In this section we introduce some of the notations and relations that will be used in the sequel; then we prove some useful lemmas and theorems. We consider the Dirac system
$$\begin{aligned}& \left . \textstyle\begin{array}{l} r(x) u_{2}^{\prime}(x)-q_{1}(x)u_{1}(x)=\lambda u_{1}(x), \\ r(x)u_{1}^{\prime}(x)+q_{2}(x)u_{2}(x)=-\lambda u_{2}(x), \end{array}\displaystyle \right \} \quad x\in[a,c_{1})\cup(c_{1},c_{2}) \cup(c_{2},c_{3})\cup\cdots\cup(c_{m},b], \end{aligned}$$
(2.1)
$$\begin{aligned}& \mathcal{U}_{1}(u):=\sin\alpha u_{1}(a)+ \cos\alpha u_{2}(a)=0, \end{aligned}$$
(2.2)
$$\begin{aligned}& \mathcal{U}_{2}(u):=\sin\beta u_{1}(b)+\cos \beta u_{2}(b)=0, \end{aligned}$$
(2.3)
and the transmission conditions
$$ \mathcal{L}_{k}(u):= \left[ \textstyle\begin{array}{@{}c@{}} \gamma_{k} u_{1}(c_{k}^{-}) \\ \gamma_{k}^{\prime}u_{2}(c_{k}^{-}) \end{array}\displaystyle \right]- \left[ \textstyle\begin{array}{@{}c@{}} \delta_{k} u_{1}(c_{k}^{+}) \\ \delta_{k}^{\prime} u_{2}(c_{k}^{+}) \end{array}\displaystyle \right]=0, \quad k=1,2,\ldots,m, $$
(2.4)
where \(\lambda\in\mathbb{C}\);
$$r(x):=\left \{ \textstyle\begin{array}{l@{\quad}l} r_{1}, &x\in[a, c_{1}), \\ r_{k}, & x\in(c_{k-1}, c_{k})\ (k=2,3,\ldots,m), \\ r_{m+1}, & x\in(c_{m}, b], \end{array}\displaystyle \right . $$
\(r_{k}>0\) (\(k=1,2,\ldots,m+1\)) are real numbers; the real-valued functions \(q_{1}(\cdot)\) and \(q_{2}(\cdot)\) are continuous in \([a,c_{1})\), \((c_{k-1},c_{k})\), and \((c_{m},b]\) (\(k=2,3,\ldots,m\)), and have finite limits \(q_{1}(c_{k}^{\pm}):=\lim _{x\rightarrow c_{k}^{\pm}}q_{1}(x)\), \(q_{2} (c_{k}^{\pm}):=\lim_{x\rightarrow c_{k}^{\pm}}q_{2}(x)\) (\(k=1,2,\ldots,m\)); \(\gamma_{k}, \delta_{k}, \gamma_{k}^{\prime}, \delta _{k}^{\prime}\in \mathbb{R}\), \(\gamma_{k}, \delta_{k}, \gamma _{k}^{\prime}, \delta_{k}^{\prime}\neq0\) and \(\alpha,\beta\in [0,\pi)\); see [26, 27].
Let \(\mathcal{H}\) be the Hilbert space
$$ \mathcal{H}:= \left \{u(x)=\left ( \textstyle\begin{array}{@{}c@{}} u_{1}(x) \\ u_{2}(x) \end{array}\displaystyle \right ), u_{1}(x), u_{2}(x)\in\bigoplus _{k=1}^{m+1} L^{2}(c_{k-1},c_{k}), a=c_{0}, b=c_{m+1} \right \}. $$
(2.5)
The inner product of \(\mathcal{H}\) is defined by
$$\begin{aligned} \bigl\langle u(\cdot),v(\cdot)\bigr\rangle _{\mathcal{H}} :=& \frac{1}{r_{1}} \int^{c_{1}}_{a}u^{\top}(x)\overline{v}(x) \,dx+\sum_{k=2}^{m}\frac{\prod_{i=1}^{k-1}D_{i}}{r_{k}} \int ^{c_{k}}_{c_{k-1}}u^{\top}(x)\overline{v}(x) \,dx \\ &{}+\frac{\prod_{i=1}^{m}D_{i}}{r_{m+1}} \int^{b}_{c_{m}}u^{\top }(x)\overline{v}(x) \,dx, \end{aligned}$$
(2.6)
where \(D_{i}=\frac{\delta_{i}\delta_{i}^{\prime}}{\gamma _{i}\gamma_{i}^{\prime}}\), \(D_{i}>0\), \(i=1,2,\ldots,m\), and ⊤ denotes the matrix transpose,
$$u(x)=\left ( \textstyle\begin{array}{@{}c@{}} u_{1}(x) \\ u_{2}(x) \end{array}\displaystyle \right ) , v(x)= \left ( \textstyle\begin{array}{@{}c@{}} v_{1}(x) \\ v_{2}(x) \end{array}\displaystyle \right ) \in\mathcal{H}, \qquad u_{i}(\cdot), v_{i}(\cdot)\in\bigoplus _{k=1}^{m+1} L^{2}(c_{k-1},c_{k}), \quad i=1,2. $$
