha_n \psi_n^{L^{\mathrm{st}}} + \alpha_{n+1} \psi_{n+1}^{L^{\mathrm{st}}} + \dots + \alpha_{n+p-1} \psi_{n+p-1}^{L^{\mathrm{st}}} , \\ \displaystyle u_2 & := & \displaystyle \alpha_{n+p} \psi_{n+p}^{L^{\mathrm{st}}} + \alpha_{n+p+1}\psi_{n+p+1}^{L^{\mathrm{st}}} + \dots . \\ \end{array} \end{aligned} $$

(11.49)

Note two important differences:

• the function $u_1$ is given by the sum of $p$ terms, where the number $p$ independent of $n$ will be chosen later, so the functions $u_1$ and $u_2$ exchange their roles;

• the function $u_2$ is given by a series, not by an increasing number of terms as the function $u_1$ in the proof of the upper estimate.

Using the fact that $q_-$ is nonnegative we may split the quadratic form

Now the first function $u_1$ is given by a finite number of terms and the following estimate can be used

$$\displaystyle \begin{aligned} {} Q_{L^{\mathrm{st}}_{-2q_-}} (u_1, u_1) \geq \left( \lambda_ n (L^{\mathrm{st}}) - c^2 p \| q_- \|{}_{L_1 (\Gamma)} \right) \| u_1 \|{}^2_{L_2(\Gamma) }. \end{aligned} $$

(11.50)

To estimate the second form we use (11.36) and the Sobolev estimate (11.12) for $\max |u(x)|{ }^2$. We get

$$\displaystyle \begin{aligned} \begin{array}{ccl} Q_{L^{\mathrm{st}}_{-2q_-}} (u_2, u_2) &\geq& \|u_2^{\prime}\|{}_{L_2}^2 -2\|q_-\|{}_{L_1}\max|u_2(x)|{}^2 \\[2mm] &\geq& \|u_2^{\prime}\|{}_{L_2}^2 -2\|q_-\|{}_{L_1}\left(\epsilon \|u_2^{\prime}\|{}_{L_2}^2+\frac{2}{\epsilon}\|u_2 \|{}_{L_2}^2\right) \\[2mm] &=& (1-2\epsilon \|q_-\|{}_{L_1})\|u_2^{\prime}\|{}_{L_2}^2 -\frac{4\|q_-\|{}_{L_1}}{\epsilon}\|u_2\|{}_{L_2}^2. \end{array}\end{aligned}$$

Taking into account

$$\displaystyle \begin{aligned} \|u_2^{\prime}\|{}_{L_2}^2 \ge \lambda_{n+p}({L_{0}^{\mathrm{st}}})\|u_2\|{}_{L_2}^2, \end{aligned} $$

(11.51)

we arrive at

$$\displaystyle \begin{aligned} {} Q_{L^{\mathrm{st}}_{-2q_-}} (u_2, u_2) \geq \left((1- 2\epsilon \| q_- \|{}_{L_1 (\Gamma)}) \lambda_{n+p} (L_0^{\mathrm{st}}) - \frac{ 4\| q_- \|{}_{L_1 (\Gamma)}}{\epsilon} \right) \| u_2 \|{}^2_{L_2}. \end{aligned} $$

(11.52)

Summing the estimates (11.50) and (11.52) and taking into account that ${\| u_2 \|{ }_{L_2}^2 = \| u \|{ }_{L_2}^2 - \| u_1 \|{ }_{L_2}^2}$ we get

$$\displaystyle \begin{aligned} \begin{array}{ccl} Q_{L^{\mathrm{st}}_q} (u,u) & \geq & \left( \lambda_{n} (L_0^{\mathrm{st}}) - 2 c^2 p \| q_- \|{}_{L_1 } \right) \| u_1 \|{}^2_{L_2 } \\[2mm] & & + \left((1- 2\epsilon \| q_- \|{}_{L_1 }) \lambda_{n+p} (L_0^{\mathrm{st}}) - \frac{4\| q_- \|{}_{L_1 }}{\epsilon} \right) \| u_2 \|{}^2_{L_2} \\[3mm] & \geq & \lambda_{n} (L_0^{\mathrm{st}}) \| u \|{}^2_{L_2} - 2 c^2 p \| q_- \|{}_{L_1} \| u \|{}^2_{L_2} \\[2mm] & & + \left( (1- 2\epsilon \| q_- \|{}_{L_1} ) \lambda_{n+p} (L_0^{\mathrm{st}}) -\frac{4\| q_- \|{}_{L_1 }}{\epsilon} - \lambda_{n-1} (L_0^{\mathrm{st}}) \right) \| u_2 \|{}^2_{L_2} . \end{array}\end{aligned}$$

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 $\epsilon$:

$$\displaystyle \begin{aligned} {} (1 - 2\epsilon \| q_- \|{}_{L_1} ) \lambda_{n+p} (L_0^{\mathrm{st}}) - \frac{4\| q_- \|{}_{L_1}}{\epsilon} - \lambda_{n} (L_0^{\mathrm{st}}) > 0. \end{aligned} $$

