Oscillation Properties for the Dirac Equation with Spectral Parameter in the Boundary Condition

Abstract

In this paper, we consider the boundary value problem for the one-dimensional Dirac system with spectral parameter in the boundary condition. We completely study the oscillatory properties of components of the eigenvector functions of this problem.

1 Introduction

In the present paper, we continue the study [7, 8] of one-dimensional Dirac equation

$$\begin{aligned} Bw'(x) - P(x)w(x) = \lambda w(x),\,\,0< x < \pi , \end{aligned}$$

(1)

with the boundary conditions

$$\begin{aligned} v (0)\cos \alpha + u(0)\sin \alpha =0, \end{aligned}$$

(2)

$$\begin{aligned} (\lambda \cos \beta + a_{1})\, v (\pi ) + (\lambda \sin \beta + b_{1})\, u\, (\pi ) = 0, \end{aligned}$$

(3)

where

$$\begin{aligned} B = \left( \begin{array}{l@{\quad }l} 0&{}1 \\ - 1&{}0 \\ \end{array} \right) ,\quad P(x) = \left( \begin{array}{l@{\quad }l} p(x)&{}0 \\ 0&{} r(x) \\ \end{array} \right) ,\quad w(x) = \left( \begin{array}{l} u(x) \\ v(x) \\ \end{array} \right) , \end{aligned}$$

$\lambda \in \mathbb {C}$ is a spectral parameter, the functions $p\,(x)$ and r(x) are continuous on the interval $[0,\, \, \pi ],$$\alpha ,\, \beta , \, a_{1}$ and $b_{1}$ are real constants such that $0\le \alpha ,\, \beta <\pi$ and

$$\begin{aligned} \sigma = a_{1} \sin \beta - b_{1} \cos \beta > 0. \end{aligned}$$

(4)

Should be noted that the boundary conditions of type (3) were first considered by Kerimov [24].

It is obvious that Eq. (1) is equivalent to the system of differential equations of first order

$$\begin{aligned} v '-\{ \lambda + p\,(x)\} \, u=0,\, \, \, u'+ \{ \lambda + r (x)\} \,v =0,\, \, 0<x<\pi \,. \end{aligned}$$

(5)

In the case when $p(x)=V(x)+m,\, \, r(x)=V(x)-m$, where V(x) is a potential function, and m is the mass of a particle, the system (1) (or -(5)) in relativistic quantum theory is known as the one-dimensional stationary Dirac system or the first canonical form of the Dirac system [28].

It is known [35] that the Dirac equation describes accurately the spectrum of the hydrogen atom and that it plays a central role in relativistic quantum mechanics and quantum field theory. The Dirac equation is related to a nonlinear wave equation (see [1, 35]), and this stimulated the increasing interest in direct and inverse problems for Dirac operators in both physical and mathematical literature (see [2] and the references therein).

Boundary value problems for ordinary differential equations with a spectral parameter in equations and boundary conditions were considered in different formulations by many authors (see [3,4,5,6,7, 10,11,12, 14, 15, 20, 24,25,26, 30,31,33, 36, 37, 39]). Problems of this type often appear in mathematical physics and mechanics (see [15, 31, 34, 36] and their bibliographies).

The oscillation properties of eigenfunctions of ordinary differential operators of second and higher orders have been studied in many papers (see, for example, [4, 5, 7, 8, 13, 14, 16, 18, 19, 21,22,23,24,25,26,27,28, 34, 39]). In the case when the spectral parameter is contained in the boundary conditions the oscillatory properties of eigenfunctions for the Sturm–Liouville problem were studied in [4, 14, 26] and for ordinary differential operators of fourth order—in [5, 25].

For the first time in paper [8], the oscillation properties of eigenvector functions of the one-dimensional Dirac operator (when the spectral parameter is not involved in the boundary conditions) are studied in detail, where the number of zeros of components of all eigenvector functions is found. It should be noted that these properties were previously investigated in [38] for the study of inverse nodal problems for Dirac operator (the inverse nodal problems for Sturm–Liouville and Dirac operators can be found in [17, 29, 39]), but the number of zeros of components of eigenvector functions found there does not correspond to reality.

Problem (1)–(3) is considered in [7], where, in particular, it was proved that the spectrum of this problem consists of an infinite set of real and simple eigenvalues with values ranging from $-\,\infty$ to $+\,\infty$ which can be numerated in increasing order. In this paper, we also studied the oscillatory properties of eigenfunctions and the location of eigenvalues on the real axis and obtained asymptotic formulas for the eigenvalues and eigenfunctions. But we could not completely investigate the oscillatory properties of components of the eigenvector functions of this problem. Moreover, the number of zeros of the components of eigenvector functions obtained earlier in [24, 39] for the Dirac problem with a spectral parameter in the boundary conditions is also inaccurate.

The subject of this paper is using Prüfer transformation to completely study the oscillation properties of components of eigenvector functions of the spectral problem (1)–(3).

This paper is organized as follows. In Sect. 2, we study the properties of the Pr$\ddot{\mathrm{u}}$fer angular function $\theta (x,\lambda )$ corresponding to the solution $w(x, \lambda ) = \left( \begin{array}{l} u (x,\lambda ) \\ v(x,\lambda ) \\ \end{array} \right)$ of the initial value problem for Eq. (1) in respect of their dependence on x and $\lambda$. In Sect. 3, we find a number of zeros of components of $w(x, \lambda )$ depending on the $\lambda \in \mathbb {R}$. In Sect. 4, we show that the eigenvalues of the problem (1)–(3) are real and simple. In Sect. 5, we prove the main result (Theorem 5) where is given the number of zeros of components of the eigenvector functions of problem (1)–(3).

2 Preliminaries

It is known [8] that there exists a unique solution $w(x,\lambda ) = \left( \begin{array}{l} u(x,\lambda ) \\ v(x,\lambda ) \\ \end{array} \right)$ of Eq. (1) satisfying the initial condition

$$\begin{aligned} u(0,\lambda ) = \cos \alpha ,\quad v(0,\lambda ) = - \sin \alpha ; \end{aligned}$$

(6)

moreover, for each fixed $x \in \left[ {0, \,\pi } \right]$, the functions $u(x,\lambda )$ and $v(x,\lambda )$ are entire functions of the argument $\lambda$ .

To study the oscillation properties of the eigenvectors of the functions of the problem (1)–(3), we will use the Pr$\ddot{\mathrm{u}}$fer angular function $\theta (x,\lambda ) = \tan ^{ - 1} ({v (x,\lambda )/ u(x,\lambda )})$, or more precisely,

$$\begin{aligned} \theta (x,\lambda ) = \arg \{ u(x,\lambda ) + iv(x,\lambda )\} \end{aligned}$$

(7)

(see [13, Ch. 8, § 4]). By virtue of (2.1), we can define $\theta$ as follows:

$$\begin{aligned} \theta (0,\lambda ) = - \alpha , \end{aligned}$$

(8)

For other x and $\lambda$, $\theta (x,\,\lambda )$ is given by (7) except for an arbitrary multiple of $2\pi$, since u and v cannot vanish simultaneously. This multiple of $2\pi$ is to be fixed so that $\theta (x,\,\lambda )$ satisfies (8) and is continuous in x and $\lambda$. Since $0 \le x \le \pi ,\, - \infty< \lambda < + \infty$, is simply connected, this defines $\theta (x,\,\lambda )$ uniquely.

