Discrete Sturm-Liouville problems: singularity of the $n$-th eigenvalue with application to Atkinson type

TL;DR

This paper characterizes the singularity of the $n$-th eigenvalue in self-adjoint discrete Sturm-Liouville problems across dimensions, revealing jump phenomena and providing methods to analyze and partition the parameter space.

Contribution

It offers a complete characterization of eigenvalue singularities, including jump phenomena, and extends analysis methods to any dimension, applying them to Atkinson type problems.

Findings

Eigenvalue singularities depend heavily on coefficients and boundary conditions.

The paper introduces a Hermitian matrix to determine parameter space divisions.

It generalizes asymptotic analysis methods to higher dimensions and applies them to Atkinson type problems.

Abstract

In this paper, we characterize singularity of the $n$-th eigenvalue of self-adjoint discrete Sturm-Liouville problems in any dimension. For a fixed Sturm-Liouville equation, we completely characterize singularity of the $n$-th eigenvalue. For a fixed boundary condition, unlike in the continuous case, the $n$-th eigenvalue exhibits jump phenomena and we describe the singularity under a non-degenerate assumption. Compared with the continuous case in [8, 12], the singular set here is involved heavily with coefficients of the Sturm-Liouville equations. This, along with arbitrariness of the dimension, causes difficulty when dividing areas in layers of the considered space such that the $n$-th eigenvalue has the same singularity in any given area. We study the singularity by partitioning and analyzing the local coordinate systems, and provide a Hermitian matrix which can determine the areas' division. To prove the asymptotic behavior of the $n$-th eigenvalue, we generalize the method developed in [21] to any dimension. Finally, by transforming the Sturm-Liouville problem of Atkinson type in any dimension to a discrete one, we can not only determine the number of eigenvalues, but also apply our approach above to obtain the complete characterization of singularity of the $n$-th eigenvalue for the Atkinson type.

Equations

$$-m_i\frac{d^2}{d{t}^2}{s}_i=c_{i-1}(s_i-s_{i-1})+c_i(s_i-s_{i+1})=-\nabla (c_i\Delta s_i),1\le i\le l,$$

$$-\nabla (c_i\Delta y_i)=\lambda m_i{y}_i,1\le i\le l,$$

$$p_{rs}(j+1)={\alpha }_{s-1}{p}_{r,s-1}(j)+{\beta }_{s+1}{p}_{r,s+1}(j)+(1-{\alpha }_s-{\beta }_s)p_{rs}(j),j\ge 0,$$

$$\displaystyle T=\left(\begin{array}[]{cccccccc}-\alpha_{1}-\beta_{1}&\alpha_{1}&&&\\ \beta_{2}&-\alpha_{2}-\beta_{2}&\alpha_{2}&&\\ &\beta_{3}&-\alpha_{3}-\beta_{3}&\ddots&\\ &&\ddots&\ddots&\alpha_{l-1}\\ &&&\beta_{l}&-\alpha_{l}-\beta_{l}\end{array}\right).$$

$$P(j+1)=P(j)(I_l+T),j\ge 0.$$

$$-\nabla (g_j\Delta y_j)=\lambda a_j{y}_j,1\le j\le l,$$

$$-\nabla (P_i\Delta y_i)+Q_i{y}_i=\lambda W_i{y}_i,1\le i\le N,$$

$$A\left(\begin{array}[]{c}-y_{0}\\ y_{N}\\ \end{array}\right)+B\left(\begin{array}[]{c}P_{0}\Delta y_{0}\\ P_{N}\Delta y_{N}\\ \end{array}\right)=0,$$

$$P_j,Q_i,W_i\mathrm{are}\mathrm{Hermitian},P_j\mathrm{is}\mathrm{invertible},W_i\mathrm{is}\mathrm{positive}\mathrm{definite},$$

$$\text{rank}(A,B)=2d,A{B}^{\ast }=B{A}^{\ast }.$$

$${\Omega }_N^{\mathbb{C}}:=\{\mathbf{\omega }=\left(\{P_j^{-1}{\}}_{j=0}^N,\{Q_i{\}}_{i=1}^N,\{W_i{\}}_{i=1}^N\right)\in {\left({\mathcal{M}}_d(\mathbb{C})\right)}^{3N+1}:(\text{ref}3)\mathrm{holds}\}$$

$${\mathcal{B}}^{\mathbb{C}}:=\text{lower}3.0pt\mathrm{GL}(2d,\mathbb{C})\backslash \text{raise}2.0pt{\mathcal{L}}_{2d,4d}(\mathbb{C}),$$

$${\mathcal{L}}_{2d,4d}(\mathbb{C}):=\{(A,B)\in {\mathcal{M}}_{2d,4d}(\mathbb{C}):\mathrm{rank}(A,B)=2d,A{B}^{\ast }=B{A}^{\ast }\}$$

$$\mathrm{GL}(2d,\mathbb{C}):=\{T\in {\mathcal{M}}_{2d}(\mathbb{C}):\det T\ne 0\}.$$

$$K_1=K\cap \{1,2,\cdots ,d\},K_2=K\cap \{d+1,d+2,\cdots ,2d\}.$$

$$E_{K}=\begin{pmatrix}E_{K,1}\\ E_{K,2}\end{pmatrix}=\left(\begin{array}[]{cccccccc}E_{1}&0&I_{d}-E_{1}&0\\ 0&E_{2}&0&I_{d}-E_{2}\\ E_{1}-I_{d}&0&E_{1}&0\\ 0&E_{2}-I_{d}&0&E_{2}\\ \end{array}\right),$$

$$\displaystyle\alpha_{i}=\left\{\begin{array}[]{cc}0&{\rm if}\;i\in K_{1},\\ e_{i}&{\rm if}\;i\notin K_{1},\end{array}\right.\quad\beta_{i}=\left\{\begin{array}[]{cc}0&{\rm if}\;d+i\in K_{2},\\ e_{i}&{\rm if}\;d+i\notin K_{2},\end{array}\right.$$

$$E_K^{\ast }{J}_{2d}{E}_K=J_{2d},E_K{E}_K^{\ast }=I_{4d},$$

$$\displaystyle{J}_{2d}=\left(\begin{array}[]{cc}0&-I_{2d}\\ I_{2d}&0\\ \end{array}\right).$$

$${\mathcal{O}}_K^{\mathbb{C}}:=\{[(S|I_{2d})E_K]:S\in {\mathcal{H}}_{2d}(\mathbb{C})\}.$$

$${\mathcal{B}}^{\mathbb{C}}=\bigcup _{K\subset \{1,2,\cdots ,2d\}}{\mathcal{O}}_K^{\mathbb{C}}.$$

$$l[0,N+1]:=\left\{y=\{y_i{\}}_{i=0}^{N+1}:y_i\in {\mathbb{C}}^d,0\le i\le N+1\right\}.$$

$$P_i\Delta y_i=P_{i-1}\Delta y_{i-1}-(\lambda W_i-Q_i)y_i,1\le i\le N.$$

$$\left(\begin{array}[]{ccc}\phi^{1}_{0}(\lambda)&\cdots&\phi^{2d}_{0}(\lambda)\\ P_{0}\Delta\phi^{1}_{0}(\lambda)&\cdots&P_{0}\Delta\phi^{2d}_{0}(\lambda)\\ \end{array}\right)=I_{2d}.$$

$$\Phi(\lambda):=\left(\begin{array}[]{ccc}-\phi^{1}_{0}(\lambda)&\cdots&-\phi^{2d}_{0}(\lambda)\\ \phi^{1}_{N}(\lambda)&\cdots&\phi^{2d}_{N}(\lambda)\\ \end{array}\right),\quad\Psi(\lambda):=\left(\begin{array}[]{ccc}P_{0}\Delta\phi^{1}_{0}(\lambda)&\cdots&P_{0}\Delta\phi^{2d}_{0}(\lambda)\\ P_{N}\Delta\phi^{1}_{N}(\lambda)&\cdots&P_{N}\Delta\phi^{2d}_{N}(\lambda)\\ \end{array}\right).$$

$${\Gamma }_{(\mathbf{\omega },\mathbf{A})}(\lambda ):=\det (A\Phi (\lambda )+B\Psi (\lambda )).$$

$$J_d\Delta \left(\begin{array}{c}{y}_i\\ x_i\end{array}\right)=\left(\left(\begin{array}{cc}-Q_{i+1}& 0\\ 0& P_i^{-1}\end{array}\right)+\lambda \left(\begin{array}{cc}{W}_{i+1}& 0\\ 0& 0\end{array}\right)\right)R\left(\begin{array}{c}{y}_i\\ x_i\end{array}\right),0\le i\le N-1,$$

$${♯}_1(\sigma (\mathbf{\omega },\mathbf{A}))=(N-2)d+\mathrm{rank}(A_1{P}_0^{-1}+B_1,B_2),$$

$$\mathbf{A}=[A|B]=[(A_1,A_2)|(B_1,B_2)].$$

$${♯}_1(\sigma (\mathbf{\omega },\mathbf{A}))=(N-2)d+\mathrm{rank}(A_1+B_1{P}_0,B_2).$$

Full text

Keywords: Discrete Sturm-Liouville problem; the $n$-th eigenvalue; singularity; boundary condition; Atkinson type.

2010 Mathematics Subject Classification: 39A12, 34B24, 39A70.

1. Introduction

Discrete Sturm-Liouville problems come from several physical models, including the vibrating string and random walk with discrete time process [4, 9]. We briefly introduce these two models. Suppose that a weightless string bears $l$ particles with masses $m_{1},\cdots,m_{l}$, and the horizontal distance between $m_{i}$ and $m_{i+1}$ is $1/c_{i}$, $1\leq i\leq l-1$. Moreover, the string extends to length $1/c_{l}$ beyond $m_{l}$ and $1/c_{0}$ beyond $m_{1}$. Let $s_{i}$, $1\leq i\leq l$, be the displacement of the particle $m_{i}$ at a fixed time. Both ends are pinned down (i.e., $s_{0}=s_{l+1}=0$). Since the particle $m_{i}$ does not move horizontally, we may assume that the horizontal component of the tension at $m_{i}$ is both unit from the left and right, respectively. Then the restoring forces, induced by the vertical component of the tension from the left and right, are $c_{i-1}(s_{i}-s_{i-1})$ and $c_{i}(s_{i}-s_{i+1})$, respectively.

m_{i-1}$$m_{i}$$m_{i+1}$$1/c_{i-1}$$1/c_{i}$$s_{i-1}$$s_{i}$$s_{i+1}

**Figure 1. **

Therefore, by Newton’s second law,

[TABLE]

where $\Delta s_{i}=s_{i+1}-s_{i}$ and $\nabla s_{i}=s_{i}-s_{i-1}$. Taking $s_{i}=y_{i}\cos(\omega t),$ where $y_{i}$ is the amplitude of $m_{i}$, we obtain from (1.1) that

[TABLE]

where $\lambda=\omega^{2}$. Since the boundary condition corresponds to the assumption that both ends are pinned down, this system becomes a self-adjoint discrete Sturm-Liouville problem.

Another model of the discrete Sturm-Liouville equation is random walking with discrete time process from probability theory.

\longleftarrow$$\beta_{1}$$\longrightarrow$$\alpha_{1}$$\longleftarrow$$\beta_{2}$$\longrightarrow$$\alpha_{2}$$\cdots\cdots$$\longleftarrow$$\beta_{i}$$\longrightarrow$$\alpha_{i}$$\longrightarrow$$\alpha_{l}$$\longleftarrow$$\beta_{l}$$\longrightarrow$$\alpha_{l-1}$$\longleftarrow$$\beta_{l-1}$$\cdots\cdots$$1$$2$$i$$l-1$$l

**Figure 2. **

Let a particle be in one of the $l$ positions $1,\cdots,l$ at $t=0$. Suppose that the particle is in position $i$ at $t=t_{0}$. The rule of this random walking is that the particle will move to $i+1$ at $t=t_{0}+1$ with a probability $\alpha_{i}$, move to $i-1$ at $t=t_{0}+1$ with a probability $\beta_{i}$, and stay in position $i$ with a probability $1-\alpha_{i}-\beta_{i}$. Moreover, if the particle moves to the left of position $1$, or to the right of position $l$, it is considered permanently lost. So it is reasonable to set $\alpha_{0}=0$ and $\beta_{l+1}=0$. Define $p_{rs}(j)$ as the probability of the particle being in position $s$ at $t=j$ and starting in position $r$ at $t=0$. Then we have $p_{rs}(0)=\delta_{rs}$, and

[TABLE]

where $\delta_{rs}=1$ if $r=s$, and $\delta_{rs}=0$ if $r\neq s$. Let $P(j)=(p_{rs}(j))_{1\leq r,s\leq l}$, $j\geq 0$, and

[TABLE]

Then $P(0)=I_{l}$ and (1.2) is equivalent to

[TABLE]

So $P(j+1)=(I_{l}+T)^{j}$, $j\geq 0$. However, the form $(I_{l}+T)^{j}$ provides little information on asymptotic form of $P(j)$ for large $j$. Instead, in the spectral theory, the eigenvalues and corresponding eigenfunctions of $T$ play an important role in studying properties of $P(j)$ for any $j\geq 1$. To find an eigenvalue $\lambda$ and the corresponding eigenfunction $(y_{1},y_{2},\cdots,y_{l})$ of $-T$, we need to study the self-adjoint discrete Sturm-Liouville equation

[TABLE]

with the boundary condition $y_{0}=y_{l+1}=0$, where $g_{j}=\alpha_{j}a_{j}$ and $g_{j-1}=\beta_{j}a_{j}$.

