An Inverse Three Spectra Problem for Parameter-Dependent and Jumps Conformable Sturm–Liouville Operators

Abstract

In this manuscript, we study the parameter-dependent conformable Sturm–Liouville problem (PDCSLP) in which its transmission conditions are arbitrary finite numbers at an interior point in $[0,\pi ]$. Also, we prove the uniqueness theorems for inverse second order of conformable differential operators by applying three spectra with jumps and eigen-parameter-dependent boundary conditions. To this end, we investigate the PDCSLP in three intervals $[0,\pi ]$, [0, p], and $[p,\pi ]$ where $p\in (0,\pi )$ is an interior point.

1 Introduction

Sturm–Liouville problem is one of the most classical and important problems in mathematics, physics and engineering. This problem arises in the modeling of many systems in vibration theory, quantum mechanics, hydrodynamics, etc. [15, 36].

In 2014, Khalil et al. [18] defined a well-behaved conformable derivative called conformable fractional derivative (CFD) that depends only on the basic limit definition of the derivative. Unlike other definitions of fractional derivative such as Riemann–Liouville and Caputo, this definition enables us to prove many properties similar to derivatives of integer order, for more information about the CFD, refer to [1, 4]. However, the CFD has its drawbacks. Its derivative has some disadvantages and some unusual properties, e.g., the zeroth derivative of a function does not return the function.

Fractional Sturm–Liouville problems (FSLPs) have attracted much attention as an important branch of fractional derivative research [19, 20, 28]. In our opinion, the most important useful property of the conformable derivative is the possibility of defining the inner product in the integral form. This capability makes the conformable Sturm–Liouville problem (CSLP) and PDCSLP well investigated in different situations. In [23], the authors investigated the existence of infinity of real eigenvalues of CSLP. So that the eigenvalues of CSLP are simple and the corresponding eigenfunctions are orthogonal.

The inverse three spectra problems to reconstruction of the potential function in the SLP were firstly discussed in [14, 26, 27]; it was shown that if these spectra are pairwise distinct, the potential function can be uniquely determined by applying the three spectra to the problems defined in the three intervals [0, 1], [0, d], and [d, 1], $(d\in (0, 1))$. Also, in [14] the authors gave a violation example to demonstrate that the pairwise disjoint conditions are necessary. Recently, in [6, 7, 9,10,11,12,13, 31], the authors discussed the inverse three spectra problems in the several cases such as reconstruction of the potential function with different boundary and transmissions conditions, with one or some turning point, and some uniqueness results.

The main purpose of this manuscript is to study the PDCSLP with an arbitrary finite number of transmission conditions at an interior point in $[0,\pi ]$, which is considered to formulated the inverse PDCSLP by using three spectra. One may consider the results of this paper as an extension of [12,13,14, 26, 27, 31] to the PDCSLP. For some related result in the inverse problems in SLP, FSLP, CSLP, PDSLP, we refer to [2, 3, 5, 8, 17, 24, 25, 29, 30, 32,33,34, 37]

2 Asymptotic Forms of PDCSLP

In this section, before introducing the asymptotic forms of PDCSLP, we give several important content of the CFD. In [18], Khalil and et al. defined the CFD as follows:

Definition 2.1

For the function $h:[0,\infty )\rightarrow {\mathbb {R}}$, the CF derivative of order $\alpha \in (0, 1]$ defined by:

$$\begin{aligned} T_{\alpha } h(x)=\lim _{\varepsilon \rightarrow 0} \frac{h(x+\varepsilon x^{1-\alpha })-h(x)}{\varepsilon },\qquad \end{aligned}$$

for all $x > 0$, and

$$\begin{aligned} T_\alpha h(0)=\lim _{x \rightarrow 0^+}T_\alpha h(x). \end{aligned}$$

If h is a differentiable function, then

$$\begin{aligned} T_\alpha h(x)=x^{1-\alpha }h'(x). \end{aligned}$$

If $T_\alpha h(x_0)$ exists and finite, then the function h is $\alpha$-differentiable at $x_0$.

Definition 2.2

The CF integral of order $\alpha \in (0,1]$ for a function $h: [0, \infty ) \rightarrow {\mathbb {R}}$ defined by:

$$\begin{aligned} J^{\alpha }h(x)=\int _0^x h(t)\textrm{d}_{\alpha } t=\int _0^x t^{\alpha -1}h(t)\textrm{d}t, \quad x > 0. \end{aligned}$$

However, the integral is the Riemann improper integral.

We use some important CFD for conformable Sturm–Liouville problems (CSLPs) in [1, 18, 34].

Let us consider the following three PDCSLPs

• I:
  $$\begin{aligned} \ell _0 y:=-T_\alpha T_\alpha y+qy=\mu y \end{aligned}$$

  (2.1)

with parameter-dependent boundary conditions

$$\begin{aligned} \textrm{B}_1(y)&:=\mu (T_\alpha y(0)+h_1 y(0))-h_2T_\alpha y(0)-h_3y(0)= 0, \end{aligned}$$

(2.2)

$$\begin{aligned} \textrm{B}_2(y)&:= \mu (T_\alpha y(\pi )+ H_1y(\pi ))-H_2 T_\alpha y(\pi )-H_3y(\pi ) =0, \end{aligned}$$

(2.3)

and the following jump conditions

$$\begin{aligned} U_{k}(y)&:= y(p_k+0)- b_ky(p_k-0)=0, \ \ k=1,2,\ldots ,m-1\\ V_{k}(y)&:=T_\alpha y(p_k+0)-c_kT_\alpha y(p_k-0)-d_k y(p_k-0)=0,\ \nonumber \end{aligned}$$

(2.4)

• II:
  $$\begin{aligned} \ell _1 y:=-T_\alpha T_\alpha y+q_1y=\mu y \end{aligned}$$

  (2.5)

with the following conditions

$$\begin{aligned} \textrm{B}_1(y)= 0, \ \ \textrm{B}_3(y):= T_\alpha y(p)+\kappa _1\,y(p) =0, \end{aligned}$$

(2.6)

by the jump conditions

$$\begin{aligned} U_{k}(y) =0, \ \ V_{k}(y)=0,\ \text {for}\ k=1,2,\ldots , p-1, \end{aligned}$$

(2.7)

and

• III:
  $$\begin{aligned} \ell _2 y:=-T_\alpha T_\alpha y+q_2y=\mu y \end{aligned}$$

  (2.8)

with Robin and parameter-dependent boundary conditions

$$\begin{aligned} \textrm{B}_4(y):= T_\alpha y(p)+\kappa _2\, y(p)= 0,\ \textrm{B}_2(y)= 0, \ \ \end{aligned}$$

(2.9)

by the following jump conditions

$$\begin{aligned} U_{k}(y) =0, \ \ V_{k}(y)=0,\ \text {for}\ k=p+1,2,\ldots , m-1, \end{aligned}$$

(2.10)

where $T_\alpha$ is the CFD of order $0 < \alpha \le 1$, $q(x)\in L^1_\alpha [0,\pi ]$, $q_1=q|_{[0,p)}$, and $q_2=q|_{(p,\pi ]}$ are real valued functions, and $\kappa _1$, $\kappa _2$, $h_j$, and $H_j$, $j=1,2,3$ are real numbers, satisfying

$$\begin{aligned} r_1:=h_3-h_1h_2>0 \quad \text{ and }\quad r_2:=H_1H_2-H_3>0. \end{aligned}$$

Also, the numbers $b_k$, $c_k$, $d_k$, and $p_k$, with $k=1,2,\dots ,m-1,\ (m\ge 2)$ are real. The parameter $\mu$ is the spectral parameter. In this note, we suppose that $c_kb_k>0$, $p_0=0<p_1<p_2<\cdots<p_{m-1}<p_m=\pi$, and $p=p_s$ for $1\le s\le m-1$. As well as, we use the notations $L_0=L(q(x); h_j;H_j;p_k)$, $L_1=L(q_1(x); h_j;\kappa _1;p_k)$, and $L_2=L(q_2(x);\kappa _2;H_j;p_k)$ for the problems (2.1)–(2.10).

