# Spectral analysis of the matrix Sturm–Liouville operator

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## Abstract

The self-adjoint matrix Sturm–Liouville operator on a finite interval with a boundary condition in general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on an analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm–Liouville operators on a star-shaped graph with two different types of matching conditions.

## Keywords

Matrix Sturm–Liouville operator General self-adjoint boundary condition Eigenvalue asymptotics Asymptotics of weight matrices Sturm–Liouville operators on graphs## MSC

34B09 34B24 34B45 34L20 34L40## 1 Introduction

The paper concerns the spectral theory of differential operators. In particular, we focus on analysis of matrix Sturm–Liouville operators in the form \(\ell Y = -Y'' + Q(x) Y\), where \(Q(x)\) is a matrix function. Operators of such form generalize scalar Sturm–Liouville operators, which have been studied fairly completely (see the monographs [1, 2, 3]). Matrix Sturm–Liouville operators have applications in various branches of physics. For example, such operators are used in elasticity theory [4], they are also applied for the description of electromagnetic waves [5] and nuclear structure [6]. Matrix Sturm–Liouville operators also generalize Sturm–Liouville operators on metric graphs (see [7, 8, 9, 10, 11, 12] are the references therein). The latter operators are often called quantum graphs. They are applied for modeling wave propagation through a domain being a thin neighborhood of a graph. Such models are used in organic chemistry, mesoscopic physics, nanotechnology and other branches of science and engineering.

\(Y(x) = [y_{j}(x)]_{j = 1}^{m}\) is a vector function;

\(Q(x) = [q_{jk}(x)]_{j, k = 1}^{m}\) is a Hermitian matrix function, called the

*potential*, such that \(Q(x) = Q^{\dagger }(x)\), \(q_{jk} \in L_{2}(0, \pi )\), \(j, k = \overline{1, m}\);*λ*is the spectral parameter;*T*is an orthogonal projector in \(\mathbb{C}^{m}\), \(T \in \mathbb{C} ^{m \times m}\), \(T^{\perp } = I_{m} - T\);\(H = H^{\dagger } \in \mathbb{C}^{m \times m}\), \(H = T H T\).

Here and below \(\mathbb{C}^{m}\) and \(\mathbb{C}^{m \times m}\) are the spaces of complex *m*-vectors and \(m \times m\)-matrices, respectively, the symbol † denotes the conjugate transpose: \(A = [a_{jk}]\), \(A^{\dagger } = [\overline{a_{kj}}]\), and \(I_{m}\) is the \(m \times m\) unit matrix.

Note that the boundary value problem *L* is self-adjoint. The boundary condition \(V(Y) = 0\) is the self-adjoint boundary condition in the most general form. For simplicity, we impose the Dirichlet boundary condition at \(x = 0\). The problem with the two boundary conditions in the general form, similar to \(V(Y) = 0\), can also be investigated by using our methods.

Spectral properties of operators with general self-adjoint boundary conditions in the form (1.2) have not been so well-studied yet, since this case is more complex for investigation. Recently Xu [19] proved uniqueness theorems for inverse problems for Eq. (1.1) with the two boundary conditions in the general self-adjoint form. Such operators are especially worth to be studied because of their applications to quantum graphs. The inverse problem theory for differential operators on graphs is a rapidly developing field nowadays (see the survey [12]). However, characterization of spectral data is an open problem even for Sturm–Liouville operators on the simplest star-shaped graphs, as well as for the matrix Sturm–Liouville operator with the general self-adjoint boundary conditions.

The goal of the present paper is to make the first step to a characterization of the spectral data of the problem (1.1)–(1.2). We aim to obtain asymptotic formulas for the eigenvalues and the weight matrices of *L*. In the future, we plan to apply these results to the inverse problem theory.