Motivated by these two interesting models and recent interest on discrete equations [5, 6, 9], in this paper we consider a general self-adjoint discrete $d$-dimensional Sturm-Liouville problem for any $d\geq 1$. It consists of a symmetric discrete Sturm-Liouville equation

[TABLE]

and a self-adjoint boundary condition

[TABLE]

where $P=\{P_{j}\}_{j=0}^{N}$, $Q=\{Q_{i}\}_{i=1}^{N}$ and $W=\{W_{i}\}_{i=1}^{N}$ are sequences of $d\times d$ complex-valued matrices and satisfy

[TABLE]

$0\leq j\leq N$, $\lambda\in\mathbb{C}$ is the spectral parameter, $N\geq 2$, $A$ and $B$ are $2d\times 2d$ complex-valued matrices such that

[TABLE]

The spectrum of a self-adjoint discrete Sturm-Liouville problem consists of real and finite eigenvalues, and thus can be arranged in the non-decreasing order. The $n$-th eigenvalue can be considered as a function defined on the space of self-adjoint discrete Sturm-Liouville problems or on its subset. This function is not continuous in general, see the $1$-dimensional case in [21]. Here we call the set of all discontinuity points in the considered space to be the singular set, and call any element in the singular set to be a singular point. The $n$-th eigenvalue exhibits jump phenomena near the singular points. Unlike only jumping to $-\infty$ in the continuous case, the $n$-th eigenvalue also blows up to $+\infty$ in the discrete case.

The aim of this paper is to determine the singular set and to completely provide the asymptotic behavior of the $n$-th eigenvalue near any fixed singular point for the discrete Sturm-Liouville problems. As applications, we consider the Sturm-Liouville problem of Atkinson type, transform it into a discrete Sturm-Liouville problem, and then apply the discrete method to completely characterize singularity of the $n$-th eigenvalue for the Atkinson type. Though the $n$-th eigenvalue jumps to $\pm\infty$ near the singular points in the Atkinson type as well as in the discrete case, the singular set in the Atkinson type is the same one as in the continuous case and is independent of coefficients of the Sturm-Liouville equations. This leads tremendous difference with the discrete case, where the singular set is involved heavily with coefficients of the equations.

Singularity of the $n$-th eigenvalue of Sturm-Liouville problems has attracted a lot of attention (see [7, 8, 10, 12, 15, 19, 21] and their references) since Rellich [17]. Let us mention three contributions to finding the singular set of the $n$-th eigenvalue and providing all the asymptotic behavior near each singular point. Kong, Wu, and Zettl completely characterized it for the continuous $1$-dimensional Sturm-Liouville problems, while Hu et al. gave the answer for the continuous $d$-dimensional case, where $d\geq 2$. Zhu and Shi obtained the desired result for the discrete $1$-dimensional case. This paper is devoted to the discrete case in any dimension. We mention here that our result in Theorem 4.4 for singularity of the $n$-th eigenvalue on the boundary conditions is complete, while the conclusion in Theorem 4.9 for singularity on the equations is partial due to the non-degenerate assumption (4.30)–(4.31).

Compared with the continuous Sturm-Liouville problems, the $n$-th eigenvalue in the discrete case is not continuously dependent on the equations, and the criterion for continuity of the $n$-th eigenvalue is different due to the finiteness of the number of eigenvalues. This makes the method used in the continuous case [8, 12] unable to apply to the discrete case. On the other hand, compared with the $1$-dimensional discrete case, the first difficulty for any dimensional case is how to divide areas in layers of the considered space such that the $n$-th eigenvalue has the same singularity in any given area. Our method in this paper is to find some invertible elementary transformations converting the matrix, which determines the number of eigenvalues of the Sturm-Liouville problems, to a Hermitian matrix. The areas’ division is then determined by the spectral information of this Hermitian matrix. The second difficulty is how to prove the asymptotic behavior of the $n$-th eigenvalue. Our approach is first to prove the asymptotic behavior in a certain direction using the monotonicity of continuous eigenvalue branches, and then combine the local topological property (geometric structure) of the considered space with the perturbation theory of eigenvalues to obtain the whole asymptotic behavior. This can be regarded as a generalization of the method developed for $1$-dimensional discrete case in [21] to any dimension. Finally, though our method for the Atkinson type is by transforming the Sturm-Liouville problem into a discrete one, it turns out to be no singularity of the $n$-th eigenvalue on the equations for the Atkinson type.

The rest of this paper is organized as follows. In Section 2, topology on the space of Sturm-Liouville equations, and that on the space of boundary conditions are presented. Properties of eigenvalues are given in Section 3. The number and multiplicity of eigenvalues are discussed in Subsection 3.1, continuous eigenvalue branches are constructed and their properties are provided in Subsection 3.2, and properties of the $n$-th eigenvalue are presented in Subsection 3.3. In Section 4, singularity of the $n$-th eigenvalue on the boundary conditions is completely characterized for a fixed equation in Subsection 4.1, while singularity of the $n$-th eigenvalue on the equations is obtained for a fixed boundary condition under a non-degenerate assumption in Subsection 4.2. Sturm-Liouville problem of the Atkinson type is transformed to a discrete one, and singularity of the $n$-th eigenvalue is provided thoroughly in Section 5. Conclusions are given in Section 6.

Notation.

By $\mathbb{R}$ and $\mathbb{C}$ denote the set of all the real and complex numbers, respectively. The set of all $m\times n$ matrices over a field $\mathbb{F}$ is denoted by $\mathcal{M}_{m,n}(\mathbb{F})$, and $\mathcal{M}_{n,n}(\mathbb{F})$ is abbreviated to $\mathcal{M}_{n}(\mathbb{F})$. $A^{*}$ is the complex conjugate transpose of $A\in\mathcal{M}_{m,n}(\mathbb{F})$, while $A^{T}$ is the transpose of $A$. $\mathcal{H}_{n}(\mathbb{F})$ is the set of all $n\times n$ Hermitian matrices, while $\mathcal{P}_{n}(\mathbb{F})$ is the set of all $n\times n$ positive definite matrices over a field $\mathbb{F}$. For a matrix $S\in\mathcal{M}_{n}(\mathbb{F})$, its entries and columns are denoted by $s_{ij}$ and $s_{j}=(s_{1j},\cdots,s_{nj})^{T}$, respectively, $1\leq i,j\leq n$. By $I_{n}$ denote the $n\times n$ unit matrix. $\sharp(K)$ is the cardinality of the set $K$. By $r^{-}(A)$, $r^{0}(A)$, and $r^{+}(A)$ denote the total multiplicity of negative, zero, and positive eigenvalues of $A\in\mathcal{H}_{n}(\mathbb{C})$, respectively.

2. Space of self-adjoint discrete Sturm-Liouville problems

In this section, we introduce the topology on the space of self-adjoint discrete Sturm-Liouville problems.

The space of discrete Sturm-Liouville equations is

[TABLE]

with the topology induced by $\mathbb{C}^{(3N+1)d^{2}}$.

Note that the space of self-adjoint boundary conditions is the same as the continuous case. Following [8], it is exactly the quotient space

[TABLE]

where

[TABLE]

and

[TABLE]

The boundary condition in $\mathcal{B}^{\mathbb{C}}$ is denoted by $[A\,|\,B]:=\{(TA\;|\;TB):T\in{\rm GL}(2d,\mathbb{C})\}$. Bold faced capital Latin letters, such as $\mathbf{A}$, are also used for boundary conditions.

Next we introduce the following form for the local coordinate systems on $\mathcal{B}^{\mathbb{C}}$. Let $K$ be any subset of $\{1,2,\cdots,2d\}$. Denote

[TABLE]

By $E_{K}$ denote the $4d\times 4d$ matrix generated from $I_{4d}$ by multiplying $-1$ to the $(k+2d)$-th column and then exchanging the $k$-th and the $(k+2d)$-th columns for each $k\in K$. Then it has the following form:

[TABLE]

where $E_{K,1},E_{K,2}\in\mathcal{M}_{2d,4d}(\mathbb{C})$, $E_{1}=\{\alpha_{1},\alpha_{2},\cdots,\alpha_{d}\}$ and $E_{2}=\{\beta_{1},\beta_{2},\cdots,\beta_{d}\}$ are $d\times d$ diagonal matrices with

[TABLE]

and $e_{i}$ is the $i$-th column of $I_{d}$. Then

[TABLE]

where

[TABLE]

We define

[TABLE]

For $\mathbf{A}=[(S\,|\,I_{2d})E_{K}]\in\mathcal{O}_{K}^{\mathbb{C}}$, we denote $S$ by $S(\mathbf{A})$ to indicate its dependence on $\mathbf{A}$ if necessary. It is clear that $\mathcal{O}_{K}^{\mathbb{C}}$ defined here coincides with that defined in (2.1) of [8]. It follows from Theorem 2.1 in [8] that

[TABLE]

Moreover, $\mathcal{B}^{\mathbb{C}}$ is a connected and compact real-analytic manifold of dimension $4d^{2}$. The readers are also referred to [2, 3, 13, 16] for more details.

The product space $\Omega_{N}^{\mathbb{C}}\times\mathcal{B}^{\mathbb{C}}$ is the space of self-adjoint discrete Sturm-Liouville problems, and $(\boldsymbol{\omega},\mathbf{A})$ is used to stand for an element in $\Omega_{N}^{\mathbb{C}}\times\mathcal{B}^{\mathbb{C}}$ in the sequel.

3. Properties of eigenvalues

In this section, we study properties of eigenvalues of the self-adjoint discrete Sturm-Liouville problems.

3.1. The number and multiplicity of eigenvalues.

Let

[TABLE]

The initial value problem of (1.3) has a unique solutions. More precisely,

Lemma 3.1**.**

Let $z_{i_{0}},\tilde{z}_{i_{0}}\in\mathbb{C}^{d}$ for some $1\leq i_{0}\leq N$. Then, for each $\lambda\in\mathbb{C}$, (1.3) has a unique solution $y(\lambda)\in l[0,N+1]$ satisfying $y_{i_{0}}(\lambda)=z_{i_{0}}$, $P_{i_{0}}\Delta y_{i_{0}}(\lambda)=\tilde{z}_{i_{0}}$.

Proof.

This can be deduced by the invertibility of $P_{j}$ for $0\leq j\leq N$ and the iteration of

[TABLE]

∎

Recall that $\lambda$ is called an eigenvalue of the discrete Sturm-Liouville problem $(\boldsymbol{\omega},\mathbf{A})$ if there exists $y\in l[0,N+1]$ which is non-trivial and solves (1.3)–(1.4). Here $y$ is called an eigenfunction corresponding to $\lambda$, and it is said to be normalized if $\sum_{i=1}^{N}y^{*}_{i}W_{i}y_{i}=1.$ By $\sigma(\boldsymbol{\omega},\mathbf{A})$ denote the spectral set of $(\boldsymbol{\omega},\mathbf{A})$. For any $\lambda\in\mathbb{C}$, let $\phi^{j}(\lambda)=\{\phi_{i}^{j}(\lambda)\}_{i=0}^{N+1}$, $j=1,\cdots,2d$, be the fundamental solutions to (1.3) determined by the initial data

[TABLE]

Denote

[TABLE]

Then the eigenvalues of $(\boldsymbol{\omega},\mathbf{A})$ can be regarded as zeros of the polynomial $\Gamma_{(\boldsymbol{\omega},\mathbf{A})}$ as follows.

Lemma 3.2**.**

$\lambda\in\sigma(\boldsymbol{\omega},\mathbf{A})$* if and only of $\lambda$ is a zero of*

[TABLE]

Proof.