Using the jump conditions (2.4) in the transmission point $p=p_s$, $(1\le s\le m-1)$, we must have $d_s=0$ and

$$\begin{aligned} \kappa _2=\frac{c_s}{b_s}\kappa _1, \ \text {for} \ \kappa _1\in (0,\infty ). \end{aligned}$$

(2.11)

Define the weighted inner products as follows:

$$\begin{aligned} \langle F , G \rangle _{{\mathcal {H}}_0}&:=\int _0^\pi f\overline{g} w_0\textrm{d}_{\alpha }x +\frac{w_0(0)}{r_1}f_1\overline{g_1} +\frac{w_0(\pi )}{r_2} f_2\overline{g_2},\nonumber \\&\qquad F=\begin{pmatrix} f(x) \\ f_1\\ f_2 \end{pmatrix}, \, G=\begin{pmatrix} g(x) \\ g_1\\ g_2 \end{pmatrix}, \end{aligned}$$

(2.12)

$$\begin{aligned}&\langle F_1 , G_1 \rangle _{\mathcal {H}_1}:= \int _0^d f\overline{g} w_1\textrm{d}_{\alpha }x +\frac{w_1(0)}{r_1}f_1\overline{g_1}, \quad F_1=\begin{pmatrix} f(x) \\ f_1\end{pmatrix}, \, G_1=\begin{pmatrix} g(x) \\ g_1 \end{pmatrix}, \end{aligned}$$

and

$$\begin{aligned}&\langle F_2 , G_2 \rangle _{\mathcal {H}_2}:=\int _d^\pi f\overline{g} w_2 \textrm{d}_{\alpha }x +\frac{w_2(\pi )}{r_2} f_2\overline{g_2}, \quad F_2=\begin{pmatrix} f(x) \\ f_2 \end{pmatrix}, \, G_2=\begin{pmatrix} g(x) \\ g_2 \end{pmatrix}. \end{aligned}$$

where

$$\begin{aligned} w_0(x) = {\left\{ \begin{array}{ll} 1, &{} 0 \le x< p_1,\\ \frac{1}{b_1 c_1}, &{} p_1<x<p_2,\\ \vdots \\ \frac{1}{b_1 c_1 \cdots b_{m-1} c_{m-1}}, &{} p_{m-1} <x \le \pi . \end{array}\right. } \end{aligned}$$

Also, $w_1=w_0|_{[0,p)}$, and $w_2=w_0|_{(p,\pi ]}$. We note that $\mathcal {H}_0:=L_\alpha ^2((0,\pi ); w_0)\oplus \mathbb {C}^2$ and $\mathcal {H}_i:=L_\alpha ^2((0,\pi ); w_i)\oplus \mathbb {C}$ $(i=1,2)$, are Hilbert spaces with norms $\Vert F\Vert _{\mathcal {H}_i}=\langle F , F \rangle _{\mathcal {H}_i}^{1/2}$, (i=0,1,2).

Next we introduce

$$\begin{aligned} \begin{array}{cc} R_1(f):=T_\alpha f(0)+h_1f(0),\ \ {} &{} R'_1(f):=h_2T_\alpha f(0)+h_3f(0), \\ \\ R_2(f):=T_\alpha f(\pi )+H_1f(\pi ),\ \ {} &{} R'_2(f):=H_2T_\alpha f(\pi )+H_3f(\pi ). \end{array} \end{aligned}$$

(2.13)

In these spaces, we have the following operators

$$\begin{aligned} A_i:\mathcal {H}_i\rightarrow \mathcal {H}_i\quad i=0,1,2 \end{aligned}$$

with domains

$$\begin{aligned} \textrm{dom}\left( A_0\right) =\left\{ F=\begin{pmatrix} f(x) \\ f_1\\ f_2\end{pmatrix} \left| \begin{array}{c} f, T_\alpha f\in AC\big (\cup _0^{m-1}(p_k,p_{k+1})\big ), \, \ell _0 f\in L_\alpha ^2(0,\pi ) \\ U_k(f)=V_k(f)=0, \ f_1=R_1(f),\ f_2=R_2(f) \end{array}\right. \right\} , \\ \textrm{dom}\left( A_1\right) =\left\{ F_1=\begin{pmatrix} f(x) \\ f_1\end{pmatrix} \left| \begin{array}{c} f, T_\alpha f\in AC\big (\cup _0^{s-1}(p_k,p_{k+1})\big ), \, \ell _1 f\in L_\alpha ^2(0,p) \\ U_i(f)=V_i(f)=0, \ f_1=R_1(f) \end{array}\right. \right\} , \end{aligned}$$

and

$$\begin{aligned} \textrm{dom}\left( A_2\right) =\left\{ F_2=\begin{pmatrix} f(x) \\ f_2\end{pmatrix} \left| \begin{array}{c} f, T_\alpha f\in AC\big (\cup _s^{m-1}(p_k,p_{k+1})\big ), \, \ell _2 f\in L_\alpha ^2(p,\pi ) \\ U_k(f)=V_k(f)=0, \ f_2=R_2(f) \end{array}\right. \right\} \end{aligned}$$

by

$$\begin{aligned} A_0F=\begin{pmatrix}\ell f \\ R'_1(f)\\ R'_2(f) \end{pmatrix}\quad \text {with}\ F= \begin{pmatrix}f(x) \\ R_1(f)\\ R_2(f)\end{pmatrix}\in \textrm{dom}\left( A_0\right) . \end{aligned}$$

(2.14)

and

$$\begin{aligned}A_iF=\begin{pmatrix}\ell f \\ R'_i(f) \end{pmatrix}\quad \text {with}\ F_i=\begin{pmatrix}f(x) \\ R_i(f)\end{pmatrix}\in \textrm{dom}\left( A_i\right) \ (i=1,2).\end{aligned}$$

By construction, the eigenvalue problems $A_0$ and $A_i$,

$$\begin{aligned} A_0Y=\mu Y, \qquad Y:= \begin{pmatrix} y(x) \\ R_1(y)\\ R_2(y) \end{pmatrix}\in \textrm{dom}\left( A_0\right) , \\ A_iY_i=\mu Y_i, \qquad Y_i:= \begin{pmatrix}y(x) \\ R_i(y) \end{pmatrix}\in \textrm{dom}\left( A_i\right) , \end{aligned}$$

are equivalent to the problems (2.1)–(2.4) for the operator L, and (2.5)–(2.7) or (2.8)–(2.10) for $L_i$, $(i=1,2)$, respectively.