Let us introduce the spectral data studied in our paper. Furthermore, we show that the problem *L* has a countable set of real eigenvalues. It is convenient to number them as \(\{ \lambda _{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\) in nondecreasing order: \(\lambda _{n_{1}, k_{1}} \le \lambda _{n_{2}, k_{2}}\) for \((n_{1}, k_{1}) < (n_{2}, k_{2})\). For pairs of integers, we mean by \((n_{1}, k_{1}) < (n_{2}, k_{2})\) that \(n_{1} < n_{2}\) or \(n_{1} = n_{2}\), \(k_{1} < k_{2}\). The double indices \((n, k)\) are used because of the eigenvalue asymptotics (see Theorem 2.1). Multiple eigenvalues occur in the sequence \(\{ \lambda _{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\) multiple times, according to the multiplicities.

*the Weyl solution*and

*the Weyl matrix*of the problem

*L*, respectively. Weyl matrices generalize Weyl functions for scalar differential operators (see [1, 3]) and play an important role in inverse problem theory of matrix Sturm–Liouville operators (see, e.g., [16, 18]). One can easily show that the Weyl matrix \(M(\lambda )\) is meromorphic in the

*λ*-plane. All its singularities are simple poles, which coincide with the eigenvalues \(\{ \lambda _{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\). Define

*the weight matrices*:

The collection \(\{ \lambda _{nk}, \alpha _{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\) is called *the spectral data* of the problem *L*.

A general approach to the derivation of eigenvalue asymptotics was suggested in [20] for arbitrary first-order differential systems. This approach is based on an analysis of analytic characteristic functions in the complex plane of the spectral parameter. Nevertheless, adaption of general methods to systems of special form (e.g., matrix Sturm–Liouville operators or differential operators on graphs) usually requires an additional investigation (see, e.g., [13, 14, 15, 16, 21]). We also mention that matrix Sturm–Liouville operators have been studied on the half-line [10, 22] and on the line [23, 24, 25]. However, operators on infinite domains usually have a finite number of eigenvalues, therefore problems of eigenvalue asymptotics are not relevant to them.

The main difficulty in the derivation of the spectral data asymptotics for the problem (1.1)–(1.2) is that its spectrum can contain infinitely many groups of multiple and/or asymptotically multiple eigenvalues (see Theorem 2.1 for details). Consequently, it is impossible to obtain separate asymptotic formulas for the sequences \(\{ \alpha _{nk} \}_{n \in \mathbb{N}}\), \(k = \overline{1, m}\). In the present paper, we overcome this difficulty, by developing the ideas of [18, 26, 27]. The eigenvalues are grouped by asymptotics, and asymptotic formulas are derived for the sums of the weight matrices, corresponding to each eigenvalue group.

The paper is organized as follows. Section 2 is devoted to the asymptotic behavior of the eigenvalues. Its main result is Theorem 2.1, proof of which is based on complex analysis of an analytic characteristic function and on the matrix Rouche’s theorem. In Sect. 3, we define the sums of the weight matrices, corresponding to eigenvalue groups. We derive asymptotic formulas for those sums (Theorems 3.1 and 3.4), by developing the methods of Sect. 3 and by using contour integration. In Sect. 4, our results for the matrix Sturm–Liouville operator are adapted to the Sturm–Liouville operators on the star-shaped graph with two types of matching conditions: *δ*-coupling and \(\delta '\)-coupling, arising in applications.

## 2 Eigenvalue asymptotics

The goal of this section is to derive asymptotic of eigenvalues of the problem *L*. First we need to introduce some notations.

*λ*-plane for each fixed \(x \in [0, \pi ]\), \(\nu = 0, 1\). The eigenvalues of the boundary value problem

*L*coincide with the zeros of the

*characteristic function*\(\Delta (\lambda ) := \det (V(S(x, \lambda )))\), counting with their multiplicities. This fact can be proved similarly to [18, Lemma 5]. Since the problem

*L*is self-adjoint, its eigenvalues are real. Obviously, the function \(\Delta (\lambda )\) is entire. In order to study its asymptotic behavior, we need the relations for \(S^{(\nu )}(\pi , \lambda )\) and \(C^{(\nu )}( \pi , \lambda )\), \(\nu = 0, 1\), provided below.