For vector-valued functions \(u(x)\), which are defined on \([a,c_{1})\cup (c_{1},c_{2})\cup(c_{2},c_{3})\cup\cdots\cup(c_{m},b]\) and have a finite limit \(u(c_{k}^{\pm}):=\lim_{x\rightarrow c_{k}^{\pm}}u(x)\) (\(k=1,2,\ldots,m\)), by \(u_{(k)}(x)\) (\(k=1,2,\ldots,m+1\)) we denote the functions
$$ \begin{aligned} &u_{(1)}(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} u(x), & x\in[a,c_{1}) , \\ u(c_{1}^{-}), & x=c_{1} , \end{array}\displaystyle \right .\qquad u_{(i)}(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} u(c_{i-1}^{+}), & x=c_{i-1}, \\ u(x), & x\in(c_{i-1},c_{i}), \\ u(c_{i}^{-}), &x=c_{i}, \end{array}\displaystyle \right .\quad i=2,3,\ldots,m, \\ &u_{(m+1)}(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} u(x), & x\in(c_{m},b], \\ u(c_{m}^{+}), & x=c_{m}, \end{array}\displaystyle \right . \end{aligned} $$
(2.7)
which are defined on \(\mathcal{I}_{1}:=[a,c_{1}]\), \(\mathcal {I}_{k}:=[c_{k-1},c_{k}]\) (\(k=2,3,\ldots,m\)) and \(\mathcal {I}_{m+1}:=[c_{m},b]\), respectively.
In the following lemma, we will prove that the eigenvalues of the problem (2.1)-(2.4) are real.
Lemma 2.1
The eigenvalues of the problem (2.1)-(2.4) are real.
Proof
Assume to the contrary that \(\lambda_{0}\) is a non-real eigenvalue of problem (2.1)-(2.4). Let \(\bigl ({\scriptsize\begin{matrix}{} u_{1}(x) \cr u_{2}(x)\end{matrix}} \bigr )\) be a corresponding (non-trivial) vector-valued eigenfunction. By (2.1), we have
$$\begin{aligned}& r(x)\frac{d}{dx} \bigl\{ u_{1}(x)\overline{u}_{2}(x)- \overline{u}_{1}(x)u_{2}(x) \bigr\} \\& \quad =(\overline{ \lambda}_{0}-\lambda_{0}) \bigl\{ \bigl\vert u_{1}(x)\bigr\vert ^{2}+\bigl\vert u_{2}(x) \bigr\vert ^{2} \bigr\} ,\quad x\in[a,c_{1})\cup (c_{1},c_{2})\cup(c_{2},c_{3})\cup \cdots\cup(c_{m},b]. \end{aligned}$$
Integrating the above equation through \([a, c_{1})\), \((c_{k-1}, c_{k})\) (\(k=2,3,\ldots,m\)), and \((c_{m}, b]\), we obtain
$$\begin{aligned}& \frac{\overline{\lambda}_{0}-\lambda_{0}}{r_{1}} \biggl[ \int_{a}^{c_{1}} \bigl(\bigl\vert u_{1}(x) \bigr\vert ^{2}+\bigl\vert u_{2}(x)\bigr\vert ^{2} \bigr) \,dx \biggr] \\& \quad =u_{1}\bigl(c_{1}^{-} \bigr)\overline{u}_{2}\bigl(c_{1}^{-}\bigr)- \overline{u}_{1}\bigl(c_{1}^{-} \bigr)u_{2}\bigl(c_{1}^{-}\bigr)- \bigl[u_{1}(a)\overline{u}_{2}(a)- \overline{u}_{1}(a)u_{2}(a) \bigr], \end{aligned}$$
(2.8)
$$\begin{aligned}& \frac{\overline{\lambda}_{0}-\lambda_{0}}{r_{k}} \biggl[ \int_{c_{k-1}}^{c_{k}} \bigl(\bigl\vert u_{1}(x) \bigr\vert ^{2}+\bigl\vert u_{2}(x)\bigr\vert ^{2} \bigr) \,dx \biggr] \\& \quad =u_{1}\bigl(c_{k}^{-} \bigr)\overline{u}_{2}\bigl(c_{k}^{-}\bigr)- \overline{u}_{1}\bigl(c_{k}^{-} \bigr)u_{2}\bigl(c_{k}^{-}\bigr) - \bigl[u_{1}\bigl(c_{k-1}^{+}\bigr) \overline{u}_{2}\bigl(c_{k-1}^{+}\bigr)- \overline{u}_{1}\bigl(c_{k-1}^{+} \bigr)u_{2}\bigl(c_{k-1}^{+}\bigr) \bigr], \end{aligned}$$
(2.9)
for \(k=2,3,\ldots,m\), and
