Unique Existence and Reconstruction of the Solution of Inverse Spectral Problem for Differential Pencil

Abstract

In this paper, the half-inverse spectral problem for energy-dependent Sturm–Liouville problems (that is, differential pencils), defined on interval $[0,\pi ]$ with the potential functions $p,q$ being a priori known on the subinterval $[0,\pi /2]$, is considered. We provide a method for the unique reconstruction of the two potential functions on $[\pi /2,\pi ]$ and the boundary condition at $x=\pi$ by using one full spectrum. Consequently, based on the reconstruction method, we also provide a necessary and sufficient condition under which the existence of the quadratic pencil of differential operators is unique.

1. Introduction

In this paper, we focus on the half-inverse spectral problem for the differential pencil denoted by

$$\mathcal{H}y:=-y^{″}+[p\left(x\right)+2\lambda q\left(x\right)]y={\lambda }^{2}y,$$

(1)

for $0<x<\pi$, satisfying the following boundary conditions:

$$\begin{array}{cc}\hfill & y^{\prime }\left(0\right)-hy\left(0\right)=0,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & y^{\prime }\left(\pi \right)+Hy\left(\pi \right)=0.\hfill \end{array}$$

(3)

Here, $q\left(x\right)\in W_2^1[0,\pi ]$ and $p\left(x\right)\in L_2[0,\pi ]$ are real-valued functions and $h,H\in \mathbb{R}$. Here, $W_2^n[0,\pi ]$ denotes the Sobolev space of real-valued functions on the interval $[0,\pi ]$, which have $n-1$ absolutely continuous derivatives and a square-summable nth derivative on $[0,\pi ]$. Let $\sigma ={\left\{{\lambda }_n\right\}}_{n\in \mathbb{Z}}$ be the eigenvalues of the differential pencil $\mathcal{H}$. Then, one knows that $\sigma$ is a countably infinite set and each eigenvalue ${\lambda }_n$ is real and simple [1].

Spectral problems for the Sturm–Liouville equation with potentials depending on the spectral parameter arise in various models of quantum and classical mechanics in [2], as well as in [3,4], where further references to applications can be found. For instance, the evolution equations that are used to model interactions between colliding relativistic spineless particles, such as the Klein–Gordon equation [5,6], can be reduced to the corresponding form of (1). Then, ${\lambda }^2$ is related to the energy of the system. In these applications, the potential functions $p\left(x\right)$ and $q\left(x\right)$ directly correspond to physical quantities such as interaction potentials and damping coefficients, yet in practical scenarios, measuring these potentials over the entire spatial domain (e.g., $[0,\pi ]$) is often infeasible (e.g., due to experimental limitations in quantum scattering or structural constraints in mechanical systems). To overcome this measurement limitation and infer potential functions and system characteristics from accessible information, research on inverse spectral problems is indispensable, and such problems also play a key role in various applications, e.g., in modeling interactions between colliding relativistic particles in quantum mechanics [7] and in recovering mechanical systems vibrating in viscous media [8].

Inverse spectral problems for classical Sturm–Liouville operators (i.e., $q\left(x\right)\equiv 0$ in (1) have been extensively studied in various publications. We refer the reader to the foundational works of Yurko, Gesztesy, and Hochstadt (see [9,10,11]) and to recent works [12,13,14,15,16] for new methods. In particular, in [15,16], one of the authors and the co-authors firstly used the Mittag–Leffler decomposition, which helped them to decompose an entire function of the exponential type into two functions of smaller types and allowed them to use Lagrange interpolation in order to reconstruct the potential in the Sturm–Liouville problem.

In contrast to classical Sturm–Liouville equation, Equation (1) possesses non-linear dependence on the spectral parameter, which complicates the study of inverse problems. By this reason, the inverse spectral theory for differential pencils is not as developed as that for the Sturm–Liouville operator [17]. In [18], by using the Hochstadt’s method, Koyunbakan proved that the potential functions $p\left(x\right)$ and $q\left(x\right)$ can be determined from two spectra uniquely. This is the basic and comprehensive result of inverse spectral problems concerning a differential pencil and is called the Borg-type theorem [19]. In [20], by using the method of spectral mappings, the authors proved that the Weyl function, or one spectrum, and the corresponding generalized weight numbers can determine the potential functions uniquely. These conclusions can be viewed as extensions of the Borg-type theorem.