*ρ*and satisfying the conditions

### Theorem 2.1

*The spectrum of the boundary value problem*

*L*

*is a countable set of real eigenvalues*\(\{ \lambda _{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\),

*counted with their multiplicities*, \(\lambda _{n_{1}, k_{1}} \le \lambda _{n_{2}, k_{2}}\),

*if*\((n_{1}, k_{1}) < (n_{2}, k_{2})\).

*Moreover*,

*the following asymptotic relations hold for*\(\rho _{nk} := \sqrt{\lambda _{nk}}\), \(n \in \mathbb{N}\):

*where*\(\{ z_{k} \}_{k = 1}^{p}\)

*and*\(\{ z_{k} \}_{k = p + 1}^{m}\)

*are the eigenvalues of the Hermitian matrices*\((\omega _{11} - H)\)

*and*\(\omega _{22}\),

*respectively*,

*numbered in the nondecreasing order according to their multiplicities*: \(z_{k} \le z_{k + 1}\), \(k = \overline{1, p-1}\)

*and*\(k = \overline{p + 1, m - 1}\),

*and*\(\{ \varkappa _{nk} \} \in l_{2}\).

For the proof of Theorem 2.1, we need several auxiliary results. The following proposition is a matrix version of Rouche’s theorem (see [15, Lemma 2.2]).

### Proposition 2.2

*Let*\(F(\lambda )\)*and*\(G(\lambda )\)*be matrix functions*, *analytic in the disk*\(|\lambda - a| \le r\)*and satisfying the condition*\(\| G(\lambda ) F^{-1}(\lambda ) \| < 1\)*on the boundary*\(|\lambda - a| = r\). *Then the scalar functions*\(\det (F)\)*and*\(\det (F + G)\)*have the same number of zeros inside the circle*\(|\lambda - a| < r\), *counting with multiplicities*.

Then we have the next proposition [27, Lemma 3].

### Proposition 2.3

*Let*\(\{ \delta _{n} \}_{n \ge 1}\)

*and*\(\{ \gamma _{n} \}_{n \ge 1}\)

*be two sequences of nonzero numbers*,

*such that*

*where*

*Then*\(\delta _{n} = O(\gamma _{n})\)

*as*\(n \to \infty \).

Now we are ready to obtain rough asymptotics for the eigenvalues.

### Lemma 2.4

*The problem*

*L*

*has a countable set of eigenvalues*\(\{ \lambda _{nk} \} _{n \in \mathbb{N}, k = \overline{1, m}}\), \(\lambda _{nk} = \rho _{nk}^{2}\),

*numbered according to their multiplicities and having the following asymptotics*:

*where*\(\varepsilon _{nk} = O ( n^{-1} )\)

*as*\(n \to \infty \).

### Proof

*L*are the zeros of \(\det (W(\lambda ))\). The asymptotics (2.3) yield

*δ*, \(0 < \delta < 1/4\). Using the standard estimates (see [3, Sect. 1.1]):

*N*and arbitrary \(\delta \in (0, 1/4)\). Taking (2.8) into account, we arrive at the assertion of the theorem with \(\varepsilon _{nk} = o(1)\) as \(n \to \infty \).

*p*with the coefficients depending on

*n*and \(\varepsilon _{nk}\). One can easily check that this polynomial satisfies the conditions of Proposition 2.3 with \(\delta _{n} = \varepsilon _{nk}\) and \(\gamma _{n} = n^{-1}\). Applying Proposition 2.3, we conclude that \(\varepsilon _{nk} = O ( n ^{-1} )\), \(n \to \infty \). □

Below the symbol *C* is used for various positive constants.

### Proof of Theorem 2.1

Theorem 2.1 follows from Lemma 2.4, if we derive the asymptotics (2.4) and (2.5) from (2.6). We focus on the proof of (2.5), since it is more complicated.

*ρ*-plane, we will work with a new

*z*-plane, obtained by the mapping

*z*lies in the circle \(|z| \le r\) of a sufficiently large radius \(r > 0\). Below we use the notation \(\{ K_{n}(z) \}_{n \in \mathbb{N}}\) for various sequences of matrix functions, such that

*C*depends on a sequence \(\{ K_{n}(z) \}_{n \in \mathbb{N}}\), but does not depend on

*z*.