Remark 1

It follows from (7) that the zeros of the functions $u(x, \lambda )$ and $v(x, \lambda )$ are the same as the occasions on which $\theta (x, \, \lambda )$ is an odd or even multiple of ${\pi /2}$, respectively.

Let us introduce the boundary condition

$$\begin{aligned} v(\pi )\cos \gamma + u(\pi )\sin \gamma = 0, \end{aligned}$$

(9)

where $\gamma \in [0, \pi )$.

Along with problem (1)–(3), consider the boundary value problem (1), (2), (9).

By $m\, (\lambda )$ and $s\, (\lambda ),\, \, \lambda \in \mathbb {R}$, we denote the number of zeros in the interval $(0,\pi )$ of functions $u(x,\lambda )$ and $v (x,\lambda ),$ respectively.

Theorem 1

(see [8, Theorem 3.1]) The eigenvalues $\mu _k = \mu _k (\alpha , \gamma ), \, k \in \mathbb {Z}$, of the problem (1), (2), (9) can be numbered in ascending order on the real axis so that the corresponding angular function $\theta (x, \mu _k)$ at $x= \pi$ satisfies the condition

$$\begin{aligned} \theta (\pi , \mu _k) = - \, \gamma + k\pi . \end{aligned}$$

(10)

The eigenvector functions $w(x,\mu _k) = \left( \begin{array}{l} u(x,\mu _k) \\ v(x,\mu _k) \\ \end{array} \right)$ have, with a suitable interpretation, the following oscillation properties (except for $k = 0$ in the cases $\alpha = \gamma = 0$ and $\alpha = \gamma = {\pi /2}$$)$:

$$\begin{aligned} \left( \begin{array}{l} m (\mu _k ) \\ s(\mu _k ) \\ \end{array}\right) = \left( \begin{array}{l} |k| - 1 + \kappa \left( {(\alpha - \frac{\pi }{2})\,\omega _ {\alpha ,\gamma } (k)} \right) + \kappa \left( ( {\frac{\pi }{2}} - \gamma )\,\omega _ {\alpha ,\gamma } (k) \right) \,\, \\ |k| - 1 + {\mathop \mathrm{sgn}} \alpha \,\, \kappa \left( {\omega _{\alpha ,\gamma } (k)} \right) + {\mathop \mathrm{sgn}} \gamma \,\,\kappa \left( { - \omega _ {\alpha ,\gamma } (k)} \right) \\ \end{array} \right) \end{aligned}$$

(11)

where the functions $\kappa (x)$ and $\omega _{\alpha ,\gamma } \left( {x} \right) ,\,x \in \mathbb {R}$, are defined as follows:

$$\begin{aligned} \begin{array}{l} \kappa (x) = \left\{ \begin{array}{l} \,0,\,\,\,\mathrm {if}\,\,\,x \le 0, \\ \,1,\,\,\,\mathrm {if}\,\,\,x> 0, \\ \end{array} \right. \,\, \,\, \omega _ {\alpha ,\gamma } \left( {x} \right) = \left\{ \begin{array}{l} - 1,\,\,\,\mathrm {if}\,\,\,x< 0\,\,\,\mathrm {or}\,\,\,x = 0,\,\,\alpha < \gamma , \\ \,\,\,\,\,1,\,\,\,\mathrm {if}\,\,\,x > 0\,\, \, \mathrm {or}\,\,\,x = 0,\,\,\alpha \ge \gamma . \\ \end{array} \right. \\ \end{array} \end{aligned}$$

(12)

Moreover, the functions $u_k (x, \mu _k)$ and $v_k (x, \mu _k),\, k \in \mathbb {Z}$, have only nodal zeros in the interval $(0,\pi )$.

Let $\eta _{k} = \lambda _k (\alpha , 0)$ and $\nu _{k} = \lambda _k (\alpha , \frac{\pi }{2}) ,\, \, k \in \mathbb {Z}$.

Remark 2

It follows from [8, Theorem 2.1] that as $\lambda$ increases, for fixed x, the function $\theta$ is increasing; in particular, $\theta (\pi ,\lambda )$ is a strictly increasing function of $\lambda$. Hence, it follows by (10) that for fixed $\alpha \in [0,\pi )$, the eigenvalues of problem are continuous, strictly decreasing functions of $\gamma$ for $\gamma \in (0,\pi )$.

Moreover, by virtue of (10), the following location of the eigenvalues of problem (1), (2), (9) on the real axis is true: If $\gamma \in (0,\pi /2)$, then

$$\begin{aligned} \cdots< \eta _{-2}< \nu _{-1}< \mu _{-1}< \eta _{-1}< \nu _{0}< \mu _{0}< \eta _{0}< \nu _{1}< \mu _{1}< \eta _{1} < \cdots , \end{aligned}$$

(13)

and if $\gamma \in (\pi /2,\pi )$, then

$$\begin{aligned} \cdots< \eta _{-2}< \mu _{-1}< \nu _{-1}<\eta _{-1}< \mu _{0}< \nu _{0}< \eta _{0}< \mu _{1}< \nu _{1}< \eta _{1} <\cdots . \end{aligned}$$

(14)

We define E to be the Banach space $C\left( {[0,\pi ];\,\mathbb {R}^2 } \right) \cap \mathrm{B.C.}$ with the usual norm $||w|| = \mathop {\max }\nolimits _{x \in [0,\pi ]} |u(x)| + \mathop {\max }\nolimits _{x \in [0,\pi ]} |v(x)|$, where $\mathrm{B.C.}$ is the set of functions satisfying the boundary conditions (2) and (9). Let $S = \left\{ {w \in E:\,|u(x) + |v(x)|} \right.$$\left. {> 0, \,x \in [0,\pi ]} \right\}$ be the subset of E with metric inherited from E.