Considering the linear differential equations, we obtain the modified fractional Wronskian as follows

$$\begin{aligned} W_\alpha (\theta ,\vartheta )= w_0(x) \big ( \theta (x) T_\alpha \vartheta (x) - T_\alpha \theta (x) \vartheta (x) \big ) \end{aligned}$$

We get that this function is constant on $x\in [0,p_1)\cup \big (\cup _1^{m-2}(p_k,p_{k+1})\big )\cup (p_{m-1},\pi ]$ for two solutions $\ell _0\theta =\mu \theta$ and $\ell _0 \vartheta =\mu \vartheta$ satisfying the discontinuous conditions (2.4).

Lemma 2.3

For $0<\alpha \le 1$, the operators $A_i$, $i=0,1,2$, are symmetric.

Proof

We prove this lemma for $i=0$. After using $\alpha$-integration by parts twice, it follows immediately and by direct calculation by (2.12)–(2.14):

$$\begin{aligned} \big \langle A_{0}F,G \big \rangle =W_{\alpha }\big (f,\bar{g}\big )\big \vert _{x=\pi }- W_{\alpha }\big (f,\bar{g}\big )\big \vert _{x=0}+\big \langle F,A_{0}G \big \rangle . \end{aligned}$$

So, from Eqs. (2.2)–(2.4) we have:

$$\begin{aligned} W_{\alpha }\big (f,\bar{g}\big )\big \vert _{x=\pi }-W_{\alpha }\big (f,\bar{g}\big )\big \vert _{x=0}=0. \end{aligned}$$

Then $A_0$ is symmetric operator on $L_\alpha ^2((0,\pi ); w_0)\oplus \mathbb {C}^2$. Similarly, the operators $A_1$ and $A_2$ are also symmetric. $\square$

By applying Lemma 2.3, the eigenvalues of the problems $A_i$ and hence of $L_i$ are simple and real. Since problem (2.1) with initial conditions $g(\nu \pm 0)=g_0$ and $g'(\nu \pm 0)=g_1$ (with $\nu \in (0,\pi ) )$) is a Cauchy problem, then it has a unique solution.

Remark 2.4

We will denote the restriction of any function g with $g\in \textrm{dom}\left( A_i\right)$, by $g_k$, $1\le k\le m$, to the subinterval $(p_{k-1},p_k)$. Also, we will set $g_k(p_{k-1})=g(p_{k-1}+0)$ and $g_k(p_k)=g(p_k-0)$.

Remark 2.5

Without losing of generality of the problem (2.1)–(2.10), by [33, Lemma 2.3], we can take $b_kc_k=1$, for $k=1,2,\ldots , m$.

Suppose $u(x,\mu )$ and $v(x,\mu )$ are solutions of (2.1) with the jump conditions (2.4) and the following conditions, respectively

$$\begin{aligned} u(0,\mu )=\mu -h_2 ,\ T_\alpha u(0,\mu )=h_3-\mu h_1, \end{aligned}$$

and

$$\begin{aligned} v(\pi ,\mu )=H_2-\mu ,\ T_\alpha v(\pi ,\mu )=\mu H_1- H_3. \end{aligned}$$

The functions $u(x,\mu ),$ $T_\alpha u(x,\mu )$, $v(x,\mu )$, and $T_\alpha v(x,\mu )$ for any fixed $x\in [0,\pi ]$ are entire functions with respect to $\mu$ of order $\frac{1}{2}$ [35]. The asymptotic form of solutions and characteristic function $\Delta (\mu )$ are discussed as follows:

Theorem 2.6

Let $\mu =\varrho ^2$ and $\varrho :=\sigma +i\tau$. The solutions $u(x,\mu )$ and $T_\alpha u(x,\mu )$ for PDCSLP (2.1)–(2.4) as $|\mu |\rightarrow \infty$, have the following asymptotic forms:

$$\begin{aligned} u(x,\mu )&= {\left\{ \begin{array}{ll} \varrho ^2\left[ \cos \left( \frac{\varrho }{\alpha }x^{\alpha }\right) \right] +O\left( {\varrho } \exp \left( \frac{\vert \tau \vert }{\alpha }x^{\alpha }\right) \right) ,\qquad \qquad 0\le x<p_1, \\ \varrho ^2\left[ a_1 \cos \left( \frac{\varrho }{\alpha }x^{\alpha }\right) + a'_1 \cos \left( \frac{\varrho }{\alpha }(x^{\alpha }-2p_1^\alpha )\right) \right] +O\left( {\varrho } \exp \left( \frac{\vert \tau \vert }{\alpha }x^{\alpha }\right) \right) ,\quad p_1<x<p_2,\\ \varrho ^2\left[ a_1a_2\cos \varrho \left( \frac{\varrho }{\alpha }x^{\alpha }\right) +a'_1a_2 \cos \left( \frac{\varrho }{\alpha }(x^{\alpha }-2p_1^\alpha )\right) +a_1a'_2\cos \left( \frac{\varrho }{\alpha }(x^{\alpha }-2p_2^\alpha )\right) \right. \\ \quad \left. +a'_1a'_2\cos \left( \frac{\varrho }{\alpha }(x^{\alpha }+2p_1^\alpha -2 p_2^\alpha )\right) \right] +O\left( {\varrho }\exp \left( \frac{\vert \tau \vert }{\alpha } x^{\alpha }\right) \right) , \qquad p_2<x<p_3,\\ \quad \vdots \\ \varrho ^2\left[ a_1a_2\ldots a_{m-1}\cos \left( \frac{\varrho }{\alpha } x^{\alpha }\right) +\right. \\ \quad + a'_1a_2\ldots a_{m-1}\cos \left( \frac{\varrho }{\alpha }(x^{\alpha } -2p_1^\alpha )\right) +\cdots \\ \quad + a_1a_2\ldots a'_{m-1} \cos \left( \frac{\varrho }{\alpha }(x^{\alpha } -2p_{m-1}^\alpha )\right) +\\ \quad + a'_1a'_2a_3\ldots a_{m-1}\cos \left( \frac{\varrho }{\alpha }(x^{\alpha } +2p_1^\alpha -2p_2^\alpha )\right) +\cdots \\ \quad +a_1\ldots a'_i\ldots a'_j\ldots a_{m-1}\cos \left( \frac{\varrho }{\alpha } (x^{\alpha }+2p_i^\alpha -2p_j^\alpha )\right) \\ \quad + a_1\ldots a'_i\ldots a'_j\ldots a'_k\ldots a_{m-1}\cos \left( \frac{\varrho }{\alpha } (x^{\alpha }-2p_i^\alpha +2p_j^\alpha -2p_k^\alpha )\right) +\cdots \\ \left. \quad + a'_1a'_2\ldots a'_{m-1}\cos \left( \frac{\varrho }{\alpha }(x^{\alpha } +2(-1)^{m-1}p_1^\alpha +2(-1)^{m-2}p_2^\alpha -2p_m^\alpha )\right) \right] \\ \quad +O\left( {\varrho }\exp \left( \frac{\vert \tau \vert }{\alpha } x^{\alpha }\right) \right) , \qquad p_{m-1}<x\le \pi , \end{array}\right. } \end{aligned}$$