*T*and \(T^{\perp }\) have different powers of

*n*. In order to overcome this difficulty, we introduce the function

*n*and \(z \in Z_{\delta }\). Applying Proposition 2.2, we conclude that, for sufficiently large

*n*, the function \(\det (R(z))\) has exactly \((m - p)\) zeros \(\{ z_{nk} \}_{k = \overline{p + 1, m}}\) (counted according to their multiplicities), having the asymptotics

Fix \(k \in \{ p + 1, \dots , m \}\). Let \(m_{k}\) be a multiplicity of the zero \(z_{k}\) of the function \(\det (R(z))\): \(m_{k} = \# \{ s \colon z _{s} = z_{k}, p + 1 \le s \le m \}\). Using (2.16) and (2.18), we derive that \(\det (H_{n}(z_{nk})) = \mathcal{P} _{nk}(\epsilon _{nk})\), where \(\mathcal{P}_{nk}\) is a polynomial of degree \(m_{k}\) with coefficients, depending on *n* and \(\epsilon _{nk}\) and satisfying Proposition 2.3 with \(\delta _{n} = \epsilon _{nk}\), \(\{ \gamma _{n} \} \in l_{2}\). Proposition 2.3 yields \(\{ \epsilon _{nk} \} \in l_{2}\). By construction, the zeros of \(\det (H_{n}(z))\) coincide with the zeros of \(W(\rho _{n}^{2}(z))\). Returning to the *ρ*-plane, we obtain (2.5).

## 3 Asymptotics of weight matrices

The goal of this section is to derive asymptotic formulas for the weight matrices \(\{ \alpha _{nk} \}\), defined by (1.3). The main difficulty in our investigation is the complicated behavior of the spectrum, which can contain infinitely many multiple and/or asymptotically multiple eigenvalues. Indeed, if there are equal numbers among \(\{ z_{k} \}_{k = 1}^{p}\) or \(\{ z_{k} \}_{k = p + 1}^{m}\) in Theorem 2.1, the corresponding eigenvalue subsequences \(\{ \lambda _{nk} \}_{n \in \mathbb{N}}\) have the same asymptotics for different values of *k*. In the latter case, one cannot obtain separate asymptotic formulas for \(\alpha _{nk}\), being the residuals of the Weyl matrix. In order to manage this problem, the approach of Refs. [18, 26, 27] is developed. We group eigenvalues by asymptotics and derive asymptotic formulas for the sums of the weight matrices, corresponding to each group.

*l*, then \(\alpha _{n_{1} k_{1}} = \alpha _{n_{2} k_{2}} = \cdots = \alpha _{n_{l} k_{l}}\). For every such group, introduce the notations \(\alpha '_{n_{1} k_{1}} = \alpha _{n_{1} k_{1}}\), \(\alpha '_{n_{j} k_{j}} = 0\), \(j = \overline{2, l}\). Define

*n*. Here \(\varGamma _{n}^{(s)}\) is a contour in the

*λ*-plane, enclosing the eigenvalues \(\{ \lambda _{nk} \}_{k = \overline{p+1, m} \colon z_{k} = z_{s}}\).

*z*and such that \(\{ \| K_{n}(z)\| \} \in l_{2}\).

Thus, we arrive at the following theorem.

### Theorem 3.1

*The matrix sequences*\(\{ \alpha _{n}^{(s)} \}_{n \in \mathbb{N}}\), \(s = \overline{1, m}\), *satisfy the asymptotic relations* (3.10) *and* (3.11), *where the matrices*\(A^{(s)}\), \(s = \overline{1, m}\), *are defined by* (3.9) *and* (3.12).

The definitions (3.9) and (3.12) immediately imply the corollary.

### Corollary 3.2

*The matrices*\(A^{(s)}\)

*in the asymptotics*(3.10)

*and*(3.11)

*are Hermitian*,

*non*-

*negative definite*: \(A^{(s)} = ( A^{(s)} )^{\dagger } \ge 0\), \(s = \overline{1, m}\).