For each $w = \left( \begin{array}{l} u \\ v \\ \end{array} \right) \in S$, we define $\theta (w, \cdot )$ to be the continuous function on $[0,\pi ]$ satisfying

$$\begin{aligned} \theta (w,x) = \arctan \frac{{v(x)}}{{u(x)}},\,\,\theta (w,0) = - \alpha \,. \ \end{aligned}$$

Let $S_k^{+},\,k \in \mathbb {Z}$, be set of $w \in S$ which satisfy the conditions:

• (i) $\theta (w,\pi ) = - \, \gamma + k \pi$;

• (ii) the function u(x) is positive in a deleted neighborhood of $x = 0$;

• (iii) if $k > 0$ or $k =0, \, \alpha \ge \gamma$ (except the cases $\alpha = \gamma = 0$ and $\alpha = \gamma = {\pi /2}$), then for fixed w, as x increases, the function $\theta$ cannot tend to a multiple of $\pi /2$ from above, and as x decreases, the function $\theta$ cannot tend to a multiple of ${\pi /2}$ from below; if $k < 0$ or $k = 0, \, \alpha < \gamma$, then for fixed w, as x increases, the function $\theta$ cannot tend to a multiple of $\pi /2$ from below, and as x decreases, the function $\theta$ cannot tend to a multiple of $\pi /2$ from above (see [5]).

Let $S_k^ - = - S_k^ +$ and $S_k = S_k^ - \cup S_k^ +$, $k \in \mathbb {Z}$.

Remark 3

It follows by [8, Theorem 2.1 and Theorem 3.1] that $w_{k} (x, \mu _k) = \left( \begin{array}{l} u_{k} (x, \mu _k) \\ v_{k} (x, \mu _k) \\ \end{array} \right)$$\in S_{k},\, k \in \mathbb {Z}$.

Let us consider the following equation (see (5) for $p\,(x) \equiv r(x) \equiv 0$)

$$\begin{aligned} v '(x) -\lambda \, u (x) = 0,\, \, \, u' (x) + \lambda v (x) = 0,\, \, 0< x < \pi . \end{aligned}$$

(15)

For fixed $\alpha \in [0,\pi )$, the eigenvalues of problem (15), (2), (9) are

$$\begin{aligned} \tau _k = \tau _k (\alpha , \gamma ) = k + {{(\alpha - \gamma )}/\pi },\,\,k \in \mathbb {Z}. \end{aligned}$$

(16)

Remark 4

From (16), we see that $\tau _k < 0$ if $k < 0$ and $\tau _k > 0$ if $k > 0$; moreover, $\tau _0 < 0$ for $\alpha < \gamma$, $\tau _0 = 0$ for $\alpha = \gamma$, $\tau _0 > 0$ for $\alpha > \gamma$.

Now consider the following boundary value problem

$$\begin{aligned} \left\{ {\begin{array}{l} v'(x) - \{ \lambda + \chi p(x)\} \,u(x) = 0,\\ u'(x) + \{ \lambda + \chi r(x)\} \,v(x) = 0,\,x \in (0,\pi ), \\ v(0)\cos \alpha + u(0)\sin \alpha = 0,\,v(\pi )\cos \gamma + u(\pi )\sin \gamma = 0, \\ \end{array} \,} \right. \end{aligned}$$

(17)

where $0\le \chi \le 1$.

Remark 5

The assertions of Theorem 1 are true for problem (17). Moreover, for the eigenvalues $\mu _{k} (\chi ),\, k \in \mathbb {Z}$, of problem (17) also are true the relations (13) and (14). By the continuous dependence of the solutions of system of differential equations on the parameter, we find that the eigenvalues $\mu _{k} (\chi ),\, k \in \mathbb {Z}$, of the problem (17) depends continuously on the parameter $\chi \in [0,\, 1]$. In this case $\mu _{k} (0)$ and $\mu _{k} (1),\, k \in \mathbb {Z}$, coincide with the eigenvalues $\tau _k$ and $\mu _k$ of the problems (15), (2), (9) and (1), (2), (9), respectively. It is obvious that transformation $\mu _{0} (\chi )$ for $\alpha = \gamma$ maps $\tau ^*$ to $\mu ^*$, where we denote by $\tau ^*$ and $\mu ^*$ the eigenvalues $\tau _{0}$ and $\mu _{0}$ of the spectral problems (15), (2), (9) and (1), (2), (9) for $\alpha = \gamma$, respectively. Hence by Remark 4, it follows that $\mu ^* = \eta _{0}$ if $\alpha = 0$, $\nu _{0}< \mu ^* < \eta _{0}$ if $\alpha \in \left( {0,\pi /2} \right)$, $\mu ^* = \nu _{0}$ if $\alpha = \pi /2$, $\eta _{- 1}< \mu ^* < \nu _{0}$ if $\alpha \in \left( {\pi /2}, \pi \right)$.

Also, we need the following result which plays a fundamental role in the study of oscillatory properties of eigenvector functions of the problem (1)–(3).

Theorem 2

(i) $\theta (x, \lambda )$ satisfies the differential equation, with respect to x,

$$\begin{aligned} \theta ' = \lambda + p\cos ^2 \theta + r\sin ^2 \theta \,; \end{aligned}$$

(18)

(ii) if $\,\lambda > \mu ^*$, then as x increases, $\theta$ cannot tend to a multiple of $\pi /2$ from above, and as x decreases, $\theta$ cannot tend to a multiple of ${\pi /2}$ from below; if $\,\lambda < \mu ^*$, then as x increases, $\theta$ cannot tend to a multiple of ${\pi /2}$ from below, and as x decreases, $\theta$ cannot tend to a multiple of $\pi /2$ from above; (iii) as $\lambda$ increases, for fixed x, $\theta$ is increasing; in particular, $\theta (\pi ,\lambda )$ is a strictly increasing function of $\lambda$.

Proof

Proof The statements (i), (iii) and first part of statement (ii) for $\lambda + p\,(x)> 0,\,\lambda + r(x) > 0$, $x \in \left[ {0,\,\pi } \right]$ and second part of statement (ii) for $\lambda + p\,(x)< 0,\,\lambda + r(x) < 0$, $x \in \left[ {0,\,\pi } \right]$ of this theorem follow from [8, Theorem 2.1] (see also [13, Ch. 8, Theorem 8.4.3]).

Let

$$\begin{aligned} m_{ - 1} = \max \left\{ {k \in \mathbb {Z}:\,\mu _k + p\,(x)< 0,\,\,\mu _k + r(x) < 0,\,\,\,x \in [0,\pi ]} \right\} , \end{aligned}$$

$$\begin{aligned} m_{1} = \min \left\{ {k \in \mathbb {Z}:\,\,\mu _k + p(x)> 0,\,\,\mu _k + r(x) > 0,\,\,\,x \in [0,\pi ]} \right\} . \end{aligned}$$

It follows by (13), (14), Remarks 4 and 5 that $\mu ^* \in (\mu _{m_{-1}}, \mu _{m_{1}})$. Therefore, by (13), (14), it remains to prove the assertion (ii) of Theorem 2 in the case when $\lambda \in [\mu _{m_{-1} - 1}, \mu _{m_{1}}]$.

Assume that $\alpha \in \left( 0,\pi /2\right)$ and $\lambda \in \left[ {\mu ^*, \mu _{m_1 }} \right]$. Then, it follows by Remark 2 that either $\lambda \in \left[ {\mu ^*, \eta _0} \right]$, or $\lambda \in \left( {\eta _{k - 1} ,\eta _k } \right] \,$ for some $\,k \in \mathbb {Z}\,$ such that $1 \le k \le m_1$.