$$\begin{aligned}& \frac{\overline{\lambda}_{0}-\lambda_{0}}{r_{m+1}} \biggl[ \int_{c_{m}}^{b} \bigl(\bigl\vert u_{1}(x) \bigr\vert ^{2}+\bigl\vert u_{2}(x)\bigr\vert ^{2} \bigr) \,dx \biggr] \\& \quad =u_{1}(b)\overline{u}_{2}(b)- \overline{u}_{1}(b)u_{2}(b)- \bigl[u_{1} \bigl(c_{m}^{+}\bigr)\overline{u}_{2} \bigl(c_{m}^{+}\bigr)- \overline{u}_{1} \bigl(c_{m}^{+}\bigr)u_{2}\bigl(c_{m}^{+} \bigr) \bigr]. \end{aligned}$$
(2.10)
Then from (2.2), (2.3), and the transmission conditions, we have, respectively,
$$\begin{aligned}& u_{1}(a)\overline{u}_{2}(a)- \overline{u}_{1}(a)u_{2}(a)=0, \\& u_{1}(b)\overline{u}_{2}(b)- \overline{u}_{1}(b)u_{2}(b)=0, \end{aligned}$$
and
$$\begin{aligned}& u_{1}\bigl(c_{k}^{-}\bigr)\overline{u}_{2} \bigl(c_{k}^{-}\bigr)- \overline{u}_{1} \bigl(c_{k}^{-}\bigr)u_{2}\bigl(c_{k}^{-} \bigr) \\& \quad =\frac{\delta_{k}\delta _{k}^{\prime}}{\gamma_{k}\gamma_{k}^{\prime }}\bigl[u_{1}\bigl(c_{k}^{+} \bigr)\overline{u}_{2}\bigl(c_{k}^{+}\bigr)- \overline{u}_{1}\bigl(c_{k}^{+} \bigr)u_{2}\bigl(c_{k}^{+}\bigr)\bigr],\quad k=1,2, \ldots,m. \end{aligned}$$
Since \(\lambda_{0}\neq\overline{\lambda}_{0}\), it follows from the last three equations and (2.8), (2.9), (2.10) that
$$\begin{aligned}& \frac{1}{r_{1}} \int_{a}^{c_{1}} \bigl(\bigl\vert u_{1}(x) \bigr\vert ^{2}+\bigl\vert u_{2}(x)\bigr\vert ^{2} \bigr) \,dx +\sum_{k=2}^{m} \frac{\prod_{i=1}^{k-1}D_{i}}{r_{k}} \int _{c_{k-1}}^{c_{k}} \bigl(\bigl\vert u_{1}(x)\bigr\vert ^{2}+\bigl\vert u_{2}(x) \bigr\vert ^{2} \bigr) \,dx \\& \quad {} +\frac{\prod_{i=1}^{m}D_{i}}{r_{m+1}} \int_{c_{m}}^{b} \bigl(\bigl\vert u_{1}(x) \bigr\vert ^{2}+\bigl\vert u_{2}(x)\bigr\vert ^{2} \bigr) \,dx=0. \end{aligned}$$
(2.11)
Then \(u_{i}(x)=0\), \(i=1,2\), and this is a contradiction. Consequently, \(\lambda_{0}\) must be real. □
Lemma 2.2
Let
\(\lambda_{1}\)
and
\(\lambda_{2}\)
be two different eigenvalues of the problem (2.1)-(2.4). Then the corresponding vector-valued eigenfunctions
\(u(x,\lambda_{1})\)
and
\(v(x,\lambda_{2})\)
of this problem satisfy the following equality:
$$\begin{aligned} \begin{aligned}[b] &\frac{1}{r_{1}} \int_{a}^{c_{1}}u^{\top}(x, \lambda_{1})v(x,\lambda _{2}) \,dx+\sum _{k=2}^{m}\frac{\prod_{i=1}^{k-1}D_{i}}{r_{k}} \int _{c_{k-1}}^{c_{k}}u^{\top}(x, \lambda_{1})v(x,\lambda_{2}) \,dx \\ &\quad {}+\frac{\prod_{i=1}^{m}D_{i}}{r_{m+1}} \int_{c_{m}}^{b}u^{\top }(x, \lambda_{1})v(x,\lambda_{2}) \,dx=0. \end{aligned} \end{aligned}$$
(2.12)
Proof
By (2.1) we obtain
$$\begin{aligned}& r(x)\frac{d}{dx} \bigl\{ u_{1}(x,\lambda_{1})v_{2}(x, \lambda_{2})- u_{2}(x,\lambda_{2})v_{1}(x, \lambda_{1}) \bigr\} \\& \quad =(\lambda _{2}-\lambda_{1}) \bigl\{ u_{1}(x,\lambda_{1})v_{1}(x, \lambda_{2}) +u_{2}(x,\lambda_{1})v_{2}(x, \lambda_{2}) \bigr\} . \end{aligned}$$
Integrating the above equation through \([a, c_{1})\), \((c_{k-1}, c_{k})\) (\(k=2,3,\ldots,m\)), and \((c_{m}, b]\), and taking into account that \(u(x,\lambda_{1})\) and \(v(x,\lambda_{2})\) satisfy (2.2)-(2.4), we obtain (2.12), where \(\lambda_{1}\neq\lambda_{2}\). □