*Furthermore*,

The following corollary is important for inverse problem theory.

### Corollary 3.3

*The spectral data*\(\{ \lambda _{nk}, \alpha _{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\)*uniquely specify the matrices*\((\omega _{11}-H)\)*and*\(\omega _{22}\).

### Proof

*r*such that \(|z_{k}| < r\), \(k = \overline{p + 1, m}\). One can easily show that for \(|z| = r\) the following relation holds:

*λ*-plane, enclosing the eigenvalues \(\{ \lambda _{nk} \}_{k = p + 1}^{m}\).

*n*has the poles \(\{ z_{nk}^{0} \}_{k = p + 1}^{m}\), such that \(z_{nk}^{0} \to z_{k}\) as \(n \to \infty \), \(k = \overline{p + 1, m}\). Hence

### Theorem 3.4

*The following asymptotic relations hold*

Theorems 2.1, 3.1 and 3.4 yield the following corollary. It is valid for an arbitrary orthogonal projector *T*, not necessarily defined by (2.1).

### Corollary 3.5

*Let*\(\hat{L} = L(\hat{Q}(x), T, \hat{H})\)

*be a boundary value problem of the same form*(1.1)

*–*(1.2)

*as*

*L*,

*but with different coefficients*\(\hat{Q}(x)\)

*and*

*Ĥ*.

*Suppose that*

*Then*

*where the values*\(\hat{\rho }_{nk}\), \(\hat{\alpha }_{n}^{(s)}\), \(\hat{\alpha }_{n}^{I}\)

*and*\(\hat{\alpha }_{n}^{II}\)

*are defined by the problem*

*L̂*

*in the same way as*\(\rho _{nk}\), \(\alpha _{n}^{(s)}\), \(\alpha _{n}^{I}\)

*and*\(\alpha _{n}^{II}\)

*are defined by the problem*

*L*,

*respectively*,

*for*\(n \in \mathbb{N}\), \(k, s = \overline{1, m}\).

## 4 Examples of Sturm–Liouville operators on a graph

In this section, we apply our results to differential operators on a star-shaped graph. Two types of matching conditions are considered in the internal vertex: *δ*-coupling and \(\delta '\)-coupling (see [28]).

Let *G* be a star-shaped graph, consisting of *m* edges \(\{ e_{j} \} _{j = 1}^{m}\) of length *π*. For every edge \(e_{j}\), we introduce a parameter \(x_{j} \in [0, \pi ]\). It is supposed that the end \(x_{j} = 0\) corresponds to the boundary vertex and \(x_{j} = \pi \) corresponds to the internal vertex.

*G*:

*δ*-type matching conditions in the internal vertex:

By a unitary transform (2.2), the problem (4.4)–(4.5) can be reduced to the form (1.1)–(1.2) with the matrices *T*, \(T^{\perp }\) and *H* in the form (2.1). Clearly, \(p = 1\) and \(h = \frac{\beta }{m}\). The matrix *U* in (2.2) has to be chosen in such a way that the columns of \(U^{\dagger }\) equal to the orthonormal eigenvectors of the matrices *T̃* and \(\tilde{T} ^{\perp }\), corresponding to the eigenvalue 1. In particular, the first column of \(U^{\dagger }\) consists of the equal numbers \(\frac{1}{\sqrt{m}}\) and the remaining columns are the orthonormal eigenvectors of \(\tilde{T}^{\perp }\). The choice of *U* is not unique, but all the possible choices lead to the same asymptotics of the spectral data.

*U*consists of the numbers \(\frac{1}{\sqrt{m}}\), we get

The unitary transform *U* does not influence the eigenvalues. Thus, we arrive at the following result.

### Theorem 4.1

*The spectrum of the boundary value problem*(4.1)

*–*(4.3)

*is formed by a countable set of real eigenvalues*\(\{ \tilde{\lambda }_{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\),

*such that*\(\tilde{\rho }_{nk} = \sqrt{\tilde{\lambda }_{nk}}\)

*satisfy the asymptotic formulas*

*where*\(z_{1}\)

*is defined by*(4.8), \(\{ z_{k} \}_{k = 2}^{m}\)

*are the zeros of the polynomial*\(\mathcal{P}(z)\),

*numbered according to their multiplicities*,

*and*\(\{ \tilde{\varkappa }_{nk} \} \in l_{2}\).