At the same time, many studies have focused on the inverse spectral problems with mixed data, and a series of conclusions have been made (see for example, [17,21,22,23,24] and the references therein). One of the most important problems introduced and studied is the half-inverse problem [21,22,23,25]. This problem for classical Sturm–Liouville equations was first examined by Hochstadt and Liberman [11] in 1978. By using an approach similar to that in [11], Yang and Guo [22] proved that if $p\left(x\right)$ (or $q\left(x\right)$) is full given on the interval $[0,\pi ]$, then a set of values of eigenfunctions at the mid-point of the interval $[0,\pi ]$ and one spectrum suffice to determine $q\left(x\right)$ (or $p\left(x\right)$) on the interval $[0,\pi ]$ and all parameters in the boundary conditions. Moreover, Yang and Zettl [24] proved the uniqueness theorem of half-inverse problems, which says that one spectrum determines the potential functions $p,q$ uniquely provided and the potential functions given a priori on the left-interval $[0,\pi /2]$ (or right-interval $[\pi /2,\pi ]$). It is also worth mentioning that the majority of those results are concerned with uniqueness problems. The principal questions of unique existence remain unaddressed. The present paper aims to fill this gap. We consider the unique existence conditions and reconstruction methods for a half-inverse problem for the quadratic pencil $\mathcal{H}$. Note that the results obtained here are new and natural generalizations of the well-known ones for the classical Sturm–Liouville operators, which were studied in [26], for a special case where $p\left(x\right)\equiv 0$ and the parameters h and H are ∞ in (1)–(3).

Define the input dataset by

$$\mathcal{D}=\{h,(p\left(x\right),q\left(x\right))\mathrm{on}[0,{\displaystyle \frac{\pi }{2}}],\sigma \}.$$

(4)

More precisely, we shall consider the following inverse problem:

Problem 1.

Given the dataset $\mathcal{D}$ defined by (4), the question remains as to how to reconstruct the functions $p\left(x\right)$ and $q\left(x\right)$ on the interval $(\pi /2,\pi ]$ and the constant H in the boundary condition (3). Under what conditions does the solution exist?

Our contribution is innovative in two regards: Firstly, we develop a direct, step-by-step algorithm to reconstruct the potential functions $p\left(x\right)$ and $q\left(x\right)$ on the unknown half-interval (e.g., $[\pi /2,\pi ]$ when $p\left(x\right),q\left(x\right)$ are known on $[0,\pi /2]$) using the input dataset $\mathcal{D}$ (comprising the known half-potentials and one spectrum of $\mathcal{H})$. The algorithm additionally recovers unknown coefficients in Robin-type boundary conditions (e.g., H in $y^{\prime }\left(\pi \right)+Hy\left(\pi \right)=0$), alongside $p\left(x\right)$ and $q\left(x\right)$, addressing a previously unmet need in pencil half-inverse problems. Secondly, we specify the necessary and sufficient conditions under which the quadratic pencil of the differential operator $\mathcal{H}$ is unique in its existence. These explicit conditions for the unique solvability of the half-inverse problem for $\mathcal{H}$ are rigid and provide a complete solvability criterion.

An outline of the contents of this paper is as follows. In Section 2, we obtain the spectral characterization of the differential pencil $\mathcal{H}$. In Section 3, we reconstruct potential functions $p\left(x\right),q\left(x\right)$ and the constant H by using the input data $\mathcal{D}$. Then, we prove the existence theorem in Section 4.

2. Preliminaries

In this section, we derive some formulations of the inverse problem for $\mathcal{H}$. We provide the asymptotic estimates of the initial solutions and the characteristic function $\Delta \left(\lambda \right)$ associated with problems (1)–(3).

Throughout this paper, let ${\mathcal{L}}_a$ always denote the entire class of functions of exponential type $\le a$, which belong to $L_2\left(\mathbb{R}\right)$ for real $\lambda$. The Nevanlinna class is defined as follows [27]:

Definition 1.

A function $f\left(z\right)$ analytic in $\left|z\right|<1$ is said to be of Nevanlinna class if the integrals

$${\int }_0^{2\pi }{log}^{+}\left|f\left(r{e}^{i\theta }\right)\right|d\theta ,$$

are bounded for $r<1$.

Let us mention that a function analytic in the unit disk belongs to the Nevanlinna class if, and only if, it is the quotient of two bounded analytic functions (see Theorem 2.1 in [27]). We consider the class of sine-type function introduced in [28]:

Definition 2.