Now we will construct a special fundamental system of solutions of (2.1) for λ not being an eigenvalue. By virtue of Theorem 1.1 in [1], we will define the two solutions of (2.1)
$$\varphi(\cdot,\lambda)=\left ( \textstyle\begin{array}{@{}c@{}} \varphi_{1}(\cdot,\lambda) \\ \varphi_{2}(\cdot,\lambda) \end{array}\displaystyle \right ), \qquad \chi(\cdot,\lambda)=\left ( \textstyle\begin{array}{@{}c@{}} \chi_{1}(\cdot,\lambda) \\ \chi_{2}(\cdot,\lambda) \end{array}\displaystyle \right ), $$
where
$$\begin{aligned}& \begin{aligned} &\varphi_{1}(x,\lambda)=\left \{ \textstyle\begin{array}{l@{\quad}l} \varphi_{1,1}(x,\lambda), & x\in[c_{0},c_{1}) , \\ \varphi_{1,2}(x,\lambda), & x\in(c_{1},c_{2}) , \\ \ldots, \\ \varphi_{1,m+1}(x,\lambda), & x\in(c_{m},c_{m+1}] , \end{array}\displaystyle \right . \\ &\varphi_{2}(x,\lambda)=\left \{ \textstyle\begin{array}{l@{\quad}l} \varphi_{2,1}(x,\lambda), & x\in[c_{0},c_{1}) , \\ \varphi_{2,2}(x,\lambda), & x\in(c_{1},c_{2}) , \\ \ldots, \\ \varphi_{2,m+1}(x,\lambda), & x\in(c_{m},c_{m+1}] , \end{array}\displaystyle \right . \end{aligned} \end{aligned}$$
(2.13)
$$\begin{aligned}& \begin{aligned} &\chi_{1}(x,\lambda)=\left \{ \textstyle\begin{array}{l@{\quad}l} \chi_{1,1}(x,\lambda), & x\in[c_{0},c_{1}) , \\ \chi_{1,2}(x,\lambda), & x\in(c_{1},c_{2}) , \\ \ldots, \\ \chi_{1,m+1}(x,\lambda), & x\in(c_{m},c_{m+1}] , \end{array}\displaystyle \right . \\ &\chi_{2}(x,\lambda)=\left \{ \textstyle\begin{array}{l@{\quad}l} \chi_{2,1}(x,\lambda), & x\in[c_{0},c_{1}) , \\ \chi_{2,2}(x,\lambda), & x\in(c_{1},c_{2}) , \\ \ldots, \\ \chi_{2,m+1}(x,\lambda), & x\in(c_{m},c_{m+1}] , \end{array}\displaystyle \right . \end{aligned} \end{aligned}$$
(2.14)
as follows, where \(c_{0}=a\) and \(c_{m+1}=b\). By virtue of Theorem 1.1 in [1] the initial-value problem
$$\begin{aligned}& \begin{aligned} &r(x)u_{2}^{\prime}(x)-q_{1}(x)u_{1}(x)= \lambda u_{1}(x), \\ &r(x) u_{1}^{\prime}(x)+q_{2}(x)u_{2}(x)=- \lambda u_{2}(x), \quad x\in (a,c_{1}), \end{aligned} \end{aligned}$$
(2.15)
$$\begin{aligned}& u_{1}(a)=\cos\alpha, \qquad u_{2}(a)=-\sin \alpha, \end{aligned}$$
(2.16)
has a unique solution \(u=\bigl ({\scriptsize\begin{matrix}{} \varphi_{1,1}(x,\lambda)\cr \varphi_{2,1}(x,\lambda)\end{matrix}} \bigr )\), which is an entire function of \(\lambda\in\mathbb{C}\) for each fixed \(x\in[a,c_{1}]\). Similarly, employing the same method as in proof of Theorem 1.1 in [1], we see that the problem
$$\begin{aligned}& \begin{aligned} &r(x)u_{2}^{\prime}(x)-q_{1}(x)u_{1}(x)= \lambda u_{1}(x), \\ &r(x)u_{1}^{\prime}(x)+q_{2}(x)u_{2}(x)=- \lambda u_{2}(x), \quad x\in(c_{m},b), \end{aligned} \end{aligned}$$
(2.17)
$$\begin{aligned}& u_{1}(b)=\cos\beta, \qquad u_{2}(b)=-\sin \beta, \end{aligned}$$
(2.18)
has a unique solution \(u=\bigl ( {\scriptsize\begin{matrix}{} \chi_{1,m+1}(x,\lambda)\cr \chi_{2,m+1}(x,\lambda) \end{matrix}} \bigr )\), which is an entire function of the parameter λ for each fixed \(x\in[c_{m},b]\).