The proof is similar as that of Lemma 3.2 in [22]. ∎

Let $\lambda\in\sigma(\boldsymbol{\omega},\mathbf{A})$. The order of $\lambda$ as a zero of $\Gamma_{(\boldsymbol{\omega},\mathbf{A})}$ is called its analytic multiplicity. The number of linearly independent eigenfunctions for $\lambda$ is called its geometric multiplicity. Let $x_{i}=P_{i}\Delta y_{i}$ for $0\leq i\leq N$. Then the Sturm-Liouville equation (1.3) can be transformed to a discrete linear Hamiltonian system:

[TABLE]

where $R(y^{T}_{i},x^{T}_{i})^{T}=(y^{T}_{i+1},x^{T}_{i})^{T}$ is the partial right shift operator. Then by Theorem 4.1 in [20], we get the relationship of analytic and geometric multiplicities of $\lambda$:

Lemma 3.3**.**

The analytic and geometric multiplicities of $\lambda\in\sigma(\boldsymbol{\omega},\mathbf{A})$ are the same.

Therefore, we do not distinguish these two multiplicities of $\lambda$. Let $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})\cap I)$ be the number of eigenvalues in $I\subset\mathbb{R}$, counting multiplicities, of $(\boldsymbol{\omega},\mathbf{A})$. Since $\sigma(\boldsymbol{\omega},\mathbf{A})\subset\mathbb{R}$ by [18], we have $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})\cap\mathbb{R})=\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))$. The next lemma determines $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))$.

Lemma 3.4**.**

[TABLE]

where $A_{1},B_{j}\in\mathcal{M}_{2d\times d}$ $(j=1,2)$ are given by

[TABLE]

Proof.

By Theorem 4.1 in [18],

[TABLE]

Then (3.1) is obtained by $(A_{1}+B_{1}P_{0},B_{2})\begin{pmatrix}P_{0}^{-1}&0\\ 0&I_{d}\end{pmatrix}=(A_{1}P_{0}^{-1}+B_{1},B_{2})$. ∎

Note that $(N-2)d\leq\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))\leq Nd$.

3.2. Continuous Eigenvalue Branch

In this subsection, we construct continuous eigenvalue branches. Then we study their derivative formulae and monotonicity in some directions.

The first lemma is the small perturbation theory of eigenvalues.

Lemma 3.5**.**

Let $(\boldsymbol{\omega}_{0},\mathbf{A}_{0})\in\mathcal{O}\subset\Omega_{N}^{\mathbb{C}}\times\mathcal{B}^{\mathbb{C}}$, and $c_{1},c_{2}\in\mathbb{R}\setminus\sigma(\boldsymbol{\omega}_{0},\mathbf{A}_{0})$ with $c_{1}<c_{2}.$ Then there exists a neighborhood $\mathcal{U}\subset\mathcal{O}$ of $(\boldsymbol{\omega}_{0},\mathbf{A}_{0})$ such that for each $(\boldsymbol{\omega},\mathbf{A})\in\mathcal{U}$, $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})\cap(c_{1},c_{2}))=\sharp_{1}(\sigma(\boldsymbol{\omega}_{0},\mathbf{A}_{0})\cap(c_{1},c_{2}))$ and $c_{1},c_{2}\notin\sigma(\boldsymbol{\omega},\mathbf{A})$.

Proof.

Using Lemma 3.2, the proof is by a standard perturbation procedure for zeros of the analytic function $\Gamma_{(\boldsymbol{\omega}_{0},\mathbf{A}_{0})}$. ∎

By Lemma 3.5 and a similar approach to Theorem 3.5 in [22], we then construct the continuous eigenvalue branches.

Lemma 3.6**.**

Let $(\boldsymbol{\omega}_{0},\mathbf{A}_{0})\in\Omega_{N}^{\mathbb{C}}\times\mathcal{B}^{\mathbb{C}}$ and $\lambda_{*}\in\sigma(\boldsymbol{\omega}_{0},\mathbf{A}_{0})$ with multiplicity $m$. Fix a small $\varepsilon>0$ such that $\sigma(\boldsymbol{\omega}_{0},\mathbf{A}_{0})\cap[\lambda_{*}-\varepsilon,\lambda_{*}+\varepsilon]=\{\lambda_{*}\}$. Then there is a connected neighborhood $\mathcal{U}$ of $(\boldsymbol{\omega}_{0},\mathbf{A}_{0})$ and continuous functions $\Lambda_{i}:\mathcal{U}\to\mathbb{R}$, $1\leq i\leq m$, such that $\lambda_{*}-\varepsilon<\Lambda_{1}(\boldsymbol{\omega},\mathbf{A})\leq\cdots\leq\Lambda_{m}(\boldsymbol{\omega},\mathbf{A})<\lambda_{*}+\varepsilon$ and $\lambda_{*}\pm\varepsilon\notin\sigma(\boldsymbol{\omega},\mathbf{A})$ for all $(\boldsymbol{\omega},\mathbf{A})\in\mathcal{U}$, where $\{\Lambda_{i}(\boldsymbol{\omega}_{,}\mathbf{A})\}_{i=1}^{m}\subset\sigma(\boldsymbol{\omega},\mathbf{A})$.

Here $\Lambda_{i}:\mathcal{U}\to\mathbb{R}$, $1\leq i\leq m$, are called the continuous eigenvalue branches through $\lambda_{*}$. We write $\Lambda_{i}(\boldsymbol{\omega})$ when $\mathbf{A}$ is fixed, and write $\Lambda_{i}(\mathbf{A})$ when $\boldsymbol{\omega}$ is fixed. Then we shall make a continuous choice of eigenfunctions for the eigenvalues along a continuous simple eigenvalue branch $(m=1)$.

Lemma 3.7**.**

Let $u_{0}$ be an eigenfunction for a simple eigenvalue $\lambda_{*}\in\sigma(\boldsymbol{\omega}_{0},\mathbf{A}_{0})$, and $\Lambda$ be the continuous eigenvalue branch defined on $\mathcal{U}$ through $\lambda_{*}$. Then there exists a neighborhood $\mathcal{U}_{1}\subset\mathcal{U}$ of $(\boldsymbol{\omega}_{0},\mathbf{A}_{0})$ such that for any $(\boldsymbol{\omega},\mathbf{A})\in\mathcal{U}_{1}$, there is an eigenfunction $u_{\Lambda(\boldsymbol{\omega},\mathbf{A})}$ for $\Lambda(\boldsymbol{\omega},\mathbf{A})$ satisfying that $u_{\Lambda(\boldsymbol{\omega},\mathbf{A})}=u_{0}$, and $u_{\Lambda(\boldsymbol{\omega},\mathbf{A})}\to u_{\Lambda(\boldsymbol{\omega}_{0},\mathbf{A}_{0})}$ in $\mathbb{C}^{(N+2)d}$ as $\mathcal{{U}}_{1}\ni(\boldsymbol{\omega},\mathbf{A})\to(\boldsymbol{\omega}_{0},\mathbf{A}_{0})$.

Proof.

The proof is similar to that of Lemma 4.3 in [22], and thus we omit the details. ∎

Besides Lemma 3.7, we also need the following lemma to deduce the derivative formulae for continuous simple eigenvalue branches.

Lemma 3.8**.**

Let $y$ be an eigenfunction for $\lambda\in\sigma(\boldsymbol{\omega},\mathbf{A})$ and $z$ be an eigenfunction for $\tilde{\lambda}\in\sigma({\boldsymbol{\tilde{\omega}}},\mathbf{A})$, where $\boldsymbol{\omega}=(P^{-1},Q,W),{\boldsymbol{\tilde{\omega}}}=(\tilde{P}^{-1},\tilde{Q},\tilde{W})$ and $\mathbf{A}=[A\,|\,B]$. Then

[TABLE]

Proof.

For convenience, denote

[TABLE]

where $\alpha_{i}\in\mathcal{M}_{1,4d}(\mathbb{C})$, $i=1,\cdots,2d$. Then

[TABLE]

Since ${\rm rank}(A,\,B)=2d$, the first equation in (3.5) yields that each solution of the equation $(A,\,B)X=0$ is a linear combination of ${J}_{2d}\alpha_{i}^{*}$, $1\leq i\leq 2d$. From the last two equations in (3.5), we know that there exists $c_{i},d_{i}\in\mathbb{C}$, $1\leq i\leq 2d$, such that $Y=\sum_{i=1}^{2d}c_{i}J_{2d}\alpha_{i}^{*}$ and $Z=\sum_{i=1}^{2d}d_{i}J_{2d}\alpha_{i}^{*}$. The first equation in (3.5) also implies that

[TABLE]

So

[TABLE]

which is equivalent to (3.3). The proof is complete. ∎

Note that the method used in Lemma 4.4 of [21] depends on separated and coupled boundary conditions, and thus can not be applied to Lemma 3.8 here for mixing boundary conditions when $d\geq 2$. With the help of Lemma 3.8, we give the derivative formulae of the continuous simple eigenvalue branch with respect to coefficients of the Sturm-Liouville equations.

Lemma 3.9**.**

Fix $\mathbf{A}\in\mathcal{B}^{\mathbb{C}}$. Let ${\boldsymbol{\omega}}=(P^{-1},Q,W)\in\Omega_{N}^{\mathbb{C}}$, $\lambda_{*}$ be a simple eigenvalue of $(\boldsymbol{\omega},\mathbf{A})$, $y\in l[0,N+1]$ be a normalized eigenfunction for $\lambda_{*}$, and $\Lambda$ be the continuous simple eigenvalue branch over $\Omega_{N}^{\mathbb{C}}$ through $\lambda_{*}$. Then

[TABLE]

for all $(H,K,L)=((H_{0},\cdots,H_{N}),(K_{1},\cdots,K_{N}),(L_{1},\cdots,L_{N}))\in(\mathcal{H}_{d}(\mathbb{C}))^{3N+1}$ and $(P^{-1}+H,Q+K,W+L)\in\Omega_{N}^{\mathbb{C}}$.

Proof.

Let $\boldsymbol{\sigma}\in\Omega_{N}^{\mathbb{C}}$ with $\boldsymbol{\sigma}=({P}^{-1}+H,{Q}+K,{W}+L)=:(\tilde{P}^{-1},\tilde{Q},\tilde{W})$. By Lemma 3.7, we can choose an eigenfunction $z=z(\cdot,\boldsymbol{\sigma})$ for $\Lambda=\Lambda(\boldsymbol{\sigma})$ with $\boldsymbol{\sigma}$ sufficiently close to $\boldsymbol{\omega}$ in $\Omega_{N}^{\mathbb{C}}$ such that $z\to y$ as $\boldsymbol{\sigma}\to\boldsymbol{\omega}$. Then it follows from (1.3) that

[TABLE]

By Lemma 3.8, we get

[TABLE]

which yields that (3.6) holds. This completes the proof. ∎

Let us fix all the components of $\boldsymbol{\omega}$ except $P_{j}^{-1}$, and write the perturbed term $P_{j}^{-1}$ by $P_{j}^{-1}(\boldsymbol{\omega})$ to indicate its dependence on $\boldsymbol{\omega}$ for a given $0\leq j\leq N-1$. $Q_{i}(\boldsymbol{\omega})$ has the similar meaning for $1\leq i\leq N$. The we get the following monotonicity result.

Corollary 3.10**.**

Fix $\mathbf{A}\in\mathcal{B}^{\mathbb{C}}$. Let $\Lambda$ be a continuous eigenvalue branch defined on $\mathcal{V}\subset\Omega_{N}^{\mathbb{C}}$. If $P_{j}^{-1}(\boldsymbol{\sigma})-P_{j}^{-1}(\boldsymbol{\omega})$ is positive semi-definite for a given $0\leq j\leq N-1$, then $\Lambda(\boldsymbol{\sigma})\leq\Lambda(\boldsymbol{\omega})$. If $Q_{i}(\boldsymbol{\sigma})-Q_{i}(\boldsymbol{\omega})$ is positive semi-definite for a given $1\leq i\leq N$, then $\Lambda(\boldsymbol{\omega})\leq\Lambda(\boldsymbol{\sigma})$.

Then we give the derivative formula of a continuous simple eigenvalue branch with respect to boundary conditions.

Lemma 3.11**.**

Fix $\boldsymbol{\omega}\in\Omega_{N}^{\mathbb{C}}$. Let $\lambda_{*}$ be a simple eigenvalue of $(\boldsymbol{\omega},\mathbf{A})$ for some $\mathbf{A}\in\mathcal{O}_{K}^{\mathbb{C}}$, $y\in l[0,N+1]$ be a normalized eigenfunction for $\lambda_{*}$, and $\Lambda$ be the continuous simple eigenvalue branch through $\lambda_{*}$. Then

[TABLE]

for $H\in\mathcal{H}_{2d}(\mathbb{C})$, where $E_{K,1}$ and $Y$ are given in (2.3) and (3.4), respectively.

Proof.