(11.53)

Again we use (11.45): we substitute $\lambda _{n+p}(L_0^{\mathrm {st}})$ with the lower bound and $\lambda _{n}(L_0^{\mathrm {st}})$ with the upper. As before we choose $\epsilon = 1/n$, so the left-hand side of (11.53) becomes

$$\displaystyle \begin{aligned} \begin{array}{cl} & (1 - 2\epsilon \| q_- \|{}_{L_1} ) \lambda_{n+p} (L_0^{\mathrm{st}}) - \frac{4}{\epsilon} \| q_- \|{}_{L_1} - \lambda_{n} (L_0^{\mathrm{st}}) \\[2mm] \geq & (1- 2 \| q_- \|{}_{L_1}/n) (n+p-M)^2 \\[2mm] & - 4n \| q_- \|{}_{L_1} - (n+N)^2 \\[2mm] = & 2n \Big(p-M-N-3 \| q_- \|{}_{L_1}\Big) + \mathcal O(1). \end{array}\end{aligned}$$

If $p > M + N + 3 \| q_- \|{ }_{L_1}$, then for sufficiently large $n$ the expression is positive, hence the following lower estimate holds

$$\displaystyle \begin{aligned} {} \lambda_n (L^{\mathrm{st}}_q) - \lambda_n (L_0^{\mathrm{st}}) \geq C , \end{aligned} $$

(11.54)

where the exact value of $C$ is determined by the difference between the first few eigenvalues as described above. Here we use (11.31) with $\epsilon = 1$. □

If we are interested in large values of $n$ only, then we have an explicit formula for the constant $C(\Gamma , q)$ given by (11.47).

Problem 47

The constant $C$ appearing in (11.32) for sufficiently large $n$ can be taken in the form (11.47). On the other hand if $q \in L_\infty (\Gamma )$, then $C$ can be taken equal to $\| q \|{ }_{L_\infty (\Gamma )}.$ Which expression provides the best value and why?

Problem 48

Show that Theorem 11.8 holds for non-Robin vertex conditions.

11.3 Spectral Estimates for General Vertex Conditions

As the title of this section indicates we are going to prove spectral estimates in the case, where the vertex conditions are not assumed to be standard. The most general class of vertex conditions described in Chap. 3 will be covered.

Assume that a finite compact metric graph $\Gamma$, a real $L_1$-potential $q \in L_1 (\Gamma )$ and unitary irreducible vertex matrices $S^m$ are given. Then the corresponding self-adjoint Schrödinger operator $L^{\mathbf {S}}_q (\Gamma )$ is defined using Definition 4.1. The spectral estimate we are going to prove resembles very much the estimate (11.32) with the only difference that the constant $C$ should obviously depend not only on the graph $\Gamma$ and potential $q$, but on the vertex conditions as well. We did not consider the most general vertex conditions in the previous section just in order to simplify our presentation. What we need now, is just to go through the proof and amend all necessary formulas.

For the estimates we used quadratic forms, hence instead of the matrices $S^m$ we need to consider the corresponding Hermitian matrices $A^m$ appearing in the generalised Robin conditions (11.7). Every Hermitian matrix can be written as a difference between two positive matrices

$$\displaystyle \begin{aligned} A^m = A^m_+ - A^m_-, \end{aligned} $$

(11.55)

where $A^m_\pm$ are defined using the spectral representation of $A^m$

$$\displaystyle \begin{aligned} A^m = \sum_{\lambda_n (A^m) \neq 0} \lambda_n (A^m) \langle \vec{e}_n, \; \cdot \; \rangle \vec{e}_n ,\end{aligned}$$

as follows

$$\displaystyle \begin{aligned} \left\{ \begin{array}{ccl} \displaystyle A^m_+ & = & \displaystyle \sum_{\lambda_n (A^m) > 0 } \lambda_n (A^m) \langle \vec{e}_n, \; \cdot \; \rangle \vec{e}_n ,\\ \displaystyle A^m_- & = & - \displaystyle \sum_{\lambda_n (A^m) < 0 } \lambda_n (A^m) \langle \vec{e}_n, \; \cdot \; \rangle \vec{e}_n . \end{array} \right. \end{aligned} $$

(11.56)

Let us analyse which ingredients were crucial for the proof of Theorem 11.8:

• 1. explicit estimates for the perturbation term from below and from above (11.48) and (11.33);

• 2. the estimate (11.36) for the perturbation term bounding it from above by $\| u \|{ }^2_{L_\infty (\Gamma )};$

• 3. the inequality (11.42) allowing to split the sum into two terms;

• 4. Sobolev type inequality (11.12) allowing to estimate the $L_\infty$-norm of a function through its $W_2^1$-norm;

• 5. the estimate (11.37) for the Laplacian eigenfunctions.

The proof can be carried out without major modifications, provided the above mentioned estimates hold. One might need to amend the constants as described above.