Now the functions \(\varphi_{i,k+1}(x,\lambda)\) and \(\chi _{i,k}(x,\lambda)\) are defined in terms of \(\varphi_{i,k}(x,\lambda)\) and \(\chi_{i,k+1}(x,\lambda)\), \(i=1,2\), \(k=1,2,\ldots,m\), respectively, as follows. The initial-value problem
$$\begin{aligned}& \begin{aligned} &r(x)u_{2}^{\prime}(x)-q_{1}(x)u_{1}(x)= \lambda u_{1}(x), \\ &r(x)u_{1}^{\prime}(x)+q_{2}(x)u_{2}(x)=- \lambda u_{2}(x), \quad x\in (c_{k},c_{k+1}), \end{aligned} \end{aligned}$$
(2.19)
$$\begin{aligned}& u_{1}(c_{k})=\frac{\gamma_{k}}{\delta_{k}}\varphi _{1,k}\bigl(c_{k}^{-},\lambda\bigr),\qquad u_{2}(c_{k})=\frac{\gamma_{k}^{\prime}}{\delta_{k}^{\prime }}\varphi_{2,k} \bigl(c_{k}^{-},\lambda\bigr), \quad k=1,2,\ldots,m, \end{aligned}$$
(2.20)
has a unique solution \(u=\bigl ( {\scriptsize\begin{matrix}{} \varphi_{1,k+1}(x,\lambda)\cr \varphi_{2,k+1}(x,\lambda) \end{matrix}} \bigr )\) for each \(\lambda\in \mathbb{C}\).
Similarly, the following problem also has a unique solution \(u=\bigl ( {\scriptsize\begin{matrix}{} \chi_{1,k}(x,\lambda)\cr \chi_{2,k}(x,\lambda)\end{matrix}} \bigr )\):
$$\begin{aligned}& \begin{aligned} &r(x)u_{2}^{\prime}(x)-q_{1}(x)u_{1}(x)= \lambda u_{1}(x), \\ &r(x)u_{1}^{\prime}(x)+q_{2}(x)u_{2}(x)=- \lambda u_{2}(x), \quad x\in (c_{k-1},c_{k}), \end{aligned} \end{aligned}$$
(2.21)
$$\begin{aligned}& u_{1}(c_{k})=\frac{\delta_{k}}{\gamma_{k}}\chi _{1,k+1}\bigl(c_{k}^{+},\lambda\bigr), \qquad u_{2}(c_{k})=\frac{\delta _{k}^{\prime}}{\gamma_{k}^{\prime}} \chi_{2,k+1} \bigl(c_{k}^{+},\lambda\bigr),\quad k=1,2,\ldots,m. \end{aligned}$$
(2.22)
By virtue of equations (2.20) and (2.22) these solutions satisfy both transmission conditions (2.4). These functions are entire in λ for all \(x\in[a,c_{1})\cup (c_{1},c_{2})\cup(c_{2},c_{3})\cup\cdots\cup(c_{m},b]\).
Let \(W(\varphi,\chi)({\cdot,\lambda})\) denote the Wronskian of \(\varphi(\cdot,\lambda)\) and \(\chi(\cdot,\lambda)\) defined in [2], p.194, i.e.,
$$W(\varphi,\chi) (\cdot,\lambda):=\left \vert \textstyle\begin{array}{@{}c@{\quad}c@{}} \varphi_{1}(\cdot,\lambda) & \varphi_{2}(\cdot,\lambda) \\ \chi_{1}(\cdot,\lambda) & \chi_{2}(\cdot,\lambda) \end{array}\displaystyle \right \vert . $$
It is obvious that the Wronskians
$$\begin{aligned} \omega_{k}(\lambda) :=&W(\varphi,\chi) (x,\lambda) \\ =& \varphi _{1,k}(x,\lambda) \chi_{2,k}(x,\lambda)- \varphi_{2,k}(x,\lambda) \chi_{1,k}(x,\lambda), \quad x\in \mathcal{I}_{k}, k=1,2,\ldots,m+1, \end{aligned}$$
(2.23)
are independent of \(x\in\mathcal{I}_{k}\) and are entire functions. Taking into account (2.20) and (2.22), a short calculation gives
$$\omega_{1}(\lambda)=D_{1}\omega_{2}(\lambda)= \cdots=\prod_{i=1}^{m}D_{i} \omega_{m+1}(\lambda), $$
for each \(\lambda\in\mathbb{C}\).