Define the weight matrices \(\{ \tilde{\alpha }_{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\) of the problem (4.1)–(4.3) on the star-shaped graph as the weight matrices of the equivalent matrix problem (4.4)–(4.5). Let \(\{ \tilde{\alpha }_{n}^{(s)} \} _{n \in \mathbb{N}, s = \overline{1, m}}\), and \(\{ \tilde{\alpha } _{n}^{II} \}_{n \in \mathbb{N}}\) be the sums of the weight matrices, defined for the problem (4.4)–(4.5) similarly to the ones in Sect. 3. Then Theorems 3.1 and (3.4) together with Eqs. (2.2) imply the following theorem.

### Theorem 4.2

*The weight matrices of the problem*(4.1)

*–*(4.3)

*satisfy the following asymptotic relations*:

*where*\(\{ A^{(s)} \}_{s = 2}^{m}\)

*are the matrices*,

*defined by*(3.9).

The eigenvalue asymptotics of Theorem 4.1 have been obtained in [21] by another method. In [29], asymptotic relations for the main diagonals of the weight matrices \(\{ \alpha _{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\) have been derived, which conform to Theorem 4.2.

*G*for the Sturm–Liouville equations (4.1) with the Dirichlet boundary conditions (4.2) and the following \(\delta '\)-type matching conditions:

*T̃*, \(\tilde{T}^{\perp }\) and

*H̃*are defined by (4.6)–(4.7). In the case \(\beta \ne 0\), we obtain the Robin-type boundary condition

Matrix Sturm–Liouville operators with such boundary conditions have been studied, e.g., in [18], so we exclude this case from consideration.

Suppose that \(\beta = 0\). Then the problem (4.1), (4.2), (4.10) can be reduced to the matrix form (1.1)–(1.2) by the unitary transform with the same matrix *U*, which has been used for the *δ*-coupling case. We obtain \(T = U \tilde{T}^{\perp } U ^{\dagger }\), \(T^{\perp } = U \tilde{T} U^{\dagger }\), \(p = m - 1\), \(H = 0\). Summarizing the arguments above, we arrive at the following theorem, providing the asymptotics for the spectral data \(\{ \tilde{\lambda }_{nk}, \tilde{\alpha }_{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\) of the problem (4.1), (4.2), (4.10).

### Theorem 4.3

*The spectrum of the boundary value problem*(4.1), (4.2), (4.10)

*with*\(\beta = 0\)

*is formed by a countable set of real eigenvalues*\(\{ \tilde{\lambda }_{nk} \}_{n \in \mathbb{N}, k = \overline{1, m}}\),

*such that*\(\tilde{\rho }_{nk} := \sqrt{ \tilde{\lambda }_{nk}}\)

*satisfy the asymptotic formulas*

*where*\(\{ z_{k} \}_{k = 1}^{m-1}\)

*are the zeros of the polynomial*\(\mathcal{P}(z)\), \(z_{m} = \frac{1}{m} \sum_{j = 1}^{m} \tilde{\omega }_{j}\), \(\{ \tilde{\varkappa }_{nk} \} \in l_{2}\).

*The sums of the weight matrices satisfy the following asymptotic relations*:

*where the matrices*\(\{ A^{(s)} \}_{s = 1}^{m-1}\)

*are defined by*(3.12)

*and*

*T̃*, \(\tilde{T}^{\perp }\)

*are defined by*(4.7).

## Notes

### Acknowledgements

Not applicable.

### Availability of data and materials

Not applicable.

### Authors’ contributions

NPB is the sole author of this paper. The author has read and approved the final manuscript.

### Funding

This work was supported by Grant 19-71-00009 of the Russian Science Foundation.

### Competing interests

The author declares that she has no competing interests.

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