By (2.10), there exists $S\in\mathcal{H}_{2d}(\mathbb{C})$ such that $\mathbf{A}=[S\;|\;I_{2d}]E_{K}$. Let $\mathbf{B}=[S+H\;|\;I_{2d}]E_{K}$ with $H\in\mathcal{H}_{2d}$. Then there exists an eigenfunction $\tilde{y}=y_{\Lambda(\mathbf{B})}$ for $\Lambda(\mathbf{B})$ such that $\tilde{y}\to y$ in $\mathbb{C}^{(N+2)d}$ as $\mathbf{B}\to\mathbf{A}$. $\tilde{Y}$ has the similar meaning as $Y$. Note that $\tilde{y}$ and ${y}$ satisfy

[TABLE]

and thus

[TABLE]

where $[y_{i},\tilde{y}_{i}]=(\Delta\tilde{y}_{i})^{*}P_{i}y_{i}-\tilde{y}_{i}^{*}P_{i}\Delta y_{i}$. From the boundary conditions $\mathbf{A}$ and $\mathbf{B}$, we have

[TABLE]

It then follows from (2.9) and (3.7) that

[TABLE]

This completes the proof. ∎

The following result is a direct consequence of Lemma 3.11.

Corollary 3.12**.**

Fix $\boldsymbol{\omega}\in\Omega_{N}^{\mathbb{C}}$. Let $\Lambda$ be a continuous eigenvalue branch defined on $\mathcal{U}\subset\mathcal{O}_{K}^{\mathbb{C}}$. Then $\Lambda(\mathbf{A})\leq\Lambda(\mathbf{B})$ if $\mathbf{A},\mathbf{B}\in\mathcal{U}$ and $S(\mathbf{B})-S(\mathbf{A})$ is positive semi-definite.

3.3. Properties of the $n$-th eigenvalue.

Based on Lemma 3.4, the eigenvalues of $(\boldsymbol{\omega},\mathbf{A})\in\Omega_{N}^{\mathbb{C}}\times\mathcal{B}^{\mathbb{C}}$ can be arranged in the following non-decreasing order:

[TABLE]

Therefore, for any $1\leq n\leq Nd$, the $n$-th eigenvalue can be regarded as a function defined on $\Omega_{N}^{\mathbb{C}}\times\mathcal{B}^{\mathbb{C}}$ or on its subset, called the $n$-th eigenvalue function. Firstly, we provide a criterion for all these functions to be continuous on a subset of $\Omega_{N}^{\mathbb{C}}\times\mathcal{B}^{\mathbb{C}}$.

Lemma 3.13**.**

Let $\mathcal{O}$ be a connected subset of $\Omega^{\mathbb{C}}_{N}\times\mathcal{B}^{\mathbb{C}}$. If $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))\equiv k_{0}$, $(\boldsymbol{\omega},\mathbf{A})\in\mathcal{O}$, for some $k_{0}>0$, then the restrictions of $\lambda_{n}$, $1\leq n\leq k_{0}$, to $\mathcal{O}$ are continuous. Moreover, they are locally continuous eigenvalue branches on $\mathcal{O}$.

Then we list several other properties of the $n$-th eigenvalue function in order to study its asymptotic behavior. The following lemma strengths the result in Theorem 2.2 of [22].

Lemma 3.14**.**

Let $\mathcal{O}\subset\Omega^{\mathbb{C}}_{N}\times\mathcal{B}^{\mathbb{C}}$, $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))=m_{1}+m_{2}+m_{3}$ for all $(\boldsymbol{\omega},\mathbf{A})\in\mathcal{O}$, and $\sharp_{1}(\sigma(\boldsymbol{\omega}_{0},\mathbf{A}_{0}))=m_{2}$ for some $(\boldsymbol{\omega}_{0},\mathbf{A}_{0})\in\bar{\mathcal{O}}\setminus\mathcal{O}$, where $m_{i}\geq 0$, $1\leq i\leq 3$. If

[TABLE]

and

[TABLE]

then

[TABLE]

Proof.

Let $c_{1},c_{2}\in\mathbb{R}$ such that $\sharp_{1}(\sigma(\boldsymbol{\omega}_{0},\mathbf{A}_{0})\cap(c_{1},c_{2}))=m_{2}$. Then we get by Lemma 3.5 that there exists a neighborhood $\mathcal{U}\subset\mathcal{O}$ of $(\boldsymbol{\omega}_{0},\mathbf{A}_{0})$ such that $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})\cap(c_{1},c_{2}))=m_{2}$ and $c_{1},c_{2}\notin\sigma(\boldsymbol{\omega},\mathbf{A})$ for all $(\boldsymbol{\omega},\mathbf{A})\in\mathcal{U}$. It follows from (3.8)–(3.9) that $\mathcal{U}$ can be shrunk such that $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})\cap(-\infty,c_{1}))=m_{1}$ and $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})\cap(c_{2},\infty))=m_{3}$ for all $(\boldsymbol{\omega},\mathbf{A})\in\mathcal{U}$. This implies that $\sigma(\boldsymbol{\omega},\mathbf{A})\cap(c_{1},c_{2})=\{\lambda_{n}(\boldsymbol{\omega},\mathbf{A}):\;m_{1}+1\leq n\leq m_{1}+m_{2}\}$ for all $(\boldsymbol{\omega},\mathbf{A})\in\mathcal{U}$. Then the conclusion holds again by Lemma 3.5. ∎

Lemma 3.15**.**

Let $\mathcal{O}$ be a connected subset of $\Omega^{\mathbb{C}}_{N}\times\mathcal{B}^{\mathbb{C}}$ and $c_{1},c_{2}\in\mathbb{R}$ with $c_{1}<c_{2}$. Assume that for all $(\boldsymbol{\omega},\mathbf{A})\in\mathcal{O}$, $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))=k$, $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})\cap(c_{1},c_{2}))=m$ with $m<k$, and $c_{1},c_{2}\notin\sigma(\boldsymbol{\omega},\mathbf{A})$. Then the other $k-m$ eigenvalues out of $[c_{1},c_{2}]$, denoted by $\hat{\lambda}_{1}(\boldsymbol{\omega},\mathbf{A})\leq\cdots\leq\hat{\lambda}_{k-m}(\boldsymbol{\omega},\mathbf{A})$, have the following properties.

• $(1)$

Let $E_{i}=\{\hat{\lambda}_{i}(\boldsymbol{\omega},\mathbf{A}):\,(\boldsymbol{\omega},\mathbf{A})\in\mathcal{O}\}$. Then for all $1\leq i\leq k-m$,

[TABLE]

and there exists $1\leq i_{0}\leq k$ such that $\hat{\lambda}_{i}=\lambda_{i_{0}}$ is continuous on $\mathcal{O}$.

• $(2)$

Let $(\boldsymbol{\omega}_{0},\mathbf{A}_{0})\in\bar{\mathcal{O}}\setminus\mathcal{O}$, $\sharp_{1}(\sigma(\boldsymbol{\omega}_{0},\mathbf{A}_{0}))=m$, and $\sigma(\boldsymbol{\omega}_{0},\mathbf{A}_{0})\subset(c_{1},c_{2})$. If $E_{i_{0}}\subset(-\infty,c_{1})$ for some $1\leq i_{0}\leq k-m$, then

[TABLE]

If $E_{j_{0}}\subset(c_{2},+\infty)$ for some $1\leq j_{0}\leq k-m$, then

[TABLE]

The following result indicates that the monotonicity of $\lambda_{n}$ in a certain direction determines its asymptotic behavior in this direction.

Lemma 3.16**.**

Let $\mathcal{O}=\{(\boldsymbol{\omega},\mathbf{A})_{\nu}\in\Omega_{N}^{\mathbb{C}}\times\mathcal{B}^{\mathbb{C}}:\nu\in(\nu_{0}-\epsilon,\nu_{0}+\epsilon)\}$, where $(\boldsymbol{\omega},\mathbf{A})_{\nu}$ is continuously dependent on $\nu\in(\nu_{0}-\epsilon,\nu_{0}+\epsilon)$ for some $\epsilon>0$. Assume that $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})_{\nu_{0}})=m\geq 0$, and for all $\nu\in(\nu_{0}-\epsilon,\nu_{0}+\epsilon)\setminus\{\nu_{0}\}$, $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})_{\nu})=k>m$.

• $(1)$

If $\lambda_{n}(\nu):=\lambda_{n}((\boldsymbol{\omega},\mathbf{A})_{\nu})$ is non-increasing on $(\nu_{0}-\epsilon,\nu_{0})$ for all $1\leq n\leq k$, then

[TABLE]

• $(2)$

If $\lambda_{n}(\nu)$ is non-decreasing on $(\nu_{0}-\epsilon,\nu_{0})$ for all $1\leq n\leq k$, then

[TABLE]

• $(3)$

If $\lambda_{n}(\nu)$ is non-increasing on $(\nu_{0},\nu_{0}+\epsilon)$ for all $1\leq n\leq k$, then

[TABLE]

• $(4)$

If $\lambda_{n}(\nu)$ is non-decreasing on $(\nu_{0},\nu_{0}+\epsilon)$ for all $1\leq n\leq k$, then

[TABLE]

Note that the analyses in the proofs of Lemmas 3.13, 3.15 and 3.16 are independent of the dimension of the Sturm-Liouville problem (1.3)–(1.4). Thus they are indeed a straightforward generalization of Theorems 2.1, 2.3 and Lemma 2.7 in [21].

4. Singularity of the $n$-th eigenvalue of discrete Sturm-Liouville problems

In this section, we completely characterize singularity of the $n$-th eigenvalue on the boundary conditions for a fixed equation. Then we characterize singularity of the $n$-th eigenvalue on the equations for a fixed boundary condition under a non-degenerate assumption.

4.1. Singularity of the $n$-th eigenvalue on the boundary conditions

Fix a Sturm-Liouville equation $\boldsymbol{\omega}=(P^{-1},Q,W)$ such that $P_{0}^{-1}\in\mathcal{P}_{d}(\mathbb{C})$ in this subsection. Let $K\subseteq\{1,2,\cdots,2d\}$. For any boundary condition $\mathbf{A}=[A\;|\;B]\in\mathcal{O}_{K}^{\mathbb{C}}$, it follows from (2.10) that there exists $S\in\mathcal{H}_{2d}(\mathbb{C})$ such that $\mathbf{A}=[(S\,|\,I_{2d})E_{K}]$. Let us write $S=S(\mathbf{A})$ in the partitioned form:

[TABLE]

where $S_{1},S_{3}\in\mathcal{H}_{d}(\mathbb{C})$ and $S_{2}\in\mathcal{M}_{d}(\mathbb{C})$. Then it follows that

[TABLE]

Recall that $A_{j},B_{j}\in\mathcal{M}_{2d\times d}$, $j=1,2$, are defined in (3.2). Then we have

[TABLE]

From the structure of $E_{1}$ and the fact that $P_{0}^{-1}\in\mathcal{P}_{d}(\mathbb{C})$, we infer that $E_{1}P_{0}^{-1}+I_{d}-E_{1}$ is invertible. Then it follows that

[TABLE]

Recall that $K_{2}$ is defined in (2.2), and $e_{i}$ is the $i$-th column of $I_{d}$. For convenience, we set $r=\sharp(K_{2})$. Let us write $K_{2}=\{k_{1}+d,k_{2}+d,\cdots,k_{r}+d\}$ with $1\leq k_{1}<k_{2}<\cdots<k_{r}\leq d$, and

[TABLE]

if $K_{2}\neq\emptyset$. For any $\mathbf{A}\in\mathcal{B}^{\mathbb{C}}$, we define

[TABLE]

For any $\mathbf{A}\in\mathcal{O}_{K}^{\mathbb{C}}$, we define

[TABLE]

Then the following result holds.

Lemma 4.1**.**

Let $\mathbf{A}\in\mathcal{O}_{K}^{\mathbb{C}}$. Then

[TABLE]

and $S^{D}_{K}(\mathbf{A})\in\mathcal{H}_{d+r}(\mathbb{C})$.

Proof.

By (2.8), ${\rm rank}(E_{2})=d-r$ and thus (4.16) holds. To prove $S^{D}_{K}(\mathbf{A})\in\mathcal{H}_{d+r}(\mathbb{C})$, it suffices to show that

[TABLE]

Direct computation gives

[TABLE]

Since $(P_{0}^{-1}E_{1}+I_{d}-E_{1})$ and $(E_{1}P_{0}^{-1}+I_{d}-E_{1})$ are invertible, we have

[TABLE]

which yields (4.17). ∎

Define

[TABLE]

for nonnegative integers $r^{0},r^{\pm}$ with $r^{0}+r^{-}+r^{+}=d+r$. (4.18) gives the $2d+1$ layers in $\mathcal{B}^{\mathbb{C}}$, while (4.19) divides $\mathcal{O}_{K}^{\mathbb{C}}$ into different areas. Theorem 4.4 below indicates that the $n$-th eigenvalue exhibits the same singularity in any given area. By Lemma 3.4, we have the following result.