Let $\lambda \in \left[ {\mu ^*, \eta _0 } \right]$. We now define the angle of $\gamma _{\lambda } \in [0, \alpha ]$ (see Remark 2) by the equality $\gamma _\lambda = - \tan ^{-1} \frac{{v (\pi ,\lambda )}}{{u(\pi ,\lambda )}}$. Then, the vector function $w (x,\lambda ) = \left( \begin{array}{l} u(x,\lambda ) \\ v(x,\lambda ) \\ \end{array} \right)$ is an eigenvector function, corresponding to the eigenvalue $\lambda = \mu _{0}$ of problem (1), (2), (9) for $\gamma = \gamma _{\lambda }$. Hence, by Remark 3 we have $w(x,\lambda ) \in S_{0}$. Then, it follows from the definition of $S_{0}$ that the assertion (ii) of Theorem 2 holds for $\theta (x, \lambda ) = \theta (w (x,\lambda ),x)$ as $\lambda \in \left( {\mu ^*, \eta _0} \right]$ .

Now let $\lambda \in \left( {\eta _{k - 1} ,\eta _k } \right]$, where $1 \le k \le m_1$. Again determine the angle of $\gamma _{\lambda } \in [0, \pi )$ from an equality $\gamma _\lambda = - \tan ^{-1} \frac{{v(\pi ,\lambda )}}{{u(\pi ,\lambda )}}\,$. Then the vector function $w (x,\lambda ) = \left( \begin{array}{l} u (x,\lambda ) \\ v (x,\lambda ) \\ \end{array} \right)$ is an eigenvector function, corresponding to the eigenvalue $\lambda = \mu _{k}, \,\,1 \le k \le m_1$, of problem (1), (2), (9) at $\gamma = \gamma _{\lambda }$. Hence, it follows by Remark 3 that $w (x,\lambda ) \in S_{k}$. Since $1 \le k \le m_{1}$, it follows from the definition of $S_{k}$ that the assertion (ii) of Theorem 2 is true for $\theta (x,\lambda ) = \theta (w (x,\lambda ),x)$ as $\lambda \in \left( {\eta _{k - 1} ,\eta _k } \right]$.

The remaining cases are considered in the same way. The proof of Theorem 2 is complete.

Remark 6

It should be noted that the Remarks 2.2, 2.3 and Theorem 2.2 from [8] are true for any $\lambda \in \mathbb {R}$.

3 Some Properties of Solution of Problem (1)–(2)

The function $F (\lambda ) = \frac{u\, (\pi ,\lambda )}{\vartheta (\pi ,\lambda )}$ is defined for

$$\begin{aligned} \lambda \in \mathrm{B} \equiv \left( \mathbb {C} \backslash \mathbb {R} \right) \bigcup \left( \bigcup _{k = -\infty }^{+ \infty }(\eta _{k-1} ,\eta _{k}) \right) \end{aligned}$$

and is meromorphic function finite order, and $\eta _{k}$ and $\nu _{k} ,\, \, k \in \mathbb {Z}$, are poles and zeros of this function, respectively.

Lemma 1

[7, Lemma 2.4] The following formula holds

$$\begin{aligned} \frac{{dF(\lambda )}}{{d\lambda }} = - \frac{\int _{0}^{\pi }\{ u^{2} (x,\lambda ) + v ^{2} (x,\lambda )\} \, \mathrm{d}x }{v^{2} (\pi ,\lambda )}, \, \, \lambda \in \mathrm{B} . \end{aligned}$$

(19)

Corollary 1

The function $F (\lambda )$ is continuous and strictly decreasing on each interval $(\eta _{k-1} ,\eta _{k} ),\, k \in \mathbb {Z}$.

We define numbers $m_k$ and $s_k$, $k \in \mathbb {Z}$, as follows:

$$\begin{aligned} m_{k} = |\,k| - 1 + \kappa \left( {(\alpha - \frac{\pi }{2})\,\omega _{\alpha ,\, \frac{\pi }{2}} (k)} \right) ,\,\,\, s_{k} = |\,k| - 1 + {\mathop \mathrm{sgn}} \alpha \,\kappa (\omega _{\alpha ,\,0} (k)). \end{aligned}$$

Let

$$\begin{aligned} I_k = \left\{ \begin{array}{l@{\quad }l} [\eta _{k}, \eta _{k + 1})&{} \mathrm{if} \;k< 0,\\ \eta _{0}&{}\mathrm{if} \; k = 0,\\ (\eta _{k - 1}, \eta _k]&{} \mathrm{if} \; k> 0, \\ \end{array} \right. \quad J_k = \left\{ \begin{array}{l@{\quad }l} [\nu _{k}, \nu _{k + 1})&{} \mathrm{if} \;k < 0,\\ \nu _{0} &{} \mathrm{if} \; k = 0,\\ (\nu _{k - 1}, \nu _k]&{} \mathrm{if} \; k > 0,\\ \end{array} \right. k \in \mathbb {Z}. \end{aligned}$$

For the investigation of oscillatory properties of components of eigenvector functions of problem (1)–(3), we need the following result.

Theorem 3

If $\,\lambda \in J_{k},\,\,k \in \mathbb {Z}$, then $m (\lambda ) = m_{k}$, and if $\,\lambda \in I_{k},\,\,k \in \mathbb {Z}$, then $s (\lambda ) = s_{k}$.

Proof

Let $\alpha \in (0, \frac{\pi }{2})$. Then, by Remark 5, we have $\mu ^* \in \mathrm{int} \,J_{1}$. Assume that $\lambda > \mu ^*$ ($\lambda \le \mu ^*$). By (8), we obtain the equality $\theta (0, \lambda ) = - \,\alpha \in \left( { - \frac{\pi }{2},\,0} \right)$. In view of (10), we have $\theta (\pi ,\nu _k) = - \frac{\pi }{2} + k \pi \,$. As is known (see statement (ii) of Theorem 2 and Remark 6), if $\lambda > \mu ^*$ ($\lambda \le \mu ^*$) the function $\theta (x,\lambda )$, which is strictly increasing (decreasing), assumes the values ${k\pi /2},\,k \ge 0$ ($\,k \le -1$). Hence, it follows by statement (iii) of Theorem 2 that $\theta (x,\lambda ) \in \left( -\pi /2,\pi /2 \right]$ for $\lambda \in \left( {\mu ^*, \nu _1} \right]$, $\theta (x,\lambda ) \in \left( { - \pi /2,(k - 1/2)\pi } \right]$ for $\lambda \in J_{k},\,k \ge 2$ ($\theta (x,\lambda ) \in \left( - \pi /2,0\right)$ for $\lambda \in \left( {\nu _0 ,\mu ^* } \right]$, $\theta (x,\lambda ) \in \left[ {\left( {k - {1/2}} \right) \pi ,\,0} \right)$ for $\lambda \in J_{k}, \,k \le 0$). This proves the result of Theorem 3 for $\alpha \in (0, \frac{\pi }{2})$ and $\lambda \in J_{k},\, k \in \mathbb {Z}$.