Corollary 2.3
The zeros of the functions
\(\omega_{k}(\lambda)\) (\(k=1,2,\ldots,m+1\)) coincide.
Then we may introduce into the consideration the characteristic function \(\omega(\lambda)\) as
$$ \omega(\lambda):=\omega_{1}(\lambda)=D_{1} \omega_{2}(\lambda )=\cdots=\prod_{i=1}^{m}D_{i} \omega_{m+1}(\lambda). $$
(2.24)
Lemma 2.4
All eigenvalues of problem (2.1)-(2.4) are just zeros of the function
\(\omega(\lambda)\).
Proof
Since the functions \(\varphi_{1}(x,\lambda)\) and \(\varphi _{2}(x,\lambda)\) satisfy the boundary condition (2.2) and the transmission conditions (2.4), to find the eigenvalues of the (2.1)-(2.4) we have to insert the functions \(\varphi _{1}(x,\lambda)\) and \(\varphi_{2}(x,\lambda)\) in the boundary condition (2.3) and find the roots of this equation. □
In the following lemma, we show that all eigenvalues of the problem (2.1)-(2.4) are simple.
Lemma 2.5
The eigenvalues of the boundary value problem (2.1)-(2.4) form an at most countable set without finite limit points. All eigenvalues of the boundary value problem (2.1)-(2.4) (of
\(\omega(\lambda)\)) are simple.
Proof
The eigenvalues are the zeros of the entire function occurring on the left-hand side, see equation (2.24), in
$$\prod_{i=1}^{m}D_{i}\bigl[\sin \beta\varphi_{1,m+1}(b,\lambda)+ \cos\beta\varphi_{2,m+1}(b,\lambda) \bigr]=0. $$
We have shown (see Lemma 2.1) that this function does not vanish for non-real λ. In particular, it does not vanish identically. Therefore, its zeros form an at most countable set without finite limit points.
By (2.1) we obtain for \(\lambda,\mu\in\mathbb{C}\), \(\lambda\neq\mu\),
$$r(x)\frac{d}{dx} \bigl\{ \varphi_{1}(x,\lambda) \varphi_{2}(x,\mu)- \varphi_{1}(x,\mu)\varphi_{2}(x, \lambda) \bigr\} =(\mu-\lambda) \bigl\{ \varphi_{1}(x,\lambda) \varphi_{1}(x,\mu) +\varphi_{2}(x,\lambda) \varphi_{2}(x,\mu) \bigr\} . $$
Integrating the above equation through \([a, c_{1})\), \((c_{k-1}, c_{k})\) (\(k=2,3,\ldots,m\)) and \((c_{m}, b]\), we obtain
$$\begin{aligned}& \frac{\mu-\lambda}{r_{1}} \biggl[ \int_{a}^{c_{1}} \bigl(\varphi_{1,1}(x, \lambda)\varphi_{1,1}(x,\mu) +\varphi_{2,1}(x,\lambda) \varphi_{2,1}(x,\mu) \bigr) \,dx \biggr] \\& \quad = \varphi_{1,1}\bigl(c_{1}^{-},\lambda\bigr) \varphi_{2,1}\bigl(c_{1}^{-},\mu\bigr)- \varphi_{1,1}\bigl(c_{1}^{-},\lambda\bigr) \varphi_{2,1}\bigl(c_{1}^{-},\mu\bigr) \\& \qquad {}- \bigl(\varphi_{1,1}(a,\lambda)\varphi_{2,1}(a,\mu)- \varphi_{1,1}(a,\lambda)\varphi_{2,1}(a,\mu) \bigr), \end{aligned}$$
(2.25)