Lemma 4.2**.**

$\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))=Nd-k$* for $\mathbf{A}\in\mathcal{B}_{k}$, and $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))=Nd-r^{0}$ for $\mathbf{A}\in\mathcal{B}_{K}^{(r^{0},r^{+},r^{-})}$.*

Lemma 4.3**.**

Let $\mathbf{A}\in\mathcal{B}_{K}^{(r_{1}^{0},r_{1}^{+},r_{1}^{-})}$. Then

[TABLE]

with $\mathcal{U}_{\varepsilon}=\{\mathbf{B}\in\mathcal{O}_{K}^{\mathbb{C}}:\|S(\mathbf{B})-S(\mathbf{A})\|_{\mathcal{M}_{2d}}<\varepsilon\}$ is path connected for any $r^{0}\leq r_{1}^{0}$, $r^{\pm}\geq r_{1}^{\pm}$ satisfying $r^{0}+r^{+}+r^{-}=d+r$, and $\varepsilon>0$ sufficiently small.

Proof.

The proof is similar as Lemma 7.2 in [8]. ∎

We are now in a position to give the complete characterization of singularity of the $n$-th eigenvalue on the boundary conditions.

Theorem 4.4**.**

Fix $\boldsymbol{\omega}\in\Omega_{N}^{\mathbb{C}}$.

(1) Let $0\leq k\leq 2d$. Then the restriction of $\lambda_{n}$ to $\mathcal{B}_{k}$ is continuous for any $1\leq n\leq Nd-k$.

(2) Consider the restriction of $\lambda_{n}$ to $\mathcal{O}_{K}^{\mathbb{C}}$, where $K\subset\{1,\cdots,2d\}$. Let $0\leq r^{0}<r^{0}_{1}\leq d+r$ and $r^{\pm}\geq r_{1}^{\pm}$. Then for any $\mathbf{A}\in\mathcal{B}_{K}^{(r_{1}^{0},r_{1}^{+},r_{1}^{-})}$, we have

[TABLE]

Consequently, the singular set is $\cup_{1\leq k\leq 2d}\mathcal{B}_{k}$.

Proof.

(1) is a direct consequence of Lemmas 3.13 and 4.2. Now, we prove (2). Note that $\sharp_{1}(\sigma({\boldsymbol{\omega}},\mathbf{A}))=Nd-r_{1}^{0}$ by Lemma 4.2. Choose $c_{1},c_{2}\in\mathbb{R}$ such that $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})\cap(c_{1},c_{2}))=Nd-r_{1}^{0}$. By Lemma 3.5, there exists $\varepsilon>0$ such that for all $\mathbf{B}\in\mathcal{U}_{\varepsilon}$ defined in Lemma 4.3, we have $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{B})\cap(c_{1},c_{2}))=Nd-r_{1}^{0}$ and $c_{1},c_{2}\notin\sigma(\boldsymbol{\omega},\mathbf{B})$. It follows from Lemma 4.3 that $\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$ is path connected. By Lemma 4.2, $\sharp_{1}(\sigma({\boldsymbol{\omega}},\mathbf{B}))=Nd-r^{0}$ for $\mathbf{B}\in\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$, and thus $\sharp_{1}\left(\sigma({\boldsymbol{\omega}},\mathbf{B})\cap((-\infty,c_{1})\cup(c_{2},+\infty))\right)=r_{1}^{0}-r^{0}$. Let $\sigma({\boldsymbol{\omega}},\mathbf{B})\cap((-\infty,c_{1})\cup(c_{2},+\infty)):=\{\hat{\lambda}_{1}(\mathbf{B})\leq\cdots\leq\hat{\lambda}_{r_{1}^{0}-r^{0}}(\mathbf{B})\}$ for $\mathbf{B}\in\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$. By Lemma 3.15 (1), either $\hat{\lambda}_{n}(\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(-\infty,c_{1})$ or $\hat{\lambda}_{n}(\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(c_{2},+\infty)$ for all $1\leq n\leq r_{1}^{0}-r^{0}$. Then we divide our proof in two steps.

Step 1. We show that

[TABLE]

for $r^{+}>r_{1}^{+},r^{-}=r_{1}^{-}$; and

[TABLE]

for $r^{+}=r_{1}^{+},r^{-}>r_{1}^{-}$.

Consider $r^{+}>r_{1}^{+}$ and $r^{-}=r_{1}^{-}$. In this case, $r_{1}^{0}-r^{0}=r^{+}-r_{1}^{+}$. Note that there exists a unitary matrix $M\in\mathcal{M}_{d+r}$ such that

[TABLE]

where $M_{+}={\rm diag}\{\mu_{1},\cdots,\mu_{r_{1}^{+}}\}$ with $\mu_{i}>0$, $1\leq i\leq r_{1}^{+}$, and $M_{-}={\rm diag}\{\nu_{1},\cdots,\nu_{r_{1}^{-}}\}$ with $\nu_{j}<0$, $1\leq j\leq r_{1}^{-}$. Recall that $S=S(\mathbf{A})$. If $K_{2}\neq\emptyset$, we define $\mathbf{B}_{t}=[S(\mathbf{B}_{t})\,|\,I_{2d}]E_{K}$ with

[TABLE]

$t\geq 0$ is sufficiently small, and $s_{ij}(\mathbf{B}_{t})=s_{ij}$ if $i\in\{d+1,\cdots,2d\}\setminus K_{2}$ or $j\in\{d+1,\cdots,2d\}\setminus K_{2}$. If $K_{2}=\emptyset$, we only modify (4.27) as

[TABLE]

in the definition of $\mathbf{B}_{t}$. Then $\mathbf{B}_{0}=\mathbf{A}$,

[TABLE]

and thus $r^{0}(S^{D}_{K}(\mathbf{B}_{t}))=r^{0}$, $r^{\pm}(S^{D}_{K}(\mathbf{B}_{t}))=r^{\pm}$, which gives $\mathbf{B}_{t}\in\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$. Moreover, $\sharp_{1}(\sigma({\boldsymbol{\omega}},\mathbf{B}_{t}))$ $=Nd-r^{0}$ for $t>0$, and $\sharp_{1}(\sigma({\boldsymbol{\omega}},\mathbf{B}_{0}))=Nd-r_{1}^{0}$. It follows from Lemma 3.13 that for any fixed $1\leq n\leq Nd-r^{0}$, $\lambda_{n}(\mathbf{B}_{\cdot})$ is locally a continuous eigenvalue branch for $t>0$. Since

[TABLE]

is a positive semi-definite matrix, we get by Corollary 3.12 that $\lambda_{n}(\mathbf{B}_{t_{1}})\leq\lambda_{n}(\mathbf{B}_{t_{2}})$ with $0<t_{1}<t_{2}$ for all $1\leq n\leq Nd-r^{0}$. Hence, by Lemma 3.16 (4), $\lim_{t\rightarrow{0}^{+}}\lambda_{n}({\mathbf{B}}_{t})=-\infty,1\leq n\leq r^{+}-r_{1}^{+}.$ Therefore, there exists $t_{0}>0$ such that $\mathbf{B}_{t_{0}}\in\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$ and ${\lambda}_{n}(\mathbf{B}_{t_{0}})<c_{1}$, $1\leq n\leq r^{+}-r_{1}^{+}$, which yields that $\hat{\lambda}_{n}(\mathbf{B}_{t_{0}})={\lambda}_{n}(\mathbf{B}_{t_{0}})$. According to Lemma 3.15 (1), $\hat{\lambda}_{n}(\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})})={\lambda}_{n}(\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(-\infty,c_{1})$, $1\leq n\leq r^{+}-r_{1}^{+}$, and (4.23) holds. Thanks to Lemma 3.14, we get (4.24).

Consider $r^{+}=r_{1}^{+}$ and $r^{-}>r_{1}^{-}$. Since (4.25)–(4.26) can be shown in a similar way, we omit the details.

Step 2. Show that (4.20)–(4.22) hold for $r^{\pm}>r_{1}^{\pm}$.

In this case, $r_{1}^{0}-r^{0}=(r^{+}-r_{1}^{+})+(r^{-}-r_{1}^{-})$. It follows from (4.23) and (4.26) that

[TABLE]

This implies that $\lambda_{n}(\tilde{\mathbf{B}}_{1})\in(-\infty,c_{1})$ with $1\leq n\leq r^{+}-r_{1}^{+}$, and $\lambda_{n}(\tilde{\mathbf{B}}_{2})\in(c_{2},+\infty)$ with $Nd-r_{1}^{0}<n\leq Nd-r_{1}^{0}+(r^{-}-r_{1}^{-})$ for any fixed $\tilde{\mathbf{B}}_{1}\in\mathcal{U}_{\varepsilon}^{(r_{1}^{0}-(r^{+}-r_{1}^{+}),r^{+},r_{1}^{-})}$ and $\tilde{\mathbf{B}}_{2}\in\mathcal{U}_{\varepsilon}^{(r_{1}^{0}-(r^{-}-r_{1}^{-}),r_{1}^{+},r^{-})}$.

Note that $(r^{0},r^{+},r^{-})=(r_{1}^{0}-(r^{+}-r_{1}^{+})-(r^{-}-r^{-}_{1}),r^{+},r_{1}^{-}+(r^{-}-r_{1}^{-}))$. Then we infer from (4.25) that

[TABLE]

Since $N\geq 2$, we get that

[TABLE]

Therefore, there exists $\tilde{\mathbf{B}}_{3}\in\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$ such that $\lambda_{n}(\tilde{\mathbf{B}}_{3})\in(-\infty,c_{1})$ with $1\leq n\leq r^{+}-r_{1}^{+}$.

On the other hand, $(r^{0},r^{+},r^{-})=(r_{1}^{0}-(r^{+}-r_{1}^{+})-(r^{-}-r_{1}^{-}),r_{1}^{+}+(r^{+}-r_{1}^{+}),r^{-})$. Thus we get by (4.24) that

[TABLE]

which, along with (4.28), yields that there exists $\tilde{\mathbf{B}}_{4}\in\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$ such that $\lambda_{n}(\tilde{\mathbf{B}}_{4})\in(c_{2},+\infty)$ with $Nd-r^{0}-(r^{-}-r_{1}^{-})<n\leq Nd-r^{0}$. Therefore, we have shown

[TABLE]

Note that $\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$ is path connected and $\sharp_{1}(\sigma(\boldsymbol{\omega},{\mathbf{B}})\cap(c_{1},c_{2}))=Nd-r_{1}^{0}=Nd-r^{0}-(r^{+}-r_{1}^{+})-(r^{-}-r_{1}^{-})$ for all $\mathbf{B}\in\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$. Thus we infer from (4.29) and Lemma 3.15 (1) that $\lambda_{n}(\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(-\infty,c_{1})$ for all $1\leq n\leq r^{+}-r_{1}^{+}$, and $\lambda_{n}(\mathcal{U}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(c_{2},+\infty)$ for all $Nd-r^{0}-(r^{-}-r_{1}^{-})<n\leq Nd-r^{0}$. Then it follows from Lemma 3.15 (2) that (4.20) and (4.22) hold. This, along with Lemma 3.14, implies that (4.21) holds. This completes the proof. ∎

4.2. Singularity of the $n$-th eigenvalue on the Sturm-Liouville equations

Fix a boundary condition $\mathbf{A}=[(A_{1},\;A_{2})|(B_{1},\;B_{2})]=[S\;|\;I_{2d}]E_{K}\in\mathcal{O}_{K}^{\mathbb{C}}$. In this subsection, we always assume that one of the following non-degenerate conditions holds:

[TABLE]

where $E_{0}$ is defined in (4.10) and $R:=S_{1}E_{1}+E_{1}-I_{d}$. In particular, the assumption holds for any $\mathbf{A}\in\mathcal{O}_{K}^{\mathbb{C}}$ when $K_{1}=\{1,\cdots,d\}$ and $K_{2}=\emptyset$. For any $\boldsymbol{\omega}=(P^{-1},Q,W)\in\Omega_{N}^{\mathbb{C}}$, we have by (4.3) that

[TABLE]

Then

[TABLE]

Next, we analyze the partitioned structure of the matrix above.

Lemma 4.5**.**

$[E_{1}-I_{d}+E_{1}R^{-1}(S_{1}(I_{d}-E_{1})+E_{1})]R^{*}=I_{d}$* and $S_{2}^{*}E_{1}R^{-1}S_{2}-S_{3}\in\mathcal{H}_{d}(\mathbb{C})$.*

Proof.