(2.15)

$$\begin{aligned} T_\alpha u(x,\mu )&= {\left\{ \begin{array}{ll} \varrho ^3\left[ -\sin \left( \frac{\varrho }{\alpha }x^{\alpha }\right) \right] +O\left( \varrho ^2\exp \left( \frac{\vert \tau \vert }{\alpha }x^{\alpha }\right) \right) ,\qquad \qquad 0\le x<p_1, \\ \varrho ^3\left[ -a_1 \sin \left( \frac{\varrho }{\alpha }x^{\alpha }\right) - a'_1 \sin \left( \frac{\varrho }{\alpha }(x^{\alpha }-2p_1^\alpha )\right) \right] +O \left( \varrho ^2\exp \left( \frac{\vert \tau \vert }{\alpha }x^{\alpha }\right) \right) , \quad p_1<x<p_2,\\ \varrho ^3\left[ -a_1a_2\sin \varrho \left( \frac{\varrho }{\alpha }x^{\alpha } \right) -a'_1a_2\sin \left( \frac{\varrho }{\alpha }(x^{\alpha }-2p_1^\alpha ) \right) -\alpha _1\alpha '_2\sin \left( \frac{\varrho }{\alpha }(x^{\alpha }-2 p_2^\alpha )\right) \right. \\ \quad \left. -a'_1 a'_2\sin \left( \frac{\varrho }{\alpha }(x^{\alpha }+2 p_1^\alpha -2p_2^\alpha )\right) \right] +O\left( \varrho ^2\exp \left( \frac{\vert \tau \vert }{\alpha }x^{\alpha }\right) \right) , \qquad p_2<x<p_3,\\ \quad \vdots \\ \varrho ^3\left[ - a_1a_2\ldots \alpha _{m-1}\sin \left( \frac{\varrho }{\alpha } x^{\alpha }\right) \right. \\ \quad - a'_1a_2\ldots a_{m-1}\sin \left( \frac{\varrho }{\alpha }(x^{\alpha } -2p_1^\alpha )\right) +\cdots \\ \quad - a_1a_2\ldots a'_{m-1} \sin \left( \frac{\varrho }{\alpha }(x^{\alpha } -2p_{m-1}^\alpha )\right) +\\ \quad - a'_1a'_2a_3\ldots a_{m-1}\sin \left( \frac{\varrho }{\alpha }(x^{\alpha }+2 p_1^\alpha -2p_2^\alpha )\right) +\cdots \\ \quad -a_1\ldots a'_i\ldots a'_j\ldots a_{m-1}\sin \left( \frac{\varrho }{\alpha } (x^{\alpha }+2p_i^\alpha -2p_j^\alpha )\right) \\ \quad - a_1\ldots a'_i\ldots a'_j\ldots a'_k\ldots a_{m-1} \sin \left( \frac{\varrho }{\alpha }(x^{\alpha }-2p_i^\alpha +2 p_j^\alpha -2p_k^\alpha )\right) +\cdots \\ \quad \left. -a'_1a'_2\ldots a'_{m-1}\sin \left( \frac{\varrho }{\alpha } (x^{\alpha }+2(-1)^{m-1}p_1^\alpha +2(-1)^{m-2}p_2^\alpha -2p_m^\alpha )\right) \right] \\ \quad +O\left( \varrho ^2\exp \left( \frac{\vert \tau \vert }{\alpha } x^{\alpha }\right) \right) , \qquad p_{m-1}<x\le \pi , \end{array}\right. } \end{aligned}$$

(2.16)

where

$$\begin{aligned} a_k=\frac{1}{2}(b_k+c_k), \qquad a'_k=\frac{1}{2}(b_k-c_k), \end{aligned}$$

(2.17)

for $k=1,2,\ldots ,m-1$.

Proof

Let $\mathcal {S}(x,\mu )$ and $\mathcal {C}(x,\mu )$ be the solutions of (2.1) and (2.4) with the following conditions

$$\begin{aligned} \mathcal {S}(0,\mu )= 0,\ T_\alpha \mathcal {S}(0,\mu )= 1,\ \mathcal {C}(0,\mu )=1,\ \text {and}\ T_\alpha \mathcal {C}(0,\mu )= 0. \end{aligned}$$

Using (2.4) for $\mathcal {C}(x,\mu )$, we obtain

$$\begin{aligned} \mathcal {C}(x,\mu )= {\left\{ \begin{array}{ll} \cos \left( \frac{\varrho }{ \alpha }x^\alpha \right) +O\left( \frac{1}{\varrho } \exp \frac{|\tau |}{\alpha }x^\alpha \right) , \qquad \qquad 0\le x<p_1,\\ b_1\mathcal {C}_1(p_1,\mu )\cos \Big (\frac{\varrho }{\alpha }(x^{\alpha }- p_1^\alpha )\Big )+\frac{c_1}{\varrho }\mathcal {C}'_1(p_1,\mu )\sin \Big (\frac{\varrho }{\alpha }(x^{\alpha }-p_1^\alpha )\Big )\\ \qquad \qquad +O\bigg (\frac{1}{\varrho }\exp \frac{\tau }{\alpha } \big (x^{\alpha }-p_1^\alpha \big )\bigg ), \qquad p_1< x<p_2, \\ b_2\mathcal {C}_2(p_2,\mu )\cos \Big (\frac{\varrho }{\alpha } (x^{\alpha }-p_2^\alpha )\Big )+\frac{c_2}{\varrho }\mathcal {C}'_2 (p_2,\mu )\sin \Big (\frac{\varrho }{\alpha }(x^{\alpha }-p_2^\alpha )\Big )\\ \qquad \qquad +O\bigg (\frac{1}{\varrho }\exp \frac{\tau }{\alpha } \big (x^{\alpha }-p_2^\alpha \big )\bigg ), \qquad p_2< x<p_3, \\ \quad \vdots &{}\\ b_{m-1}\mathcal {C}_{m-1}(p_{m-1},\mu )\cos \Big (\frac{\varrho }{\alpha } (x^{\alpha }-p_{m-1}^\alpha )\Big ) +\\ \qquad + \frac{c_{m-1}}{\varrho } \mathcal {C}'_{m-1}(p_{m-1},\mu )\sin \Big (\frac{\varrho }{\alpha }(x^{\alpha }-p_{m-1}^\alpha )\Big ) +\\ \qquad +O\bigg (\frac{1}{\varrho }\exp \frac{\tau }{\alpha }\big (x^{\alpha } -p_{m-1}^\alpha \big )\bigg ), \qquad p_{m-1}< x\le \pi .\\ \end{array}\right. } \end{aligned}$$