$$\begin{aligned}& \frac{\mu-\lambda}{r_{k}} \biggl[ \int_{c_{k-1}}^{c_{k}} \bigl(\varphi_{1,k}(x, \lambda)\varphi _{1,k}(x,\mu) +\varphi_{2,k}(x,\lambda) \varphi_{2,k}(x,\mu) \bigr) \,dx \biggr] \\& \quad =\varphi_{1,k}\bigl(c_{k}^{-},\lambda\bigr) \varphi _{2,k}\bigl(c_{k}^{-},\mu\bigr)- \varphi_{1,k}\bigl(c_{k}^{-},\lambda\bigr) \varphi_{2,k}\bigl(c_{k}^{-},\mu\bigr) \\& \qquad {}- \bigl[\varphi_{1,k}\bigl(c_{k-1}^{+}, \lambda\bigr)\varphi _{2,k}\bigl(c_{k-1}^{+},\mu \bigr)- \varphi_{1,k}\bigl(c_{k-1}^{+},\lambda\bigr) \varphi_{2,k}\bigl(c_{k-1}^{+},\mu \bigr) \bigr], \end{aligned}$$
(2.26)
for \(k=2,3,\ldots,m\), and
$$\begin{aligned}& \frac{\mu-\lambda}{r_{m+1}} \biggl[ \int_{c_{m}}^{b} \bigl(\varphi_{1,m+1}(x, \lambda)\varphi _{1,m+1}(x,\mu) +\varphi_{2,m+1}(x,\lambda) \varphi_{2,m+1}(x,\mu) \bigr) \,dx \biggr] \\& \quad =\varphi_{1,m+1}(b,\lambda)\varphi_{2,m+1}(b,\mu) - \varphi_{1,m+1}(b,\lambda)\varphi_{2,m+1}(b,\mu) \\& \qquad {}- \bigl[\varphi _{1,m+1}\bigl(c_{m}^{+}, \lambda\bigr)\varphi_{2,m+1}\bigl(c_{m}^{+},\mu\bigr)- \varphi_{1,m+1}\bigl(c_{m}^{+},\lambda\bigr) \varphi_{2,m+1}\bigl(c_{m}^{+},\mu \bigr) \bigr]. \end{aligned}$$
(2.27)
Then from (2.16) and the transmission conditions (2.20), we have
$$\begin{aligned}& \frac{\mu-\lambda}{r_{1}} \biggl[ \int_{a}^{c_{1}} \bigl(\varphi_{1,1}(x, \lambda)\varphi_{1,1}(x,\mu) +\varphi_{2,1}(x,\lambda) \varphi_{2,1}(x,\mu) \bigr) \,dx \biggr] \\& \qquad {} +(\mu-\lambda)\sum_{k=2}^{m} \frac{\prod_{i=1}^{k-1}D_{i}}{r_{k}} \biggl[ \int_{c_{k-1}}^{c_{k}} \bigl(\varphi_{1,k}(x, \lambda)\varphi _{1,k}(x,\mu) +\varphi_{2,k}(x,\lambda) \varphi_{2,k}(x,\mu) \bigr) \,dx \biggr] \\& \qquad {} +(\mu-\lambda) \biggl[\frac{\prod_{i=1}^{m}D_{i}}{r_{m+1}} \int_{c_{m}}^{b} \bigl(\varphi_{1,m+1}(x, \lambda)\varphi _{1,m+1}(x,\mu) +\varphi_{2,m+1}(x,\lambda) \varphi_{2,m+1}(x,\mu) \bigr) \,dx \biggr] \\& \quad =\prod_{i=1}^{m}D_{i} \bigl[\varphi_{1,m+1}(b,\lambda)\varphi _{2,m+1}(b,\mu)- \varphi_{1,m+1}(b,\lambda)\varphi_{2,m+1}(b,\mu)\bigr]. \end{aligned}$$
(2.28)
Dividing both sides of (2.28) by \((\lambda-\mu)\) and by letting \(\mu \rightarrow \lambda\), we arrive at the relation
$$\begin{aligned}& -\frac{1}{r_{1}} \biggl[ \int_{a}^{c_{1}} \bigl(\bigl\vert \varphi_{1,1}(x,\lambda)\bigr\vert ^{2}+\bigl\vert \varphi _{2,1}(x,\lambda)\bigr\vert ^{2} \bigr) \,dx \biggr] \\& \qquad {}-\sum _{k=2}^{m}\frac{\prod_{i=1}^{k-1}D_{i}}{r_{k}} \biggl[ \int_{c_{k-1}}^{c_{k}} \bigl(\bigl\vert \varphi_{1,k}(x,\lambda)\bigr\vert ^{2} +\bigl\vert \varphi_{2,k}(x,\lambda)\bigr\vert ^{2} \bigr) \,dx \biggr] \\& \qquad {} -\frac{\prod_{i=1}^{m}D_{i}}{r_{m+1}} \biggl[ \int_{c_{m}}^{b} \bigl(\bigl\vert \varphi_{1,m+1}(x,\lambda)\bigr\vert ^{2} +\bigl\vert \varphi_{2,m+1}(x,\lambda)\bigr\vert ^{2} \bigr) \,dx \biggr] \\& \quad =\prod_{i=1}^{m}D_{i} \biggl(\varphi_{2,m+1}(b,\lambda)\frac {\partial \varphi_{1,m+1}(b,\lambda)}{\partial\lambda}- \varphi_{1,m+1}(b, \lambda)\frac{\partial \varphi_{2,m+1}(b,\lambda)}{\partial\lambda} \biggr). \end{aligned}$$