• 1. The lower and upper bounds (11.48) and (11.33) for the perturbation term can be modified as follows:

  

  (11.57)

  Note that the quadratic forms $B^+_{L^{\mathbf S}_q}$ do not depend on the negative part $q_-$ of the potential $q$ and on the negative part $A^m_-$ of the matrix $A^m.$ Similarly, $B^-_{L^{\mathbf S}_q}$ is independent of $q_+$ and $A^m_+ .$

• 2. Taking into account that all functions from the space $W_2^1 (\Gamma \setminus \mathbf V)$ are continuous on every edge in $\Gamma$ we may modify (11.36) as

  $$\displaystyle \begin{aligned} \begin{array}{ccl} \displaystyle \left\vert B^\pm_{L^s_q} (u,u) \right\vert & \leq & \displaystyle \left( \| q_\pm \|{}_{L_1(\Gamma)} + \sum_{m=1}^M d_m \| A^m_\pm \| \right) \| u \|{}^2_{L_\infty (\Gamma)} \\ & \leq & \displaystyle \left( \| q \|{}_{L_1(\Gamma)} + \sum_{m=1}^M d_m \| A^m \| \right) \| u \|{}^2_{L_\infty (\Gamma)}. \end{array} \end{aligned} $$

  (11.58)

• 3. Taking into account nonnegativity of $q_\pm$ and $A^m_\pm$ we generalise estimate (11.42) as

  $$\displaystyle \begin{aligned} B^\pm_{L^{\mathbf{S}}_q} (u_1+u_2,u_1+u_2) \leq 2 B^\pm_{L^{\mathbf{S}}_q} (u_1,u_1) +2 B^\pm_{L^{\mathbf{S}}_q} (u_2,u_2). \end{aligned} $$

  (11.59)

• 4. The estimate (11.12) is valid without any modification, since the vertex conditions were not used in its proof.

• 5. The estimate (11.37) is valid as well for the same reason.

Therefore we may just repeat the proof of Theorem 11.8 to get:

Theorem 11.9

Let $L_q^{\mathbf {S}} (\Gamma )$ be an arbitrary Schrödinger operator on a compact finite metric graph $\Gamma$ with absolutely integrable potential $q \in L_1 (\Gamma )$ and unitary irreducible vertex matrices $S^m , m = 1,2, \dots , M$ . Consider the high energy limit of the vertex scattering matrices $S_{\mathbf {v}}^m(\infty )$ and the corresponding reference Laplace operator $L^{\mathbf S_{\mathbf {v}}(\infty )}$ defined on the graph $\Gamma ^\infty .$ Then the Schrödinger and the refernce Laplace operators are asymptotically isospectral, moreover the difference between their eigenvalues is uniformly bounded

$$\displaystyle \begin{aligned} {} \vert \lambda_n (L^{\mathbf S_{\mathbf{v}}(\infty)}(\Gamma^\infty) ) - \lambda_n (L_q^{\mathbf{S}} (\Gamma)) \vert \leq C (\Gamma, q, \mathbf S), \end{aligned} $$

(11.60)

where the constant $C (\Gamma , q, \mathbf S)$ depends on the graph $\Gamma$ , the potential $q$ and the vertex matrix $\mathbf S$ , but is independent of $n.$

Proof

As we already mentioned the proof of Theorem 11.8 can be repeated without major modifications. As a result one arrives at the conclusion that the spectrum of $L^{\mathbf {S}}_q$ is close to the spectrum of the self-adjoint operator corresponding to the quadratic form

$$\displaystyle \begin{aligned} \int_{\Gamma} \vert u' (x) \vert^2 dx\end{aligned}$$

with the domain of functions from $W_2^1 (\Gamma \setminus \mathbf V)$ satisfying the generalised Dirichlet conditions (11.3) at the vertices. The self-adjoint operator corresponding to this quadratic form is the Laplace operator defined on the functions from $W_2^2 (\Gamma \setminus \mathbf V)$ satisfying both generalised Dirichlet (11.3) and generalised Neumann conditions (11.23) at the vertices. To write such vertex conditions in the form (3.21) one has to substitute the unitary matrix $\mathbf S$ with the high energy limit of the vertex scattering matrix $\mathbf S_{\mathbf {v}} (\infty ).$ □

The main difference between the Theorems 11.8 and 11.9 is that the reference Laplacian is not the standard Laplacian on the same metric graph $\Gamma$, but is the Laplace operator on the graph $\Gamma ^\infty$ with the vertex conditions given by (11.3) and (11.23). Hence the reference Laplacian is not necessarily a standard Laplacian on $\Gamma ^\infty .$ Of course, if the vertex conditions in $L^{\mathbf S}_q$ are asymptotically properly connecting and standard, then the reference operator is just the standard Laplacian on $\Gamma .$

Problem 49

Consider the cycle of length $\pi$ with a single degree two vertex. Assume that the vertex conditions are given by the matrix $\mathbf S = \frac {1}{2} \left ( \begin {array}{cc} 1 + i & 1-i \\ 1-i & 1+ i \end {array} \right )$ from Example 11.2. Show that the spectrum tends to the spectrum of the Neumann Laplacian on the interval $[0, \pi ].$