So, we insert the k’th statement into the $(k+1)$’th statement to get

$$\begin{aligned} \mathcal {C}(x,\mu )= {\left\{ \begin{array}{ll} \cos \Big (\frac{\varrho }{\alpha }x^{\alpha }\Big )+O\Big (\frac{1}{\varrho } \exp \big (\frac{\tau }{\alpha }x^{\alpha }\big )\Big ),\qquad \qquad 0\le x<p_1, \\ a_1 \cos \Big (\frac{\varrho }{\alpha }x^{\alpha }\Big ) + a'_1 \cos \Big (\frac{\varrho }{\alpha }(x^{\alpha }-2p_1^\alpha )\Big )+O \Big (\frac{1}{\varrho }\exp \big (\frac{\tau }{\alpha }x^{\alpha } \big )\Big ),\quad p_1<x<p_2,\\ a_1a_2\cos \Big (\frac{\varrho }{\alpha }x^{\alpha }\Big ) +a'_1 a_2\cos \Big (\frac{\varrho }{\alpha }(x^{\alpha }-2p_1^\alpha )\Big ) +a_1a'_2\cos \Big (\frac{\varrho }{\alpha }(x^{\alpha }-2p_2^\alpha )\Big )\\ \quad +a'_1a'_2\cos \Big (\frac{\varrho }{\alpha }(x^{\alpha }+2 p_1^\alpha -2p_2^\alpha )\Big )+O\Big (\frac{1}{\varrho }\exp \big (\frac{\tau }{\alpha }x^{\alpha }\big )\Big ), \qquad p_2<x<p_3,\\ \quad \vdots \\ a_1a_2\ldots a_{m-1}\cos \Big (\frac{\varrho }{\alpha }x^{\alpha }\Big )+\\ \quad + a'_1a_2\ldots a_{m-1}\cos \Big (\frac{\varrho }{\alpha } (x^{\alpha }-2p_1^\alpha )\Big )+\cdots \\ \quad + a_1a_2\ldots a'_{m-1} \cos \Big (\frac{\varrho }{\alpha } (x^{\alpha }-2p_{m-1}^\alpha )\Big )+\\ \quad + a'_1a'_2 a_3\ldots a_{m-1}\cos \Big (\frac{\varrho }{\alpha } (x^{\alpha }+2p_1^\alpha -2p_2^\alpha )\Big )+\cdots \\ \quad +a_1\ldots a'_j\ldots a'_k\ldots a_{m-1}\cos \Big (\frac{\varrho }{\alpha }(x^{\alpha }+2p_j^\alpha -2p_k^\alpha )\Big )\\ \quad + a_1\ldots a'_j\ldots a'_k\ldots a'_s\ldots a_{m-1}\cos \Big (\frac{\varrho }{\alpha }(x^{\alpha }-2 p_j^\alpha +2p_k^\alpha -2p_s^\alpha )\Big )+\cdots \\ \quad + a'_1a'_2\ldots a'_{m-1}\cos \Big (\frac{\varrho }{\alpha } (x^{\alpha }+2(-1)^{m-1}p_1^\alpha +2(-1)^{m-2}p_2^\alpha -2p_m^\alpha )\Big )\\ \quad +O\Big (\frac{1}{\varrho }\exp \big (\frac{\tau }{\alpha } x^{\alpha }\big )\Big ), \qquad p_{m-1}<x\le \pi , \end{array}\right. } \end{aligned}$$

where $a_i$ and $a'_i$ are defined in (2.17) and $j<k<s$, $j,k,s=1,2,\ldots ,m-1$. Similarly, we can obtain the asymptotic formula for $\mathcal {S}(x,\mu )$. Applying Definition 2.1, we calculate the asymptotic form of $T_\alpha \mathcal {S}(x,\mu )$ and $T_\alpha \mathcal {C}(x,\mu )$. This completes the proof by using $u(x,\mu )=(\mu -h_2)\mathcal {C}(x,\mu )+(h_3-\mu h_1)\,\mathcal {S}(x,\mu )$. $\square$

From Theorem 2.6 and Definition 2.1, we get that

$$\begin{aligned} | u(x,\mu )|&=O\left( |\varrho |^2\exp \bigg (\frac{\vert \tau \vert }{\alpha } x^{\alpha }\bigg )\right) ,\nonumber \\ |T_\alpha u(x,\mu )|&=|x^{1-\alpha } u'(x,\mu )|=O\left( |\varrho |^3 \exp \bigg (\frac{\vert \tau \vert }{\alpha }x^{\alpha }\bigg )\right) , \ 0\le x \le \pi . \end{aligned}$$

By changing x to $\pi -x$ in (2.1) and using the jump condition (2.4), we get a new problem. Applying the definition of 2.1, we obtain the asymptotic form of $w(x,\mu )v(x, \tau )$ and $T_\alpha v(x,\mu )$. Specially,

$$\begin{aligned} | v(x,\mu )|&=O\left( |\varrho |^2\exp \bigg (\frac{\vert \tau \vert }{\alpha }(\pi -x)^{\alpha }\bigg )\right) ,\nonumber \\ |T_\alpha v(x,\mu )|&=|x^{1-\alpha } v'(x,\mu )|=O\left( |\varrho |^3\exp \bigg (\frac{\vert \tau \vert }{\alpha }(\pi -x)^{\alpha }\bigg )\right) ,\quad 0\le x \le \pi . \end{aligned}$$

In addition, using (2.2) and Remark 2.4, we obtain

$$\begin{aligned} \Delta (\mu ):&= W_\alpha ( u(\mu ), v(\mu )) \nonumber \\&= B_1( v(\mu )) \nonumber \\&= - w_0(\pi ) B_2( u(\mu ))\nonumber \\&=w_0(p)\left( c_s u(p,\mu )T_\alpha v(p,\mu )-b_s T_\alpha u(p,\mu ) v(p,\mu )\right) . \end{aligned}$$

(2.18)

It follows from equation (2.18) that the characteristic function $\Delta (\mu )$ is the combination of the solutions and from [16] it is clear that every solution is an entire function of order $\frac{1}{2}$. As a result, $\Delta (\mu )$ is an entire function of order $\frac{1}{2}$, so that its roots are, $\mu _n$, the eigenvalues of L. The asymptotic form of the characteristic function will be:

$$\begin{aligned} \Delta (\mu )&=\varrho ^5 w_0(\pi )\left[ a_1a_2\ldots a_{m-1}\sin \left( \frac{\varrho }{\alpha }\pi ^{\alpha }\right) +a'_1a_2\ldots a_{m-1}\sin \left( \frac{\varrho }{\alpha }(\pi ^{\alpha }-2p_1^\alpha )\right) +\cdots \right. \nonumber \\&\quad +a_1a_2\ldots a'_{m-1}\sin \left( \frac{\varrho }{\alpha }(\pi ^{\alpha }-2p_{m-1}^\alpha )\right) +a'_1a'_2a_3\ldots a_{m-1}\sin \left( \frac{\varrho }{\alpha }(\pi ^{\alpha }+2p_1^\alpha -2p_2^\alpha )\right) \nonumber \\&\quad +\cdots +a_1\ldots a'_i\ldots a'_j\ldots a_{m-1}\sin \left( \frac{\varrho }{\alpha }(\pi ^{\alpha }+2p_i^\alpha -2p_j^\alpha )\right) \nonumber \\&\quad +a_1\ldots a'_i\ldots a'_j\ldots a'_k\ldots a_{m-1}\sin \left( \frac{\varrho }{\alpha }(\pi ^{\alpha }-2p_i^\alpha +2p_j^\alpha -2p_k^\alpha )\right) +\cdots \nonumber \\&\quad \left. +a'_1a'_2\ldots a'_{m-1}\sin \left( \frac{\varrho }{\alpha }(\pi ^{\alpha } +2(-1)^{m-1}d_1^\alpha +2(-1)^{m-2}p_2^\alpha -2p_m^\alpha )\right) \right] \nonumber \\&\quad +O\left( \varrho ^4\exp \left( \frac{\vert \tau \vert }{\alpha }\pi ^{\alpha }\right) \right) .\nonumber \\ \end{aligned}$$

(2.19)

As a result of Valiron’s theorem ([22, Thm. 13.4]) and (2.19), we obtain the following asymptotic form.