Direct computation gives

[TABLE]

Since

[TABLE]

we have

[TABLE]

$S_{2}^{*}E_{1}R^{-1}S_{2}-S_{3}\in\mathcal{H}_{d}(\mathbb{C})$ follows directly from (4.33). ∎

By Lemma 4.5, we have $S_{2}^{*}[E_{1}-I_{d}+E_{1}R^{-1}(S_{1}(I_{d}-E_{1})+E_{1})]=(R^{-1}S_{2})^{*}$. In the case that $K_{2}\neq\emptyset$, we define

[TABLE]

Then by the assumption (4.30) and Lemma 4.5, $\hat{T}_{2}\in\mathcal{H}_{r}(\mathbb{C})$ is invertible. Direct computation implies that

[TABLE]

In the case that $K_{2}=\emptyset$, we have $E_{2}=I_{d}$. Then the next transformation after (4.32) is

[TABLE]

where $\tilde{R}=S_{2}^{*}[E_{1}-I_{d}+E_{1}R^{-1}(S_{1}(I_{d}-E_{1})+E_{1})]$. For any $\boldsymbol{\omega}\in\Omega_{N}^{\mathbb{C}}$, we define

[TABLE]

and $F(\boldsymbol{\omega}):=(A_{1}P_{0}^{-1}+B_{1},\,B_{2})$ in both cases. Then ${\rm rank}\,F(\boldsymbol{\omega})={\rm rank}\,T(\boldsymbol{\omega})+d.$ Moreover, we have the following result.

Lemma 4.6**.**

Let $\boldsymbol{\omega}\in\Omega_{N}^{\mathbb{C}}$. Then $T(\boldsymbol{\omega})\in\mathcal{H}_{d}(\mathbb{C})$.

Proof.

Since

[TABLE]

and $R$ is invertible, we have

[TABLE]

This implies that $P_{0}^{-1}+R^{-1}(S_{1}(I_{d}-E_{1})+E_{1})\in\mathcal{H}_{d}(\mathbb{C})$. Since $\hat{T}_{2}\in\mathcal{H}_{r}(\mathbb{C})$ when $K_{2}\neq\emptyset$, we get $T(\boldsymbol{\omega})\in\mathcal{H}_{d}(\mathbb{C})$.∎

Let

[TABLE]

Then $l_{1}\leq d$. Define

[TABLE]

for attainable nonnegative integers $r^{0},r^{+},r^{-}$ with $r^{0}+r^{+}+r^{-}=d$. The $l_{1}+1$ layers of $\Omega_{N}^{\mathbb{C}}$ are given in (4.38), while the areas’ division is provided in (4.39). Note here that not all the nonnegative integers $0\leq r^{0},r^{\pm}\leq d$ satisfying $r^{0}+r^{+}+r^{-}=d$ can be achievable in general, since $P_{0}^{-1}\in\mathcal{P}_{d}(\mathbb{C})$ while it is not necessary that $P_{0}^{-1}-T(\boldsymbol{\omega})\in\mathcal{P}_{d}(\mathbb{C})$. Similarly, it is possible that $l_{1}<d$. The following result is a direct consequence of Lemma 3.4.

Lemma 4.7**.**

$\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))=Nd-k$* for any $\boldsymbol{\omega}\in\mathcal{E}_{k}$, and $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))=Nd-r^{0}$ for any $\boldsymbol{\omega}\in\mathcal{E}^{(r^{0},r^{+},r^{-})}$.*

Note that the transformations (4.32), (4.34) and (4.35) are independent of $\boldsymbol{\omega}\in\Omega_{N}^{\mathbb{C}}$. Moreover, the following result holds by the construction of $T(\boldsymbol{\omega})$ and a similar argument as that in the proof of Lemma 7.2 in [8].

Lemma 4.8**.**

Let $\boldsymbol{\omega}\in\mathcal{E}^{(r_{1}^{0},r_{1}^{+},r_{1}^{-})}$. Then $\mathcal{V}_{\varepsilon}^{(r^{0},r^{+},r^{-})}:=\{\boldsymbol{\sigma}\in\Omega_{N}^{\mathbb{C}}:\|\boldsymbol{\sigma}-\boldsymbol{\omega}\|_{\mathbb{C}^{(3N+1)d^{2}}}<\varepsilon\}\cap\mathcal{E}^{(r^{0},r^{+},r^{-})}$ is path connected for any $0\leq r^{0}\leq r_{1}^{0}$, $r^{\pm}\geq r_{1}^{\pm}$ satisfying $r^{0}+r^{+}+r^{-}=d$, and $\varepsilon>0$ sufficiently small.

Theorem 4.9**.**

Fix $\mathbf{A}\in\mathcal{O}_{K}^{\mathbb{C}}$.

(1) Let $0\leq k\leq l_{1}$. Then the restriction of $\lambda_{n}$ to $\mathcal{E}_{k}$ is continuous for any $1\leq n\leq Nd-k$.

(2) Consider the restriction of $\lambda_{n}$ to $\Omega_{N}^{\mathbb{C}}$. Let $0\leq r^{0}<r^{0}_{1}\leq l_{1}$ and $r^{\pm}\geq r_{1}^{\pm}$. Then for any $\boldsymbol{\omega}\in\mathcal{E}^{(r_{1}^{0},r_{1}^{+},r_{1}^{-})}$, we have

[TABLE]

Consequently, the singular set is $\cup_{1\leq k\leq l_{1}}\mathcal{E}_{k}$.

Proof.

By Lemma 4.7, $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A}))=Nd-k$ for any $\boldsymbol{\omega}\in\mathcal{E}_{k}$. It follows from Lemma 3.13 that (1) holds. Choose $c_{1},c_{2}\in\mathbb{R}$ such that $\sharp_{1}(\sigma(\boldsymbol{\omega},\mathbf{A})\cap(c_{1},c_{2}))=Nd-r_{1}^{0}$. Then Lemmas 3.5, 3.15 (1) and 4.8 ensure that for $\boldsymbol{\sigma}\in\mathcal{V}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$ with $\varepsilon>0$ small enough, all the eigenvalues of $(\boldsymbol{\sigma},\mathbf{A})$ outside $[c_{1},c_{2}]$, denoted by $\tilde{\lambda}_{1}(\boldsymbol{\sigma})\leq\cdots\leq\tilde{\lambda}_{r_{1}^{0}-r^{0}}(\boldsymbol{\sigma})$, satisfy that either $\tilde{\lambda}_{n}(\mathcal{V}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(-\infty,c_{1})$ or $\tilde{\lambda}_{n}(\mathcal{V}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(c_{2},+\infty)$, $1\leq n\leq r_{1}^{0}-r^{0}$. Then we divide our proof in two steps.

Step 1. We show that

[TABLE]

for $r^{-}>r_{1}^{-},r^{+}=r_{1}^{+}$; and

[TABLE]

for $r^{-}=r_{1}^{-},r^{+}>r_{1}^{+}$.

We only prove (4.43)–(4.44), since (4.45)–(4.46) can be proved similarly. Let $L\in\mathcal{M}_{d}(\mathbb{C})$ be a unitary matrix such that $T(\boldsymbol{\omega})=L\,{\rm diag}\{\tilde{\mu}_{1},\cdots,\tilde{\mu}_{d}\}L^{*},$ where $\tilde{\mu}_{i}$, $1\leq i\leq d$, are the eigenvalues of $T(\boldsymbol{\omega})$ and $\tilde{\mu}_{1}=\cdots=\tilde{\mu}_{r_{1}^{0}}=0$. Recall that $P_{0}^{-1}(\boldsymbol{\omega})$ is used to indicate its dependence on $\boldsymbol{\omega}$, while all the components of $\boldsymbol{\omega}$ except $P_{0}^{-1}$ are fixed. Define

[TABLE]

with $t\leq 0$ small enough. Then $\boldsymbol{\sigma}_{0}=\boldsymbol{\omega}$ and $\boldsymbol{\sigma}_{t}\in\mathcal{V}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$, $t<0$. Since $P_{0}^{-1}(\boldsymbol{\sigma}_{t_{2}})-P_{0}^{-1}(\boldsymbol{\sigma}_{t_{1}})$ is a positive semi-definite matrix for $t_{1}<t_{2}<0$, we infer from Corollary 3.10 and Lemma 3.13 that $\lambda_{n}(\boldsymbol{\sigma}_{t_{1}})\geq\lambda_{n}(\boldsymbol{\sigma}_{t_{2}})$ for each $1\leq n\leq Nd-r^{0}$. Hence, by Lemma 3.16 (1), $\lim_{t\rightarrow{0}^{-}}\lambda_{n}(\boldsymbol{\sigma}_{t})=-\infty,1\leq n\leq r^{-}-r_{1}^{-}.$ Then we get by Lemma 3.15 that $\tilde{\lambda}_{n}(\mathcal{V}_{\varepsilon}^{(r^{0},r^{+},r^{-})})={\lambda}_{n}(\mathcal{V}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(-\infty,c_{1})$, $1\leq n\leq r^{-}-r_{1}^{-}$, satisfy (4.43). This, along with Lemma 3.14, yields (4.44).

Step 2. Show that (4.40)–(4.42) hold for $r^{\pm}>r_{1}^{\pm}$.

By (4.43) and (4.46), we have $\lambda_{n}(\tilde{\boldsymbol{\sigma}}_{1})\in(-\infty,c_{1})$ with $1\leq n\leq r^{-}-r_{1}^{-}$, and $\lambda_{n}(\tilde{\boldsymbol{\sigma}}_{2})\in(c_{2},+\infty)$ with $Nd-r_{1}^{0}<n\leq Nd-r_{1}^{0}+(r^{+}-r_{1}^{+})$ for any fixed $\tilde{\boldsymbol{\sigma}}_{1}\in\mathcal{V}_{\varepsilon}^{(r_{1}^{0}-(r^{-}-r_{1}^{-}),r_{1}^{+},r^{-})}$ and $\tilde{\boldsymbol{\sigma}}_{2}\in\mathcal{V}_{\varepsilon}^{(r_{1}^{0}-(r^{+}-r_{1}^{+}),r^{+},r_{1}^{-})}$. Then we infer from (4.44)–(4.45) that

[TABLE]

Since $Nd-r^{0}-(r^{+}-r_{1}^{+})\geq r^{-}-r_{1}^{-}$, we obtain that there exists $\tilde{\boldsymbol{\sigma}}_{3}\in\mathcal{V}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$ such that $\lambda_{n}(\tilde{\boldsymbol{\sigma}}_{3})\in(-\infty,c_{1})$ with $1\leq n\leq r^{-}-r_{1}^{-}$, and there exists $\tilde{\boldsymbol{\sigma}}_{4}\in\mathcal{V}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$ such that $\lambda_{n}(\tilde{\boldsymbol{\sigma}}_{4})\in(c_{2},+\infty)$ with $Nd-r^{0}-(r^{+}-r_{1}^{+})<n\leq Nd-r^{0}$. This implies that

[TABLE]

for all $\boldsymbol{\sigma}\in\mathcal{V}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$. Then Lemma 3.15 (2) ensures that (4.40) and (4.42) hold. Finally, (4.41) is obtained by Lemma 3.14. ∎

5. Applications to $d$-dimensional Sturm-Liouville problems of Atkinson type

Consider the $d$-dimensional Sturm-Liouville problem of Atkinson type with $d\geq 1$. The continuous Sturm-Liouville equation is

[TABLE]

where $\hat{P},\hat{Q}$ and $\hat{W}$ are $d\times d$ Hermitian matrix-valued functions on $[a,b]$, and

[TABLE]

The self-adjoint boundary condition is given by

[TABLE]

where $A$ and $B$ are $2d\times 2d$ complex matrices, where $A$ and $B$ satisfy (1.6).

(5.1) is said to be of Atkinson type if there exists a partition of the interval $(a,b)$,

[TABLE]

for some $N>1$ such that

[TABLE]

and

[TABLE]

A $d$-dimensional Sturm-Liouville problem is said to be of Atkinson type if it consists of (5.1) of Atkinson type and a self-adjoint boundary condition. $1$-dimensional case has been studied in [1, 4, 11, 14]. In this section, we always assume that (5.1)–(5.2) is of Atkinson type. The space of Sturm-Liouville equations of Atkinson type is

[TABLE]

with topology induced by $(L((a,b),\mathbb{C}^{d\times d}))^{3}$. $\boldsymbol{\hat{\omega}}=\left(\hat{P}^{-1},\hat{Q},\hat{W}\right)$ is used for an element in $\hat{\Omega}$. Note that the space of self-adjoint boundary conditions is also $\mathcal{B}^{\mathbb{C}}$ defined by (2.1). Set

[TABLE]

Let $u=y$ and $v=\hat{P}y^{\prime}$. Then (5.1) is transformed to

[TABLE]

on $(a,b)$. It follows from (5.3)–(5.4) that if $(u,v)$ is a solution of (5.7), then $u(t)\equiv u_{i}\in\mathcal{M}_{d\times 1}$ is a constant vector on $[a_{i},b_{i}]$, $0\leq i\leq N$, and $v(t)\equiv v_{j}\in\mathcal{M}_{d\times 1}$ is a constant vector on $[b_{j-1},a_{j}]$, $1\leq j\leq N$. Furthermore, we define

[TABLE]

We construct a $d$-dimensional discrete Sturm-Liouville problem as follows:

[TABLE]

where $\hat{P}_{N+1}=\hat{P}_{0}=I_{d}$, and a boundary condition

[TABLE]

where $A$ and $B$ are given in (5.2). By writing $A$ and $B$ into the form (3.2), direct computation implies that (5.10) is equivalent to the standard discrete boundary condition:

[TABLE]

Now we show that (5.1)–(5.2) is equivalent to the constructed discrete Sturm-Liouville problem above.