The remaining cases are considered similarly. The proof of Theorem 3 is complete.

4 The Main Properties of Eigenvalues of Problem (1)–(3)

Theorem 4

The eigenvalues of the boundary value problem (1)–(3) are real and simple and form an at most countable set without finite limit points.

Proof

The eigenvalues of problem (1)–(3) are the roots of the equation

$$\begin{aligned} (\lambda \cos \beta + a_{1}) v(\pi , \lambda ) + (\lambda \sin \beta + b_{1}) u\, (\pi , \lambda ) = 0. \end{aligned}$$

(20)

Let $\lambda$ be a nonreal eigenvalue of problem (1)–(3). Then, $\bar{\lambda }$ is also an eigenvalue of this problem, since the coefficients $p(x), r(x), \alpha , \beta , \,a_{1}$ and $b_{1}$ are real; moreover, $w (x, \bar{\lambda }) = \overline{w (x, \lambda )}$, i.e., $u (x, \bar{\lambda }) = \overline{u (x, \lambda )}$ and $v (x, \bar{\lambda }) = \overline{v (x, \lambda )}$.

By [7, formula (2.10)], we have

$$\begin{aligned}&v(\pi ,\bar{\lambda })u\, (\pi ,\lambda )-u\, (\pi ,\bar{\lambda })\, v(\pi , \lambda )\nonumber \\&\quad =(\bar{\lambda }-\lambda )\int _{0}^{\pi }\left\{ |u\, (x, \lambda )|^2 + |v(x, \lambda )|^2 \,\right\} \mathrm{d}x. \end{aligned}$$

(21)

It follows from (20) that $v (\pi ,\lambda ) = - \frac{{\lambda \sin \beta + b_1 }}{{\lambda \cos \beta + a_1 }} \, u\,(\pi ,\lambda )$. Taking this relation into account we obtain

$$\begin{aligned} v(\pi ,\bar{\lambda })\,u\,(\pi ,\lambda ) - u(\pi ,\bar{\lambda })\,v\,(\pi ,\lambda ) = \frac{\left( {\bar{\lambda }- \lambda } \right) \sigma }{{|\lambda \cos \beta + a_1 |^2 }}\,|u\,(\pi ,\lambda )|^2. \end{aligned}$$

(22)

Since $\bar{\lambda }\ne \lambda$, it follows by (21) and (22) that

$$\begin{aligned} - \frac{\sigma }{{|\lambda \cos \beta + a_1 |^2 }}|u\,(\pi ,\lambda )|^2 = \int _{0}^{\pi }\left\{ |u\, (x, \lambda )|^2 + |v(x, \lambda )|^2 \,\right\} \mathrm{d}x. \end{aligned}$$

which contradicts condition (4). Therefore, $\lambda \in \mathbb {R}$.

The entire function occurring on the left-hand side in Eq. (20) does not vanish for nonreal $\lambda$. Consequently, it does not vanish identically. Therefore, its zeros form an at most countable set without finite limit points.

Let us show that Eq. (20) has only simple roots. Indeed, if $\lambda$ is a multiple root of Eq. (20), then

$$\begin{aligned}&v(\pi ,\lambda )\cos \beta + \left( {\lambda \cos \beta + a_1 } \right) \frac{\partial }{{\partial \lambda }}v(\pi ,\lambda ) \nonumber \\&\quad +\, u(\pi ,\lambda )\sin \beta + \left( {\lambda \sin \beta + b_1 } \right) \frac{\partial }{{\partial \lambda }}u(\pi ,\lambda ) = 0. \end{aligned}$$

(23)

We rewrite relation (15) from [7] in the form

$$\begin{aligned} \{ v(\pi ,\mu ) - v\,(\pi ,\lambda )\} \,u\,(\pi ,\lambda ) - \nu \,(\pi ,\lambda )\{ u\,(\pi ,\mu ) - u\,(\pi ,\lambda )\}\\ = (\mu -\lambda )\int _{0}^{\pi }\left\{ u\, (x,\mu )\, u\, (x,\lambda ) + v(x,\mu )\, v(x,\lambda )\right\} \mathrm{d}x. \end{aligned}$$

Dividing both sides of this relation by $\mu -\lambda$ ($\mu \ne \lambda$) and by passing to the limit as $\mu \rightarrow \lambda$, we obtain

$$\begin{aligned} u\,(\pi ,\lambda )\frac{\partial }{{\partial \lambda }}v\,(\pi ,\lambda )\, - \nu \,(\pi ,\lambda )\frac{\partial }{{\partial \lambda }}u(\pi ,\lambda )\, = \int _{0}^{\pi }\{ u^{2} (x,\lambda ) + v^{2} (x,\lambda )\} \, \mathrm{d}x. \end{aligned}$$

(24)

Since $\sigma \ne 0$ (see condition (5)), we have $(\lambda \cos \beta + a_{1})^2 +(\lambda \sin \beta + b_{1})^2 \ne 0$. Suppose that $(\lambda \cos \beta + a_{1}) \ne 0$. Then, by expressing $v (\pi , \lambda )$ and $\frac{\partial }{{\partial \lambda }}v\,(\pi ,\lambda )\,$ from (20) and (23), respectively, and by substituting them into relation (24), we obtain

$$\begin{aligned} -\,\frac{\sigma }{{(\lambda \cos \beta + a_1 )^2 }} \, u^2 \,(\pi ,\lambda ) = \int _{0}^{\pi }\{ u^{2} (x,\lambda ) + v^{2} (x,\lambda )\} \, \mathrm{d}x, \end{aligned}$$

which is impossible in view of condition (4).

The case in which $(\lambda \sin \beta + b_{1}) \ne 0$ can be considered in a similar way. The proof of Theorem 4 is complete.

5 Oscillation Properties of the Eigenvector Functions of Problem (1)–(3)

For $\beta \ne 0$, let $N_{1}$ be the integer such that $- \frac{{b_1 }}{{\sin \beta }} \in I_{N_1}$, and for $\beta \ne {\pi / 2}$, let $N_{2}$ be the integer such that $- \frac{{a_1 }}{{\cos \beta }} \in J_{N_2}$.

Remark 7

It follows by (5) that $N_{2} \le N_1 + 2 - \mathrm{sgn}\, |N_1 + 1|$ if $\beta \in \left( {0,\pi /2} \right)$, and $N_{2} \ge N_1 - 1$ if $\beta \in \left( {\pi /2}, \pi \right)$.