Theorem 2.7

Let $\mu _n = \varrho _n^2$ be the eigenvalues of the problem $L_\alpha$, then we have the following asymptotic formula

$$\begin{aligned} {\varrho _n}= \alpha \pi ^{1-\alpha }n+O(1) \end{aligned}$$

(2.20)

as $n\rightarrow \infty$.

Remark 2.8

The asymptotic forms of the solutions and characteristic function for the operators $L_1$ and $L_2$ are similar to the operator $L_0$.

Example 2.9

Consider the following PDCSLP with $h_1=0,\,h_2=0,\,H_1=0,\ H_2=0$ and $h_3=H_3=1$ with one jump point in $p=\frac{\pi }{4}$,

$$\begin{aligned}&-T_\alpha T_\alpha y=\mu y \nonumber \\&\mu T_\alpha y(0)- y(0) = 0,\quad \mu T_\alpha y(\pi )+ y(\pi )=0,\\&y(\frac{\pi }{4}+)-{2} y(\frac{\pi }{4}-)=0, \quad T_\alpha y(\frac{\pi }{4}+)-\frac{1}{2} T_\alpha y(\frac{\pi }{4}-)=0.\nonumber \end{aligned}$$

(2.21)

The characteristic function and the eigenfunctions are

$$\begin{aligned} \Delta (\mu )=&-\varrho ^5\left[ \frac{5}{4}\sin (\frac{\varrho }{\alpha }\pi ^\alpha )+\frac{3}{4}\sin (\frac{\varrho }{\alpha }(\pi ^\alpha -2(\frac{\pi }{4})^\alpha ))\right] +\frac{5}{2}\varrho ^2\cos (\frac{\varrho }{\alpha }\pi ^\alpha )\\&+\frac{1}{\varrho }\left( \frac{5}{4}\sin (\frac{\varrho }{\alpha }\pi ^\alpha )-\frac{3}{4}\sin (\frac{\varrho }{\alpha }(\pi ^\alpha -2(\frac{\pi }{4})^\alpha ))\right) \\ u_{n,\alpha }(x)&= {\left\{ \begin{array}{ll} \varrho _n^2\cos (\frac{\varrho _n}{\alpha }x^\alpha )+\frac{1}{\varrho _n} \sin (\frac{\varrho _n}{\alpha }x^\alpha ), &{} 0\le x <\frac{\pi }{4},\\ \varrho _n^2(\frac{5}{4}\cos (\frac{\varrho _n}{\alpha }x^\alpha )+ \frac{3}{4}\cos (\frac{\varrho _n}{\alpha }(x^\alpha -2(\frac{\pi }{4})^\alpha ))) +\frac{5}{4\varrho _n}\sin (\frac{\varrho _n}{\alpha }x^\alpha ), &{}\\ \qquad -\frac{3}{4\varrho _n}\sin (\frac{\varrho _n}{\alpha } (x^\alpha -2(\frac{\pi }{4})^\alpha )), &{} \frac{\pi }{4}\le x \le \pi . \end{array}\right. } \end{aligned}$$

Table 1 Eigenvalues and asymptotic results for Example 2.9

Fig. 1

Eigenfunctions of Example 2.9 for different values of n and $\alpha$

Example 2.10

Consider the following PDCSLP with $h_1=0,\,h_2=0,\,H_1=0,\ H_2=0$ and $h_3=H_3=1$ with one jump point in $p=\frac{\pi }{2}$

$$\begin{aligned}&-T_\alpha T_\alpha y=\mu y \nonumber \\&\mu T_\alpha y(0)- y(0) = 0,\quad \mu T_\alpha y(\pi )+ y(\pi )=0,\nonumber \\&y(\frac{\pi }{2}+)-{3} y(\frac{\pi }{2}-)=0,\quad T_\alpha y(\frac{\pi }{2}+)-\frac{1}{3} T_\alpha y(\frac{\pi }{2}-)=0. \end{aligned}$$

(2.22)

The characteristic function and the eigenfunctions are

$$\begin{aligned} \Delta (\mu )=&-\varrho ^5\left[ \frac{5}{3}\sin (\frac{\varrho }{\alpha }\pi ^\alpha )+\frac{4}{3}\sin (\frac{\varrho }{\alpha }(\pi ^\alpha -2(\frac{\pi }{2})^\alpha ))\right] +\frac{10}{3}\varrho ^2\cos (\frac{\varrho }{\alpha }\pi ^\alpha )\\&+\frac{1}{\varrho }\left( \frac{5}{3}\sin (\frac{\varrho }{\alpha }\pi ^\alpha )-\frac{4}{3}\sin (\frac{\varrho }{\alpha }(\pi ^\alpha -2(\frac{\pi }{2})^\alpha ))\right) \\ u_{n,\alpha }(x)&= {\left\{ \begin{array}{ll} \varrho _n^2\cos (\frac{\varrho _n}{\alpha }x^\alpha )+ \frac{1}{\varrho _n}\sin (\frac{\varrho _n}{\alpha }x^\alpha ), &{} 0\le x <\frac{\pi }{2},\\ \varrho _n^2(\frac{5}{3}\cos (\frac{\varrho _n}{\alpha }x^\alpha ) +\frac{4}{3}\cos (\frac{\varrho _n}{\alpha }(x^\alpha -2 (\frac{\pi }{2})^\alpha )))+\frac{5}{3\varrho _n}\sin (\frac{\varrho _n}{\alpha }x^\alpha ), &{}\\ \qquad -\frac{4}{3\varrho _n}\sin (\frac{\varrho _n}{\alpha } (x^\alpha -2(\frac{\pi }{2})^\alpha )), &{} \frac{\pi }{2}\le x \le \pi . \end{array}\right. } \end{aligned}$$

Fig. 2

Eigenfunctions of Example 2.10 for different values of n and $\alpha$

Table 2 Eigenvalues and asymptotic results for Example 2.10

The eigenvalues and eigenfunctions are presented in Table 1 and Fig. 1. We use the Roots function in Maple 2021, to compute the zeros $\varrho _{n,\alpha }$ of the function $\Delta (\mu )$. We compared the eigenvalues with first term of asymptotic form (2.20) as $\xi _{n,\alpha }=\frac{\varrho _{n,\alpha }}{n\alpha \pi ^{1-\alpha }}$. The eigenvalues and ratios $\xi _{n,\alpha }$ are presented in Tables 1 and 2. According to asymptotic form (2.20), the values of $\xi _{n,\alpha }$ must tend to one, that hold for results of $\xi _{n,\alpha }$ in Tables 1 and 2. The first four eigenfunctions for different values of $\alpha$ are plotted in Figs. 1 and 2. It is well known that the nth eigenfunction of classical Sturm–Liouville problem defined on $[0,\pi ]$, has $(n-1)$ zero in interval $(0,\pi )$. The graphs in Figs. 1 and 2 indicate that this result holds also for PDCSLP with jump conditions.