Lemma 5.1**.**

(1) (5.9)–(5.10) is a self-adjoint discrete Sturm-Liouville problem.

(2) (5.1)–(5.2) is equivalent to (5.9)–(5.10).

Proof.

Firstly, we show that (1) holds. Since $A$ and $B$ satisfy (1.6), we have

[TABLE]

Thus

[TABLE]

and

[TABLE]

It follows that (5.11) is a self-adjoint boundary condition. Since $\hat{P}_{j+1},\hat{Q}_{i},\hat{W}_{i}$ are Hermitian, $\hat{P}_{j+1}$ is invertible, and $\hat{W}_{i}\in\mathcal{P}_{d}(\mathbb{C})$ for $0\leq i\leq N$ and $-1\leq j\leq N$, we have

[TABLE]

Hence, (1) holds.

Next, we prove (2). It suffices to show that (5.7) with (5.2) is equivalent to (5.9)–(5.10). Let $(u,v)$ be a solution of (5.7). Since $v\equiv v_{i}$ is a constant vector on $[b_{i-1},a_{i}]$, we have

[TABLE]

for any $1\leq i\leq N$, which, together with (5.8) and the fact that $\hat{P}_{N+1}=\hat{P}_{0}=I_{d}$, yields that

[TABLE]

Since $u\equiv u_{j}$ is a constant vector on $[a_{j},b_{j}]$, we obtain

[TABLE]

for any $0\leq j\leq N$. Then (5.9) is obtained by combining (5.12)–(5.13).

Conversely, let $\{u_{i}\}_{i=-1}^{N+1}$ be a solution of (5.9) and define $v_{i}=\hat{P}_{i}(u_{i}-u_{i-1})$ for $0\leq i\leq N+1$. Let $u(t)=u_{i}$ for all $t\in[a_{i},b_{i}]$ and $0\leq i\leq N$, $v(t)=v_{j}$ for all $t\in[b_{j-1},a_{j}]$ and $1\leq j\leq N$, $v(a)=v_{0}$, and

[TABLE]

Then $(u,v)$ is a solution of (5.7).

Moreover, $y(a)=u_{0}$, $y(b)=u_{N}$, $(\hat{P}y^{\prime})(a)=v(a)=u_{0}-u_{-1}=\Delta u_{-1}$ and $(\hat{P}y^{\prime})(b)=v(b)=u_{N+1}-u_{N}=\Delta u_{N}$. Thus (5.2) is equivalent to (5.10). ∎

By transforming the Sturm-Liouville problem of Atkinson type to the discrete case, we can now determine the number of eigenvalues in the following lemma, which generalizes Theorems 2.1 and 3.1 in [14] for 1-dimension to any dimension.

Lemma 5.2**.**

Let $(\boldsymbol{\hat{\omega}},\mathbf{A})\in\hat{\Omega}\times\mathcal{B}^{\mathbb{C}}$ with $\mathbf{A}$ given in (3.2). Then the eigenvalues of $(\boldsymbol{\hat{\omega}},\mathbf{A})$, including multiplicities, are the same as those of $(\boldsymbol{\tau},\mathbf{A})=(\boldsymbol{\tau},\mathbf{C})$, and

[TABLE]

where $\boldsymbol{\tau}\in\Omega_{N+1}^{\mathbb{C}}$ is the transformed discrete Sturm-Liouville equation by $\boldsymbol{\hat{\omega}}$, and

[TABLE]

Remark 5.3**.**

Note that here $\mathbf{A}$ is under the basis $(-u_{0},u_{N},\Delta u_{-1},\Delta u_{N})$ when we write $(\boldsymbol{\tau},\mathbf{A})$, while $\mathbf{C}$ is under the standard basis $(-u_{-1},u_{N},\Delta u_{-1},\Delta u_{N})$ when we write $(\boldsymbol{\tau},\mathbf{C})$. In this sense, $(\boldsymbol{\tau},\mathbf{A})=(\boldsymbol{\tau},\mathbf{C})$.

Proof.

By Lemma 5.1 (2), $(\boldsymbol{\hat{\omega}},\mathbf{A})$ is equivalent to $(\boldsymbol{\tau},\mathbf{A})=(\boldsymbol{\tau},\mathbf{C})$, and $\hat{P}_{0}=I_{d}$. Then applying Lemma 3.4 to $(\boldsymbol{\tau},\mathbf{C})$, we have

[TABLE]

This completes the proof. ∎

Then we study singularity of the $n$-th eigenvalue of $d$-dimensional Sturm-Liouville problems of Atkinson type. We first claim in Proposition 5.4 below that there is no singularity of the $n$-th eigenvalue on the equations. In fact, for a fixed $\mathbf{A}\in\mathcal{B}^{\mathbb{C}}$, we infer from Lemma 5.2 that $\sharp_{1}(\sigma(\boldsymbol{\hat{\omega}},\mathbf{A}))$ is independent of $\boldsymbol{\hat{\omega}}\in\hat{\Omega}$. This, together with Lemma 3.13, implies the following result.

Proposition 5.4**.**

Fix $\mathbf{A}=[A\;|\;B]\in\mathcal{B}^{\mathbb{C}}$. Then the $n$-th eigenvalue is continuous on the whole space of Sturm-Liouville equations of Atkinson type $\hat{\Omega}$ for all $1\leq n\leq\sharp_{1}(\sigma(\boldsymbol{\hat{\omega}},\mathbf{A}))=(N-1)d+\textrm{rank}\;(B).$

Next, we consider singularity of the $n$-th eigenvalue on the boundary conditions. Lemma 5.2 indicates that it suffices to study singularity of the $n$-th eigenvalue of $(\boldsymbol{\tau},\mathbf{C})=(\boldsymbol{\tau},\mathbf{A})$ for the fixed $\boldsymbol{\tau}$. The coupled term $B_{1}-A_{1}$ in the standard boundary condition $\mathbf{C}$ makes it hard to apply Theorem 4.4 to $(\boldsymbol{\tau},\mathbf{C})$ directly. We shall apply the method developed in Sections 3–4 to the discrete Sturm-Liouville problem $(\boldsymbol{\tau},\mathbf{A})$ and provide a direct proof here. In order to study singularity in a certain direction, we need the derivative formula of a continuous simple eigenvalue branch.

Lemma 5.5**.**

Fix $\boldsymbol{\tau}\in\Omega_{N+1}^{\mathbb{C}}$. Let $\lambda_{*}$ be a simple eigenvalue of $(\boldsymbol{\tau},\mathbf{A})$ for $\mathbf{A}\in\mathcal{O}_{K}^{\mathbb{C}}$, $u\in l[-1,N+1]$ be a normalized eigenfunction for $\lambda_{*}$, and $\Lambda$ be the continuous simple eigenvalue branch through $\lambda_{*}$. Then we have the following derivative formula

[TABLE]

for $H\in\mathcal{H}_{2d}(\mathbb{C})$, where

[TABLE]

Remark 5.6**.**

Lemma 3.11 is unable to be directly applied here due to the different basis. Note carefully that $Z$ in Lemma 5.5 is different from $Y$ in Lemma 3.11.

Proof.

Recall that there exists $S\in\mathcal{H}_{2d}(\mathbb{C})$ such that $\mathbf{A}=[S\;|\;I_{2d}]E_{K}$. Let $\mathbf{B}=[S+H\;|\;I_{2d}]E_{K}$ with $H\in\mathcal{H}_{2d}(\mathbb{C})$. Then by Lemma 3.7, there exists an eigenfunction $\tilde{u}=\{\tilde{u}_{i}\}_{i=-1}^{N+1}$ for $\Lambda(\mathbf{B})$ such that $\tilde{u}\to u$ in $\mathbb{C}^{(N+2)d}$ as $\mathbf{B}\to\mathbf{A}$. Note that $\tilde{u}$ and ${u}$ satisfy

[TABLE]

and thus

[TABLE]

where $\lfloor u_{i},\tilde{u}_{i}\rfloor=(\Delta\tilde{u}_{i})^{*}\hat{P}_{i+1}u_{i}-\tilde{u}_{i}^{*}\hat{P}_{i+1}\Delta u_{i}.$ $\mathbf{A}$ and $\mathbf{B}$ tell us that $SE_{K,1}Z+E_{K,2}Z=0$ and $(S+H)E_{K,1}\tilde{Z}+E_{K,2}\tilde{Z}=0$. Then we infer from (2.9) that

[TABLE]

This completes the proof. ∎

As a consequence, we get the following conclusion.

Corollary 5.7**.**

Let $\Lambda$ be a continuous eigenvalue branch defined on $\mathcal{U}\subset\mathcal{O}_{K}^{\mathbb{C}}$. Then $\Lambda(\mathbf{A})\leq\Lambda(\mathbf{B})$ if $\mathbf{A},\mathbf{B}\in\mathcal{U}$ and $S(\mathbf{B})-S(\mathbf{A})$ is positive semi-definite.

Let the $k$-th layer in $\mathcal{B}^{\mathbb{C}}$ be defined as

[TABLE]

Then the following result is a direct consequence of Lemma 5.2.

Corollary 5.8**.**

Fix $\boldsymbol{\hat{\omega}}\in\hat{\Omega}$. Then

(1) $\sharp_{1}(\sigma(\boldsymbol{\hat{\omega}},\mathbf{A}))=(N+1)d-k$ for all $\mathbf{A}\in\hat{\Sigma}_{k}$.

(2) $\sharp_{1}(\sigma(\boldsymbol{\hat{\omega}},\mathbf{A}))=(N+1)d$ for all $\mathbf{A}\in\mathcal{O}_{\emptyset}^{\mathbb{C}}$.

For a nonempty subset $K=\{n_{1},\cdots,n_{m_{0}}\}\subset\{1,\cdots,2d\}$, we define $\hat{E}_{0}=(\hat{e}_{n_{1}},\cdots,\hat{e}_{n_{m_{0}}})$, and

[TABLE]

where $\mathbf{A}\in\mathcal{O}_{K}^{\mathbb{C}}$, and $\hat{e}_{i}$ is the $i$-th column of $I_{2d}.$ The divided area is defined by

[TABLE]

for three nonnegative integers $r^{0},$ $r^{+}$ and $r^{-}$ satisfying $r^{0}+r^{+}+r^{-}=m_{0}$. Then we are ready to provide the complete characterization of singularity of the $n$-th eigenvalue for the Atkinson type.

Theorem 5.9**.**

Fix $\boldsymbol{\hat{\omega}}\in\hat{\Omega}$.

(1) Let $0\leq k\leq 2d$. Then the restriction of $\lambda_{n}$ to $\hat{\Sigma}_{k}$ is continuous for any $1\leq n\leq(N+1)d-k$. Moreover, the restriction of $\lambda_{n}$ to $\mathcal{O}_{\emptyset}^{\mathbb{C}}$ is continuous for any $1\leq n\leq(N+1)d$.

(2) Consider the restriction of $\lambda_{n}$ to $\mathcal{O}_{K}^{\mathbb{C}}$, where $\emptyset\neq K\subset\{1,\cdots,2d\}$. Let $0\leq r^{0}<r^{0}_{1}\leq\sharp(K)$ and $r^{\pm}\geq r_{1}^{\pm}$. Then for any $\mathbf{A}\in J^{(r_{1}^{0},r_{1}^{+},r_{1}^{-})}_{{K}}$, we have

[TABLE]

Consequently, the singular set is $\cup_{1\leq k\leq 2d}\hat{\Sigma}_{k}$.

Remark 5.10**.**

Note that $S_{K}^{A}(\mathbf{A})$ is independent of the Sturm-Liouville equations of Atkinson type, while $S_{K}^{D}(\mathbf{A})$ defined in (4.14)–(4.15) is indeed involved heavily with the coefficient $P_{0}^{-1}$ of the discrete equations.

Proof.