We define the functions $\tilde{\kappa }(x), \, \kappa _{1} (x)$ and $\tilde{\kappa }_{1} (x),\,x \in \mathbb {R}$, as follows:

$$\begin{aligned} \tilde{\kappa }(x) = 1 - \kappa (- x),\,\kappa _{1} (x) = 2 \kappa (x) - 1, \tilde{\kappa }_{1} (x) = 2 \tilde{\kappa }(x) - 1. \end{aligned}$$

Theorem 5

The eigenvalues $\lambda _{k},\, k \in \mathbb {Z}$, of the boundary value problem (1)–(3) are the values ranging from $- \infty$ to $+ \infty$ and can be numerated in increasing order:

$$\begin{aligned} \cdots< \lambda _{ - k}< \cdots< \lambda _{ - 1}< \lambda _0< \lambda _1< \cdots< \lambda _k < \cdots , \end{aligned}$$

where $\lambda _{0}$ is contained in $\left( {\mu _{ - 1} ,\mu _0 } \right]$ and is the closest to $\mu _{0}$.

Eigenvector functions $w_k (x) = w(x,\lambda _k ) = \left( \begin{array}{l} u_k (x,\lambda _k ) \\ v_k (x,\lambda _k ) \\ \end{array} \right) = \left( \begin{array}{l} u_k (x) \\ v_k (x) \\ \end{array} \right)$, $k \in \mathbb {Z}$, have, with a suitable interpretation, the following oscillation properties:

(a) if $\beta =0$, then

$$\begin{aligned} m(\lambda _k ) = \left\{ \begin{array}{l} m_k \,\,\,for\,\,\,k \kappa _{1} (N_2) < 0\,\,and\,\,\, k \kappa _{1} (N_2) \ge N_{2} \kappa _{1}(N_2) ,\, \\ m_{k - \kappa _{1} (N_2 )} \,\,\,for\,\,\, 0 \le k \kappa _{1} (N_2) \le (N_2 - \kappa _{1} (N_2)) \kappa _{1} (N_2),\, \\ \end{array} \right. \end{aligned}$$

$$\begin{aligned} s(\lambda _k ) = s_{k - 1 + \kappa (k)}; \end{aligned}$$

(b) if $\beta \in \left( {0,\frac{\pi }{2}} \right)$, then

$$\begin{aligned} m(\lambda _k ) = \left\{ \begin{array}{l} \left\{ \begin{array}{l} m_{k + 1} \,\,\,\,for\,\,\,\,k \le \,N_2 - 1\,\,and\,\,\,k \ge 0, \\ m_k \,\,\,\,\,\,\,\,\,for\,\,\,N_2 \le k< 0 \\ \end{array} \right. \,\,\,if\,\,N_2 < 0, \\ \\ \,\,\,m_{k + 1} \,\,\,if\,\,\,N_2 = 0,\,\, \\ \\ m_{k + 2 (1 - |sgn (k + 1)|)} \,\,\,if\,\,N_2 = 1,\\ \end{array} \right. \end{aligned}$$

in the case $\,N_{1} \le 0$;

$$\begin{aligned} m(\lambda _k ) = \left\{ \begin{array}{l} m_k \,\,\,for\,\,\,k \tilde{\kappa }_{1} (N_2) < 0\,\,and\,\,\, k \tilde{\kappa }_{1} (N_2) \ge N_{2} \tilde{\kappa }_{1}(N_2) ,\, \\ m_{k - \tilde{\kappa }_{1} (N_2)} \,\,\,for\,\,\, \tilde{\kappa }_{1} (N_2) (N_2 - \tilde{\kappa }_{1} (N_2)) \le k \tilde{\kappa }_{1} (N_2) \le 0,\, \\ \end{array} \right. \end{aligned}$$

in the case $\,N_{1} > 0$;

$$\begin{aligned} s(\lambda _k ) = s_{k - \chi (N_1 )} \,\,for\,\, k \kappa _{1} (N_1) < \kappa (N_{1}) -1 \,\,and \,\,k \kappa _{1} (N_1 ) \ge (N_1 + \kappa (N_1 )) \kappa _{1} (N_1), \end{aligned}$$

$$\begin{aligned} s(\lambda _k ) = s_{k - 1 + \kappa (N_1 )} \,\, for \,\,\kappa (N_1) \le k \kappa _{1} (N_1) \le (N_1 + 1 - \kappa (N_1)) \kappa _{1} (N_1); \\ \end{aligned}$$

(c) if $\beta = {\frac{\pi }{2}}$, then

$$\begin{aligned} m(\lambda _k ) = m_{k + (1 - \kappa (N_{1}))(1 - \kappa (- k)) - \kappa (N_{1}) (1 -\kappa (k))}, \end{aligned}$$

$$\begin{aligned} s(\lambda _k ) = s_{k - \chi (N_1 )} \,\,for \,\, k \kappa _{1} (N_1) < \kappa (N_{1}) -1\,\,and \,\,k \kappa _{1} (N_1 ) \ge (N_1 + \kappa (N_1 )) \kappa _{1} (N_1), \end{aligned}$$

$$\begin{aligned} s(\lambda _k ) = s_{k - 1 + \kappa (N_1 )} \,\,\,for\,\,\,\kappa (N_1) \le k \kappa _{1} (N_1) \le (N_1 + 1 - \kappa (N_1)) \kappa _{1} (N_1); \end{aligned}$$

(d) if $\beta \in \left( {\frac{\pi }{2},\pi } \right)$, then

$$\begin{aligned} m(\lambda _k ) = \left\{ \begin{array}{l} m_k \,\,for\,\,k \kappa _{1} (N_2) < 0\,\,and\,\, k \kappa _{1} (N_2) \ge N_{2} \kappa _{1}(N_2) ,\, \\ m_{k - \kappa _{1} (N_2 )} \,\,for\,\, (N_2 - \kappa _{1} (N_2)) \kappa _{1} (N_2) \le k \kappa _{1} (N_2) \le 0,\, \\ \end{array} \right. \end{aligned}$$

in the case $\,N_{1} \le 0$,

$$\begin{aligned} m(\lambda _k ) = \left\{ \begin{array}{l} m_{k - 1} \,\,\,\,for\,\,\,k \le 0\,\,and\,\,k \ge N_2 + 1, \\ m_k \,\,\,\,\,\,\,\,\,for\,\,0 < k \le N_2 , \\ \end{array} \right. \end{aligned}$$

in the case $\,N_{1} >0$;

$$\begin{aligned} s(\lambda _k) = s_{k - \chi (N_1 )} \,\,for\,\, k \kappa _{1} (N_1) < \kappa (N_{1}) -1 \,and\,\,k \kappa _{1} (N_1 ) \ge (N_1 + \kappa (N_1 )) \kappa _{1} (N_1), \end{aligned}$$

$$\begin{aligned} s(\lambda _k ) = s_{k - 1 + \kappa (N_1 )} \,\,for\,\,\kappa (N_1) \le k \kappa _{1} (N_1) \le (N_1 + 1 - \kappa (N_1)) \kappa _{1} (N_1). \end{aligned}$$