3 Uniqueness Result

In this section, we formulated the inverse PDCSLPs. To this end, first we necessity the following lemma about poles, residues, and asymptotic formulas to determine a meromorphic Herglotz–Nevanlinna function, see [14, Thm 2.3].

Lemma 3.1

Suppose that the functions $h_1(z)$ and $h_2(z)$ are two meromorphic Herglotz–Nevanlinna function with the similar sets of residues and poles. If

$$\begin{aligned} h_1(it)- h_2(it)\rightarrow 0,\quad \text {as} \ t\rightarrow \infty , \end{aligned}$$

then $h_1= h_2$.

Define the Weyl–Titchmarsh $\mathfrak {m}$-functions

$$\begin{aligned} \mathfrak {m}_-(\mu )=-\frac{T_\alpha u(p,\mu )}{ u(p,\mu )},\qquad \mathfrak {m}_+(\mu )=\frac{T_\alpha v(p,\mu )}{ v(p,\mu )}. \end{aligned}$$

(3.1)

As a consequence of theorem [14, Thms. 2.1 and 2.2] we obtain:

Lemma 3.2

The functions ${\mathfrak {m}}_- (\mu )$ and ${\mathfrak {m}}_+ (\mu )$ satisfy in the conditions of Herglotz–Nevanlinna’s functions.

Proof

Suppose that the functions u and $\bar{u}$ are solutions of $\ell _1 u=\mu u$ and ${\overline{\ell _1 u}}=\ell _1 \bar{u}=\bar{\mu } \bar{u}$. It is easy to see that

$$\begin{aligned} (\mu -\bar{\mu })\int _0^x u(t) \bar{u}(t) w_1(t)\textrm{d}_\alpha t=W_\alpha (u,\bar{u})(x)-W_\alpha (u,\bar{u})(0). \end{aligned}$$

From definition of $\mathfrak {m}_-(\mu )$ in the point $x=p$ and the condition (2.6), we get

$$\begin{aligned} \textrm{Im}(\mu ) \Vert u\Vert _{T_1}^2=\textrm{Im}(\mathfrak {m}_-(\mu )) |u(p)|^2. \end{aligned}$$

Then, the function $\mathfrak {m}_- (\mu )$ is Herglotz–Nevanlinna function. Similarly the function $\mathfrak {m}_+(\mu )$ is also Herglotz–Nevanlinna function. $\square$

Lemma 3.3

For every arbitrary $\upsilon >0$ with $\upsilon<\arg \mu <2\pi -\upsilon$, the following asymptotic formula for $\mathfrak {m}_-(\mu )$ and $\mathfrak {m}_+(\mu )$ hold:

$$\begin{aligned} \mathfrak {m}_+(\mu )=i\sqrt{\mu }+o(\sqrt{\mu }),\quad \mathfrak {m}_-(\mu )=i\sqrt{\mu }+o(\sqrt{\mu }),\quad \text {as}\ \mu \rightarrow \infty . \end{aligned}$$

(3.2)

Specially, when $\mu \rightarrow -\infty$, we have

$$\begin{aligned} \mathfrak {m}_+(\mu )=-\sqrt{|\mu |}+o(\sqrt{|\mu |}),\quad \mathfrak {m}_-(\mu )=-\sqrt{|\mu |}+o(\sqrt{|\mu |})\quad \text {as}\ \mu \rightarrow -\infty . \end{aligned}$$

(3.3)

Proof

Using the asymptotic forms of $u(x,\mu )$ and $T_\alpha u(x,\mu )$ in (2.15) and (2.16) and similar asymptotic forms for $v(x,\mu )$ and $T_\alpha v(x,\mu )$, it can be checked by direct calculations that the asymptotic forms of $\mathfrak {m}_-(\mu )$ and $\mathfrak {m}_+(\mu )$ are satisfying in (3.2)–(3.3). $\square$

Suppose that the eigenvalues of the PDCSLPs (2.5)–(2.7) and PDCSLPs (2.6)–(2.10) are denoted by $\{\mu _n\}_{n=1}^\infty$ and $\{\nu _n\}_{n=1}^\infty$, respectively. In this part, we express the fundamental theorem of uniqueness result for the problems (2.1)–(2.10). For the uniqueness theorem we need using the similar operators $\tilde{L}_i$, with operators $L_i$ but with different coefficients $\tilde{q}(x)$, $\tilde{h}$, $\tilde{H}$, $\tilde{H}_1$, $\tilde{b}_k$, $\tilde{c}_k$, $\tilde{d}_k$, $\tilde{p}_k$. Given a function

$$\begin{aligned} f(\mu ):=\left\{ \begin{array}{ll} -\frac{\Delta (\mu )}{w_0(p) u(p,\,\mu ) v(p,\,\mu )}, &{} \hbox {} H_1=\infty ,\\ &{} \hbox {}\\ -\frac{\Delta (\mu )}{w_0(p)[T_\alpha u(p,\mu )+H_1 u(p,\mu )] [T_\alpha v(p,\mu )+H_2 v(p,\mu )]}, &{} \hbox {}H_1\ne \infty . \end{array} \right. \end{aligned}$$

(3.4)

It is easy to check that $f(\mu )$ is a meromorphic function and the set of poles of $f(\mu )$ is all values of $\{\mu _n\}_{n=1}^\infty \cup \{\nu _n\}_{n=1}^\infty$. Using Eq. (2.18) and $H_2=\frac{c_s}{b_s}H_1$, we have

$$\begin{aligned} f(\mu )=&\left\{ \begin{array}{ll} -c_s\frac{T_\alpha v(p,\,\mu )}{ v(p,\,\mu )}+b_s \frac{T_\alpha u(p,\,\mu )}{ u(p,\,\mu )}, &{} \hbox {} H_1=\infty ,\\ &{} \hbox {}\\ -c_s\frac{ u(p,\,\mu )}{T_\alpha u(p,\,\mu )+H_1\, u(p,\,\mu )}+b_s\frac{ v(p,\,\mu )}{T_\alpha v(p,\,\mu )+H_2\, v(p,\,\mu )}, &{} \hbox {}H_1\ne \infty ,\\ \end{array} \right. \\ :=&{M}_+(\mu )+M_-(\mu ), \end{aligned}$$

where from (3.1)

$$\begin{aligned} \begin{array}{cc} M_+(\mu )=\left\{ \begin{array}{ll} -c_s \mathfrak {m}_+(\mu ), &{} \hbox {}H_2=\infty , \\ &{} \hbox {}\\ \frac{b_s}{H_2+\mathfrak {m}_+(\mu )}, &{} \hbox {}H_2\in {\mathbb {R}}, \end{array} \right. &{} M_-(\mu )=\left\{ \begin{array}{ll} -b_s \mathfrak {m}_-(\mu ), &{} \hbox {}H_1=\infty ,\\ &{} \hbox {}\\ \frac{c_s}{\mathfrak {m}_-(\mu )-H_1}, &{} \hbox {}H_1\in {\mathbb {R}}. \end{array} \right. \end{array} \end{aligned}$$