We study the equivalent discrete Sturm-Liouville problem $(\boldsymbol{\tau},\mathbf{A})$. (1) is straightforward by Lemma 3.13 and Corollary 5.8. Next, we show that (2) holds. Choose $c_{1},c_{2}\in\mathbb{R}$ such that $\sharp_{1}(\sigma({\boldsymbol{\tau}},\mathbf{A})\cap(c_{1},c_{2}))=(N+1)d-r_{1}^{0}$. It follows from Lemma 3.5 that $\sharp_{1}(\sigma({\boldsymbol{\tau}},\mathbf{B})\cap(c_{1},c_{2}))=(N+1)d-r_{1}^{0}$ with $c_{1},c_{2}\notin\sigma({\boldsymbol{\tau}},\mathbf{B})$ for all $\mathbf{B}\in\mathcal{U}_{\varepsilon}$, defined in Lemma 4.3, and $\varepsilon>0$ small enough. Lemma 7.2 in [8] implies that $\hat{\mathcal{U}}_{\varepsilon}^{(r^{0},r^{+},r^{-})}=\mathcal{U}_{\varepsilon}\cap J_{K}^{(r^{0},r^{+},r^{-})}$ is path connected. Note that

[TABLE]

for $\mathbf{B}\in\hat{\mathcal{U}}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$.

We show that if $r^{+}>r_{1}^{+}$ and $r^{-}=r_{1}^{-}$, then (4.23′)–(4.24′) hold. Similarly, if $r^{+}=r_{1}^{+}$ and $r^{-}>r_{1}^{-}$, then (4.25′)–(4.26′) hold. Here (4.23′)–(4.26′) are defined as (4.23)–(4.26) with $\mathcal{B}_{K}^{(r^{0},r^{+},r^{-})}$ and $N$ replaced by ${J}_{K}^{(r^{0},r^{+},r^{-})}$ and $N+1$. Let $\hat{M}\in\mathcal{M}_{m_{0}}$ be the unitary matrix such that $S^{A}_{K}(\mathbf{A})=\hat{M}{\rm diag}\{\tilde{\nu}_{1},\cdots,\tilde{\nu}_{m_{0}}\}\hat{M}^{*}$, where $m_{0}=\sharp(K)$ and $\tilde{\nu}_{1}=\cdots=\tilde{\nu}_{r_{1}^{0}}=0$. Define $\hat{\mathbf{B}}_{t}=[S(\hat{\mathbf{B}}_{t})\,|\,I_{2d}]E_{K}$ with

[TABLE]

$t\geq 0$ is sufficiently small, and $s_{ij}(\hat{\mathbf{B}}_{t})=s_{ij}(\mathbf{A})$ if $i\in\{1,\cdots,2d\}\setminus K$ or $j\in\{1,\cdots,2d\}\setminus K$. Since $S(\hat{\mathbf{B}}_{t_{2}})-S(\hat{\mathbf{B}}_{t_{1}})$ is a positive semi-definite matrix for $t_{2}>t_{1}>0$, it follows from Lemma 3.13 and Corollary 5.7 that $\lambda_{n}(\hat{\mathbf{B}}_{\cdot})$ is non-decreasing on $t\in(0,\varepsilon)$ for each $1\leq n\leq(N+1)d-r^{0}$, where $\varepsilon>0$ is small enough. Hence, by Lemma 3.16 (4), $\lim_{t\rightarrow{0}^{+}}\lambda_{n}(\hat{\mathbf{B}}_{t})=-\infty,1\leq n\leq r^{+}-r_{1}^{+}=r_{1}^{0}-r^{0}.$ This, along with Lemma 3.15 and (5.18), yields that ${\lambda}_{n}(\hat{\mathcal{U}}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(-\infty,c_{1})$, $1\leq n\leq r^{+}-r_{1}^{+}$, and (4.23′) holds. Then we get by Lemma 3.14 that (4.24′) holds.

Finally, we prove (5.15)–(5.17) for $r^{\pm}>r_{1}^{\pm}$. It follows from (4.23′) and (4.26′) that $\lambda_{n}(\tilde{\mathbf{A}}_{1})\in(-\infty,c_{1})$ with $1\leq n\leq r^{+}-r_{1}^{+}$, and $\lambda_{n}(\tilde{\mathbf{A}}_{2})\in(c_{2},+\infty)$ with $(N+1)d-r_{1}^{0}<n\leq(N+1)d-r_{1}^{0}+(r^{-}-r_{1}^{-})$ for any fixed $\tilde{\mathbf{A}}_{1}\in\hat{\mathcal{U}}_{\varepsilon}^{(r_{1}^{0}-(r^{+}-r_{1}^{+}),r^{+},r_{1}^{-})}$ and $\tilde{\mathbf{A}}_{2}\in\hat{\mathcal{U}}_{\varepsilon}^{(r_{1}^{0}-(r^{-}-r_{1}^{-}),r_{1}^{+},r^{-})}$. Furthermore, we have by (4.25′) that there exists $\tilde{\mathbf{A}}_{3}\in\hat{\mathcal{U}}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$ such that $\lambda_{n}(\tilde{\mathbf{A}}_{3})\in(-\infty,c_{1})$ with $1\leq n\leq r^{+}-r_{1}^{+}$. It follows from (4.24′) that there exists $\tilde{\mathbf{A}}_{4}\in\hat{\mathcal{U}}_{\varepsilon}^{(r^{0},r^{+},r^{-})}$ such that $\lambda_{n}(\tilde{\mathbf{A}}_{4})\in(c_{2},+\infty)$ with $r^{+}-r_{1}^{+}\leq(N+1)d-r^{0}-(r^{-}-r_{1}^{-})<n\leq(N+1)d-r^{0}$. Then we get by Lemma 3.15 (1) that $\lambda_{n}(\hat{\mathcal{U}}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(-\infty,c_{1})$ for $1\leq n\leq r^{+}-r_{1}^{+}$, and $\lambda_{n}(\hat{\mathcal{U}}_{\varepsilon}^{(r^{0},r^{+},r^{-})})\subset(c_{2},+\infty)$ for $(N+1)d-r^{0}-(r^{-}-r_{1}^{-})<n\leq(N+1)d-r^{0}$. Thanks to Lemma 3.15 (2), we get (5.15) and (5.17). Then (5.16) is a direct consequence of Lemma 3.14. The proof is complete. ∎

6. Conclusions and comparisons of singularity of the $n$-th eigenvalue among continuous case, discrete case, and Atkinson type

In this section, we compare singularity of the $n$-th eigenvalue among the Sturm-Liouville problems for the continuous case (1.1)–(1.2) in [8], the discrete case (1.3)–(1.4), and the Atkinson type (5.1)–(5.2).

(i). Comparison of singularity on boundary conditions.

According to Theorem 7.1 in [8], Theorem 4.4, and Theorem 5.9, the singularity on the boundary conditions is determined by the constructed Hermitian matrices, which are $S_{K}^{C}(\mathbf{A})$ given in (4.2) of [8] for the continuous case, $S_{K}^{D}(\mathbf{A})$ defined in (4.14)–(4.15) for the discrete case, and $S_{K}^{A}(\mathbf{A})$ defined in (5.14) for the Atkinson type, where $\mathbf{A}\in\mathcal{O}_{K}^{\mathbb{C}}$.

For the continuous case, Theorem 7.1 in [8] tells us that the first $m_{c}$ eigenvalues jump to $-\infty$ as a path of boundary conditions from the lower layer of $\mathcal{O}_{K}^{\mathbb{C}}$ tends to a given boundary condition in the upper layer. Here the jump number $m_{c}$ is exactly the number of transitional eigenvalues (from positive to zero) of the determined Hermitian matrices. It is further shown that this number is the Maslov index of the path of boundary conditions in a forthcoming paper.

For the discrete case, Theorem 4.4 indicates that not only the first $m_{d}^{-}$ eigenvalues jump to $-\infty$, but the last $m_{d}^{+}$ eigenvalues also blow up to $+\infty$ as a path of boundary conditions from the lower layer tends to a given boundary condition in the upper layer. Here the jump number $m_{d}^{-}$ has the similar meaning as $m_{c}$ in the continuous case, while $m_{d}^{+}$ is the number of transitional eigenvalues (from negative to zero) of the determined Hermitian matrices.

For the Atkinson type, Theorem 5.9 renders both similar jump phenomena to $\pm\infty$ with numbers $m_{a}^{\pm}$ as in the discrete case. However, $m_{d}^{\pm}\neq m_{a}^{\pm}$ in general, which is due to the fact that the determined Hermitian matrices are different, i.e., $S_{K}^{D}(\mathbf{A})\neq S_{K}^{A}(\mathbf{A})$. It is also interesting to see that the determined Hermitian matrices for the continuous case and the Atkinson type are the same, i.e., $S_{K}^{C}(\mathbf{A})=S_{K}^{A}(\mathbf{A})$. The singular set in the Atkinson type coincides with that in the continuous case. This, in particular, provides a direct consequence: $m_{c}=m_{a}^{-}$.

The determined Hermitian matrix is independent of coefficients of the Sturm-Liouville equations for the continuous case and the Atkinson type, while the coefficient $P_{0}^{-1}$ involves heavily in the Hermitian matrix for the discrete case. In addition, the order of the determined Hermitian matrix is $d+\sharp(K_{2})$ for the discrete case, while it is $\sharp(K)$ for the continuous case and the Atkinson type. This implies that the maximal jump number in the discrete case is always no less than that in the continuous case and the Atkinson type.

(ii). Comparison of singularity on the equations.

Based on Theorem 6.1 in [8], Theorem 4.9, and Proposition 5.4, the $n$-th eigenvalue has no singularity on coefficients of the Sturm-Liouville equations for the continuous case and the Atkinson type, while indeed exhibits jump phenomena when coefficients of the Sturm-Liouville equations vary for the discrete case.

For the discrete case, Theorem 4.9 also provides jump phenomena to both $\pm\infty$ with jump numbers $\tilde{m}_{d}^{\pm}$ as a path of equations from the lower layer of $\Omega_{N}^{\mathbb{C}}$ tends to a given equation in the upper layer. The determined Hermitian matrix is given by $T(\boldsymbol{\omega})$ defined in (4.36)–(4.37). $\tilde{m}_{d}^{-}$ is the number of transitional eigenvalues (from negative to zero) of the determined Hermitian matrices, while $\tilde{m}_{d}^{+}$ is the number of transitional eigenvalues (from positive to zero) of the determined Hermitian matrices. Here the reverse direction for the transitional eigenvalues in the definitions of $m_{d}^{\pm}$ and $\tilde{m}_{d}^{\pm}$ is essentially due to the opposite monotonicity of the continuous eigenvalue branches, see Corollaries 3.10 and 3.12.

(iii). Comparison of the method in the proof of singularity.

Compared with the continuous case in [8] and [12], the singular set in the discrete case is involved heavily with coefficients of the Sturm-Liouville equations. Moreover, the finiteness of spectrum for the discrete case or the Atkinson type makes the method for the continuous case (e.g., continuity principle in [8, 12]) invalid here. Compared with the $1$-dimensional discrete case in [21], the first difficulty is how to divide areas in each layer of the considered space such that the $n$-th eigenvalue has the same singularity in a given area. We study singularity by partitioning and analyzing the local coordinate systems, and provide a Hermitian matrix which can determine the areas’ division. As mentioned in Introduction, our approach to proving the asymptotic behavior of the $n$-th eigenvalue here should be taken as a generalization of the method developed for $1$-dimensional discrete case in [21] to any dimension.

Finally, we list several determined Hermitian matrices as follows in $2$-dimension to exhibit how the difference is between the continuous case (Atkinson type) and the discrete case. The orders of the determined Hermitian matrices for the discrete case are larger than those for the continuous case (Atkinson type) in (2)–(3), (5)–(6) and (8). On the other hand, these orders are the same in (4), (7) and (9). Even though, for example, the maximal jump number is $1$ in the continuous case and the Atkinson type, while it is $3$ in the discrete case when $K=\{3\}$. However, it is both $2$ in any case when $K=\{1,2\}$.

Let

[TABLE]

for the discrete case.

(1) $K=\emptyset$.

[TABLE]

and there are no $S^{C}_{K}(\mathbf{A})$ and $S^{A}_{K}(\mathbf{A})$, since there is no singularity for the continuous case and the Atkinson type when $K=\emptyset$.

(2) $K=\{1\}$.

[TABLE]

(3) $K=\{3\}$.

[TABLE]

(4) $K=\{1,2\}$.

[TABLE]

(5) $K=\{1,3\}$.

[TABLE]

(6) $K=\{3,4\}$.

[TABLE]

(7) $K=\{1,2,3\}$.

[TABLE]

(8) $K=\{1,3,4\}$.

[TABLE]

(9) $K=\{1,2,3,4\}$.

[TABLE]

Acknowledgement

H. Zhu sincerely thanks Prof. Yiming Long for his consistent support and great help. Part of this work was done while H. Zhu was visiting the IMS of CUHK; he sincerely thanks Prof. Zhouping Xin for his invitation. G. Ren is partially supported by NSFC (No. 11571202). H. Zhu is partially supported by NSFC (No. 11790271), PITSP (No. BX20180151) and CPSF (No. 2018M630266).