Proof

By Lemma 1 and Corollary 1 (see (19)), $F (\lambda ) = {{u (\pi ,\lambda )} /{v (\pi ,\lambda )}}$ is decreasing function in the interval $(\mu _{k-1}, \mu _{k}),\,k \in \mathbb {Z}$. Taking account of the relations $v (\pi , \mu _{k}) = 0, \,\, k \in \mathbb {Z}$, we obtain

$$\begin{aligned} \mathop {\lim }\limits _{\lambda \rightarrow \mu _{k - 1} + \,0} F (\lambda ) = + \,\,\infty ,\,\,\mathop {\lim }\limits _{\lambda \rightarrow \mu _k - \,\,0} F (\lambda ) = - \,\,\infty \,. \end{aligned}$$

Hence, the function $F (\lambda )$ takes each value in $(- \infty , + \infty )$ at a unique point in the interval $(\mu _{k - 1}, \mu _{k}), \, k \in \mathbb {Z}$. For the function $G (\lambda ) = - {{(\lambda \cos \beta + a_1 )}/{(\lambda \sin \beta + b_1 )}}$, we have $G '(\lambda ) = {\sigma /{(\lambda \sin \beta + b_1 )}}^2$. Since $\sigma > 0$, it follows that for $\beta = 0$ the function $G (\lambda )$ is strictly increasing in the interval $(- \infty , + \infty )$; for $\beta \in (0,\,\pi )$, the function $G (\lambda )$ is increasing in both intervals $(- \infty , - {{b_1 }/{\sin \beta }})$ and $(- {{b_1 }/ {\sin \beta }},\, + \,\infty )$, and we have

$$\begin{aligned} \mathop {\lim }\limits _{\lambda \rightarrow - \,{{b_1 } / {\sin \beta }} - \,0} G(\lambda ) = + \,\,\infty ,\,\,\mathop {\lim }\limits _{\lambda \rightarrow - \,{{b_1 }/ {\sin \beta }}\, + \,0} G (\lambda ) = - \,\,\infty \,. \end{aligned}$$

Assume that either $\beta = 0$, or $\beta \in (0,\pi )$ and $N_{1} =0$, or $\beta \in (0, \pi )$ and $k \ne N_{1},\, N_{1} \ne 0$. It follows from the above that in the interval ${\mathop \mathrm{int}} I_k = (\mu _{k-\kappa (k)}, \mu _{k + 1-\kappa (k)}),\, k \in \mathbb {Z} \backslash \{0\}$, there exists a unique $\lambda = \lambda ^*_{k}\,$ such that

$$\begin{aligned} F (\lambda ) = G (\lambda ), \end{aligned}$$

(25)

so that (3) holds. Therefore, $\lambda ^*_{k}\,$ is an eigenvalue of the boundary value problem (1)–(3), and $w (x, \lambda ^*_{k}) = \left( \begin{array}{l} u (x, \lambda ^{*}_{k}) \\ v (x, \lambda ^*_{k}) \\ \end{array} \right)$ is the corresponding eigenvector function. In this case we denote by $\lambda _{0}$ the eigenvalue of the problem (1)–(3) contained in the half-open interval $\left( {\mu _{-1} ,\mu _{0}} \right]$ and is the closest to $\mu _{0}$. Hence, one can readily see that (i) if $\beta = 0$, then $\lambda _{k}^*$ is the $(k + |\,\kappa (k) - 1|)$th eigenvalue of the boundary value problem (1)–(3), i.e., $\lambda _{k} = \lambda _{k - |\kappa (k) - 1|}^*$; (ii) if $\beta \ne 0$ and $N_{1} = 0$, then $\lambda _{k}^*$ is the kth eigenvalue of problem (1)–(3) for $k \ne 0$, and here, we use the fact that $\mu _{N_{1}} = \mu _{0}$ is also an eigenvalue of this problem which by above we denote by $\lambda _{0}$ and therefore $\lambda _{k} = \lambda _{k}^*$ for $k \ne 0$ and $\lambda _{0} = \mu _{0}$; (iii) if $\beta \in (0,\pi )$ and $k \ne N_{1},\, N_{1} \ne 0$, then in the case $N_{1} < 0$ we have that $\lambda _{k}^*$ for $k > 0$ is the kth eigenvalue, for $N_{1}< k < 0$ is the $(k + 1)$th eigenvalue of problem (1)–(3), in the case $N_{1} > 0$ we have that $\lambda _{k}^*$ for $k < 0$ is the $(k + 1)\,$th eigenvalue, for $0< k < N_{1}$ is the kth eigenvalue of problem (1)–(3); therefore,

$$\begin{aligned} \lambda _k = \left\{ \begin{array}{l} \lambda _k^ * \,\,\quad \mathrm{for}\,\,\,k > 0, \\ \lambda _{k - 1}^* \,\mathrm{for}\,\,\,N_1 + 1< k \le 0, \\ \end{array} \right. \,\,{ \mathrm if}\,\, N_1 < 0; \end{aligned}$$

$$\begin{aligned} \lambda _k = \left\{ \begin{array}{l} \lambda _{k - 1}^* \,\,\mathrm{for}\,\,\,k \le 0, \\ \lambda _k^ * \,\,\,\,\,\,\,\, \mathrm{for}\,\,\,0< k < N_1, \\ \end{array} \right. \,\,\,{ \mathrm if}\,\,\, N_1 > 0. \end{aligned}$$

Assume that $\beta \in (0, \pi )$ and $- \frac{b_{1}}{{\sin \beta }} \in \mathrm{int} I_{N_{1}}$ for $N_{1} \in \mathbb {Z} \backslash \{0\}$. In a similar way, one can show that in each of the intervals $(\mu _{N_{1} - \kappa (N_{1})}, {- {{b_1 } / {\sin \beta }}})$ and ${( - {{b_1 }/ {\sin \beta }},\,\mu _{N_{1} - \kappa (N_{1}) + 1 } )}$ there exists a unique point ($\,\lambda _{N_{1}}$ and $\lambda _{N_{1} + 1}$, respectively) such that relation (3) holds.

The case of $\beta \in (0, \pi )$ and $- \frac{{b_1 }}{{\sin \beta }} = \mu _{N_{1}}$ is perfectly similar; here, we use the fact that $\mu _{N_{1}}$ is also an eigenvalue of the boundary value problem (1)–(3). In this case, we have $\lambda _{N_{1}} = \mu _{N_{1}}$ and $\lambda _{N_{1} + 1} \in \mathrm{int} I_{N_{1}}$ if $N_{1} < 0$, $\lambda _{N_{1}} \in \mathrm{int} I_{N_{1}}$ and $\lambda _{N_{1} + 1} = \mu _{N_{1}}$ if $N_{1} > 0$.