(3.5)

Lemma 3.4

Fixed $H_1\in {\mathbb {R}}\cup \{\infty \}$. For every arbitrary $\upsilon >0$ with $\upsilon<\arg \mu <2\pi -\upsilon$, the following asymptotic formula for $M_-(\mu )$ and $M_+(\mu )$ hold:

$$\begin{aligned} M_-(\mu )=\left\{ \begin{array}{ll} i\, b_s \sqrt{\mu }+o(\sqrt{\mu }), &{} \hbox {}H_1=\infty ,\\ \\ \frac{i\, c_s}{\sqrt{\mu }}+o \left( \frac{1}{\sqrt{\mu }}\right) , &{} \hbox {}H_1\in {\mathbb {R}}, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} M_+(\mu )=\left\{ \begin{array}{ll} i\, c_s \sqrt{\mu }+o(\sqrt{\mu }), &{} \hbox {}H_2=\infty , \\ &{} \hbox {}\\ \frac{i\, b_s}{\sqrt{\mu }}+o\left( \frac{1}{\sqrt{\mu }}\right) , &{} \hbox {}H_2\in {\mathbb {R}}. \end{array} \right. \end{aligned}$$

Proof

The proof is similar to Lemma 3.3. $\square$

Theorem 3.5

If $\mu _n=\tilde{\mu }_n$, $\omega _n=\tilde{\omega }_n$, and $\nu _n=\tilde{\nu }_n$ for $n\ge 0$, and $w_0(x)=\tilde{w}_0(x)$, $h_j=\tilde{h}_j$, and $H_j=\tilde{H}_j$, $(j=1,2,3)$ and if $\{\omega _n\}_{n=1}^{+\infty }$ and $\{\nu _n\}_{n=1}^{+\infty }$ are pairwise disjoint, then $L=\tilde{L}$.

Proof

From Lemma 3.2, $\mathfrak {m}_-(\mu )$ and $\mathfrak {m}_+(\mu )$ are Herglotz–Nevanlinna functions. Therefore, one can easily check that the function $M_+(\mu )$ and $M_-(\mu )$ are Herglotz–Nevanlinna functions. The functions $\tilde{\mathfrak {m}}_-(\mu )$, $\tilde{M}_-(\mu )$, $\tilde{\mathfrak {m}}_+(\mu )$, $\tilde{M}_+(\mu )$, and $\tilde{ f}(\mu )$ defined by a similar way by replacing L to $\tilde{L}$. We define

$$\begin{aligned} \mathcal {G}(\mu ):=\frac{f(\mu )}{\tilde{f}(\mu )}. \end{aligned}$$

Since the functions $f(\mu )$ and $\tilde{f}(\mu )$ have the same poles and zeros, $\mathcal {G}(\mu )$ is an entire function. Applying Lemmas 3.3 and 3.4, we have

$$\begin{aligned} \mathcal {G}(\mu )=\frac{ f(\mu )}{\tilde{f}(\mu )}=1+o(1) \end{aligned}$$

for any $\upsilon >0$ in the area of $\upsilon \le \arg \mu \le 2\pi -\upsilon$. By applying Liouville’s theorem, we get

$$\begin{aligned} \mathcal {G}(\mu )=1 \end{aligned}$$

then

$$\begin{aligned} f(\mu )=\tilde{f}(\mu ). \end{aligned}$$

From (3.4) and (3.5), the poles of $M_-(\mu )$ and $M_+(\mu )$ are exactly the same $\{\omega _n\}_{n=1}^\infty$ and $\{\nu _n\}_{n=1}^\infty$, respectively. Then, we get

$$\begin{aligned} \underset{{{\mu =\omega _{\textrm{n}}}}}{\text {Res}}\, M_-(\mu )= \underset{{{\mu =\omega _{\textrm{n}}}}}{\text {Res}} f(\mu )\, \text { and } \underset{{{\mu =\nu _{\textrm{n}}}}}{\text {Res}} M_+(\mu )= \underset{{\mu =\nu _{\textrm{n}}}}{\text {Res}} f(\mu ),\ \text { for }\ n=1,2,3,\ldots . \end{aligned}$$

which means that

$$\begin{aligned} \underset{{\mu =\omega _{\textrm{n}}}}{\text {Res}}M_-(\mu )= \underset{{\mu =\omega _{\textrm{n}}}}{\text {Res}}\tilde{M}_-(\mu ) \,\text { and } \underset{{\mu =\nu _{\textrm{n}}}}{\text {Res}} M_+(\mu )= \underset{{\mu =\nu _{\textrm{n}}}}{\text {Res}}\tilde{M}_+(\mu ), \ \text { for }\ n=1,2,3,\ldots . \end{aligned}$$

Applying the Borg’s theorem [21] for the M-Weyl–Titchmarsh functions $M_+(\mu )$ and $M_-(\mu )$, we get

$$\begin{aligned} L=\tilde{L}. \end{aligned}$$

$\square$

Assuming $b_s=c_s=1$ in Eq. (2.11) we have $\kappa _1=\kappa _2$. From this assumption, the main result (Theorem 3.5) can be extended to the case $p\in (p_s-1,p_{s+1})$.

Corollary 3.6

Let $\mu _n=\tilde{\mu }_n$, $\omega _n=\tilde{\omega }_n$, and $\nu _n=\tilde{\nu }_n$ for $n\ge 0$, and $w_0(x)=\tilde{w}_0 (x)$, $h_j=\tilde{h}_j$, $H_j=\tilde{H}_j, \ (j=0,1,2)$, $b_s=c_s=1$, and if $\{\omega _n\}_{n=0}^{+\infty }$ and $\{\nu _n\}_{n=0}^{+\infty }$ are separate in pairs, then $L_0=\tilde{L}_0$.

Let $b_k=c_k=1$, $d_k=0$ for $k=1,2,\ldots , m-1$ in Eqs. (2.4), then our PDCSLP changes to the continuous case equation.

Corollary 3.7

If $\mu _n=\tilde{\mu }_n$, $\omega _n=\tilde{\omega }_n$, and $\nu _n=\tilde{\nu }_n$ for $n\ge 0$, $h_j=\tilde{h}_j$, $H_j=\tilde{H}_j, \ (j=0,1,2)$, and $b_k=c_k=1$, $d_k=0$ for $k=1,2,\ldots , m-1$, if $\{\omega _n\}_{n=0}^{+\infty }$ and $\{\nu _n\}_{n=0}^{+\infty }$ are separate in pairs, then $L_0=\tilde{L}_0$.