(2.29)
We show that the equation
$$ \omega(\lambda)=-\prod_{i=1}^{m}D_{i} \bigl(\sin\beta\varphi _{1,m+1}(b,\lambda)+ \cos\beta\varphi_{2,m+1}(b, \lambda) \bigr)=0 $$
(2.30)
has only simple roots. Assume the converse, i.e., equation (2.30) has a double root \(\lambda^{\ast}\), say. Then the following two equations hold:
$$\begin{aligned}& \sin\beta\varphi_{1,m+1}\bigl(b,\lambda^{\ast} \bigr)+ \cos\beta\varphi_{2,m+1}\bigl(b,\lambda^{\ast}\bigr)=0, \end{aligned}$$
(2.31)
$$\begin{aligned}& \sin\beta\frac{\partial\varphi_{1,m+1}(b,\lambda^{\ast })}{\partial\lambda} +\cos\beta\frac{\partial\varphi_{2,m+1}(b,\lambda^{\ast })}{\partial\lambda}=0. \end{aligned}$$
(2.32)
Equations (2.31) and (2.32) imply that
$$ \varphi_{2,m+1}\bigl(b,\lambda^{\ast}\bigr) \frac{\partial \varphi_{1,m+1}(b,\lambda^{\ast})}{\partial\lambda}- \varphi_{1,m+1}\bigl(b,\lambda^{\ast}\bigr) \frac{\partial \varphi_{2,m+1}(b,\lambda^{\ast})}{\partial\lambda}=0. $$
(2.33)
Combining (2.33) and (2.29), with \(\lambda=\lambda^{\ast}\), we obtain
$$\begin{aligned}& \frac{1}{r_{1}} \biggl[ \int_{a}^{c_{1}} \bigl(\bigl\vert \varphi_{1,1}(x,\lambda)\bigr\vert ^{2}+\bigl\vert \varphi _{2,1}(x,\lambda)\bigr\vert ^{2} \bigr) \,dx \biggr] \\& \quad {} +\sum_{k=2}^{m}\frac{\prod_{i=1}^{k-1}D_{i}}{r_{k}} \biggl[ \int_{c_{k-1}}^{c_{k}} \bigl(\bigl\vert \varphi_{1,k}(x,\lambda)\bigr\vert ^{2} +\bigl\vert \varphi_{2,k}(x,\lambda)\bigr\vert ^{2} \bigr) \,dx \biggr] \\& \quad {} +\frac{\prod_{i=1}^{m}D_{i}}{r_{m+1}} \biggl[ \int_{c_{m}}^{b} \bigl(\bigl\vert \varphi_{1,m+1}(x,\lambda)\bigr\vert ^{2} +\bigl\vert \varphi_{2,m+1}(x,\lambda)\bigr\vert ^{2} \bigr) \,dx \biggr]=0. \end{aligned}$$
(2.34)
It follows that \(\varphi_{1}(x,\lambda^{\ast})=\varphi_{2}(x,\lambda^{\ast})=0\), which is impossible. This proves the lemma. □
Here \(\{\varphi(\cdot,\lambda_{n})\}_{n=-\infty}^{\infty}\) will be a sequence of vector-valued eigenfunctions of (2.1)-(2.4) corresponding to the eigenvalues \(\{\lambda_{n}\}_{n=-\infty}^{\infty}\). Since \(\chi(\cdot,\lambda)\) satisfies (2.3) and (2.4), then the eigenvalues are also determined via
$$ \sin\alpha\chi_{1,1}(a,\lambda)+\cos\alpha\chi_{2,1}(a,\lambda )=\omega(\lambda). $$
(2.35)
Therefore \(\{\chi(\cdot,\lambda_{n})\}_{n=-\infty}^{\infty}\) is another set of vector-valued eigenfunctions which is related by \(\{\varphi(\cdot,\lambda_{n})\}_{n=-\infty}^{\infty}\) with
$$ \chi(x,\lambda_{n})=\tau_{n}\varphi(x, \lambda_{n}),\quad x\in[a,c_{1})\cup(c_{1},c_{2}) \cup(c_{2},c_{3})\cup\cdots\cup (c_{m},b], n \in\mathbb{Z}, $$
(2.36)
where \(\tau_{n}\neq0\) are non-zero constants, since all eigenvalues are simple. Since the eigenvalues are all real, we can take the vector-valued eigenfunctions to be real valued.