We observe that for $\beta \in (0, \pi )$ and $k < N_{1}$ in the case $N_{1} < 0$ ($k > N_{1}$ in the case $N_{1} > 0$) the unique solution $\lambda ^*_{k}$ of equation (25) in $I_{k}$ is the kth ($(k+1)$th) eigenvalue of the boundary value problem (1)–(3), i.e., $\lambda _{k} = \lambda _{k}^*$ ($\lambda _{k} = \lambda _{k - 1}^*$).

Next, taking into account definition of number $N_{2}$, we obtain the following location of eigenvalues of problem (1)–(3) on the real axis:

$\mathrm (i)$ if $\beta = 0$, then

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} \left\{ \begin{array}{l} J_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,\;k \le N_2 \,\,\,\mathrm{and}\,\,k> 0, \\ J_{k - 1} \,\,\mathrm{for}\,\,N_2< k \le 0 \\ \end{array} \right. \,\,\,\,\mathrm{if}\,\,N_2 \le 0, \\ \\ \left\{ \begin{array}{l} J_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,k< 0\,\,\,\mathrm{and}\,\,k \ge N_2 , \\ J_{k + 1} \,\,\mathrm{for}\,\,\;0 \le \,\,k < N_2 \\ \end{array} \right. \,\,\,\,\mathrm{if}\,\,N_2 > 0, \\ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} I_{k - 1}, \,\,\mathrm{for}\,\,\;k \le 0, \\ I_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,\,k > 0 \\ \end{array} \right. ; \end{aligned}$$

$\mathrm (ii)$ if $\beta \in \left( {0,\frac{\pi }{2}} \right)$, then

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} \left\{ \begin{array}{l} J_{k + 1} \,\,\,\,\,\mathrm{for}\,\,\, k< N_2 \,\,\,\mathrm{and}\,\,k \ge 0, \\ J_k \,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,N_2 \le k< 0, \\ \end{array} \right. \,\,\,\,\,\mathrm{if}\,\,N_2 < 0, \\ \\ \,\,\,J_{k + 1} \,\,\,\,\mathrm{if}\,\,N_2 = 0, \\ \\ \left\{ \begin{array}{l} J_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,\,k \ne - 1, \\ J_{k + 2} \,\,\mathrm{for}\,\,\,k = - 1, \\ \end{array} \right. \,\,\,\,\,\mathrm{if}\,\,N_2 = 1, \\ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} I_{k - 1} \,\,\mathrm{for}\,\,\,k \le N_1 \,\,\,\mathrm{and}\,\,\,k > 0, \\ I_k \,\,\,\,\,\,\,\mathrm{for}\,\,\;N_1 < k \le 0,\, \\ \end{array} \right. \end{aligned}$$

in the case $N_{1} \le 0$;

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} \left\{ \begin{array}{l} J_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,\,k \le N_2 \,\,\,\mathrm{and}\,\,k > 0, \\ J_{k - 1} \,\,\mathrm{for}\,\,N_2< k \le 0 \\ \end{array} \right. \,\,\,\,\,\mathrm{if}\,\,N_2< 0, \\ \\ \left\{ \begin{array}{l} J_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,\,k< 0\,\,\,\mathrm{and}\,\,k \ge N_2 , \\ J_{k + 1} \,\,\mathrm{for}\,\,\,0 \le \,\,k < N_2 \\ \end{array} \right. \,\,\,\,\,\mathrm{if}\,\,N_2 \ge 0, \\ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} I_{k - 1} \,\,\mathrm{for}\,\,\,k \le 0\,\,\,\mathrm{and}\,\,\,k > N_1 , \\ I_k \,\,\,\,\,\,\,\mathrm{for}\,\,\,0 < \,\,k \le N_1 ,\, \\ \end{array} \right. \end{aligned}$$

in the case $N_{1} > 0$;

$\mathrm (iii)$ if $\beta = \frac{\pi }{2}$, then

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} J_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,\,k< 0, \\ J_{k + 1} \,\,\mathrm{for}\,\,\;k \ge 0,\, \\ \end{array} \right. \,\mathrm{and}\, \lambda _k \in \left\{ \begin{array}{l} I_k \,\,\,\,\,\,\,\mathrm{for}\,\,\,k \le N_1 \,\mathrm{and}\,\,k > 0, \\ I_{k - 1} \,\mathrm{for}\,\,\,N_1 < k \le 0,\, \\ \end{array} \right. \end{aligned}$$

in the case $N_{1} \le 0$;

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} J_{k - 1} \,\,\,\mathrm{for}\,\,\,k \le 0, \\ J_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,\,k> 0,\, \\ \end{array} \right. \,\, \mathrm{and}\,\, \lambda _k \in \left\{ \begin{array}{l} I_{k - 1} \,\,\mathrm{for}\,\,\,k \le 0\,\,\,\mathrm{and}\,\,\,k > N_1 , \\ I_{k} \,\,\,\,\,\,\,\mathrm{for}\,\,\,0 < k \le N_1 ,\, \\ \end{array} \right. \end{aligned}$$

in the case $N_{1} > 0$; $\mathrm (iv)$ if $\beta \in \left( {\frac{\pi }{2},\pi } \right)$, then

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} \left\{ \begin{array}{l} J_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,\,k \le N_2 \,\,\,\mathrm{and}\,\,k> 0, \\ J_{k - 1} \,\,\mathrm{for}\,\,N_2< k \le 0 \\ \end{array} \right. \,\,\,\,\,\mathrm{if}\,\,N_2 \le 0, \\ \\ \left\{ \begin{array}{l} J_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,\,k< 0\,\,\,\mathrm{and}\,\,k \ge N_2 , \\ J_{k + 1} \,\,\mathrm{for}\,\,\,0 \le \,\,k < N_2 \\ \end{array} \right. \,\,\,\,\,\mathrm{if}\,\,N_2 > 0, \\ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} I_k \,\,\,\,\,\,\,\mathrm{for}\,\,\,k \le N_1 \,\,\,\mathrm{and}\,\,\,k > 0, \\ I_{k - 1} \,\,\mathrm{for}\,\,\,N_1 < k \le 0,\, \\ \end{array} \right. \end{aligned}$$

in the case $N_{1} \le 0$;

$$\begin{aligned} \lambda _k \in \left\{ \begin{array}{l} J_{k - 1} \,\,\mathrm{for}\,\,k \le 0\,\,\mathrm{and}\,\,k> N_2 , \\ J_k \,\,\,\,\,\,\,\,\mathrm{for}\,\,\,0< k \le N_2 ,\, \\ \end{array} \right. \,\,\,\mathrm{and}\,\,\, \lambda _k \in \left\{ \begin{array}{l} I_{k - 1} \,\,\mathrm{for}\,\,k \le 0\,\,\mathrm{and}\,\,k > N_1 , \\ I_{k} \,\,\,\,\,\,\,\,\mathrm{for}\,\,0 < k \le N_1 ,\, \\ \end{array} \right. \end{aligned}$$

in the case $N_{1} > 0$.

Now the statements of this theorem follow from Theorem 3. The proof of Theorem 5 is complete.