An entire function $f\left(\lambda \right)$ of exponential type $a>0$ is said to be a sine-type function if

(1)

The zeros of $f\left(\lambda \right)$ are separated;

(2)

There exist positive constants $A,B,$ and C, such that

$$A{e}^{a\left|y\right|}\le \left|f(x+iy)\right|\le B{e}^{a\left|y\right|},$$

whenever x and y are real and $\left|y\right|\ge C$.

The following interpolation theorem (see, for example, Theorem A in [29]) and Theorem 1, Lecture 21 in [30]) corresponds to sine-type functions.

Theorem 1.

(Lagrange Interpolation Theorem). Let $F\left(z\right)$ be a sine-type function with an indicator diagram of width $2\sigma$ and ${\left\{z_k\right\}}_{k\in \mathbb{Z}}$ be its zero set. Then, the mapping

$${\left\{c_k\right\}}_{k\in \mathbb{Z}}\mapsto f\left(z\right)=F\left(z\right)\sum _{k=-\infty }^{+\infty }{\displaystyle \frac{c_k}{F^{\prime }\left(z_k\right)(z-z_k)}},$$

(5)

is an isomorphism between $l^2$ and ${\mathcal{L}}_{\sigma }$. The series on the right-hand side of (5) converges in the $L_2\left(\mathbb{R}\right)$-norm. The inverse mapping is defined by the relation

$$f\mapsto {\left\{f\left(z_k\right)\right\}}_{k\in \mathbb{Z}}.$$

Let $\phi (x,\lambda )$ and $\psi (x,\lambda )$ be the solutions of (1) that satisfy the following initial conditions:

$$\begin{array}{cc}\hfill & \phi (0,\lambda )=1,{\phi }^{\prime }(0,\lambda )=h,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \psi (\pi ,\lambda )=1,{\psi }^{\prime }(\pi ,\lambda )=H.\hfill \end{array}$$

(7)

The solution $\phi (x,\lambda )$ can be expressed in the integral form [31]

$$\phi (x,\lambda )=cos(\lambda x-\alpha \left(x\right))+{\int }_0^{x}A(x,t)cos\left(\lambda t\right)dt+{\int }_0^{x}B(x,t)sin\left(\lambda t\right)dt,$$

(8)

where

$$\alpha \left(x\right)=xq\left(0\right)+2{\int }_0^x(A(\xi ,\xi )sin\alpha \left(\xi \right)+B(\xi ,\xi )cos\alpha \left(\xi \right))d\xi ,$$

(9)

and the kernel functions $A(x,t)$ and $B(x,t)$ are the solutions of the problem

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial }^{2}A(x,t)}{\partial x^2}}-2q\left(x\right){\displaystyle \frac{\partial B(x,t)}{\partial x}}-p\left(x\right)A(x,t)={\displaystyle \frac{{\partial }^{2}A(x,t)}{\partial t^2}},\hfill \\ {\displaystyle \frac{{\partial }^{2}B(x,t)}{\partial x^2}}+2q\left(x\right){\displaystyle \frac{\partial A(x,t)}{\partial x}}-p\left(x\right)B(x,t)={\displaystyle \frac{{\partial }^{2}B(x,t)}{\partial t^2}},\hfill \\ p\left(x\right)=-q^2\left(x\right)+2{\displaystyle \frac{d}{dx}}[A(x,x)cos\alpha \left(x\right)+B(x,x)sin\alpha \left(x\right)],\hfill \\ A(0,0)=h,B(x,0)=0,{\displaystyle \frac{\partial A(x,t)}{\partial t}}{|}_{t=0}=0.\hfill \end{array}\right.$$

(10)

From (8) and (9), the derivative of $\phi (x,\lambda )$ can be expressed in the following integral form

$$\begin{array}{cc}\hfill {\phi }^{\prime }(x,\lambda )=& - (\lambda -q(0\left)\right)sin(\lambda x-\alpha (x\left)\right)\hfill \\ \hfill & + A(x,x)cos(\lambda x-2\alpha (x\left)\right)+B(x,x)sin(\lambda x-2\alpha (x\left)\right)\hfill \\ \hfill & + {\int }_0^x{\displaystyle \frac{\partial A(x,t)}{\partial x}}cos\left(\lambda t\right)dt+{\int }_0^x{\displaystyle \frac{\partial B(x,t)}{\partial x}}sin\left(\lambda t\right)dt.\hfill \end{array}$$

(11)

We denote by

$$Q\left(x\right)={\int }_0^{x}q\left(t\right)dt,W\left(x\right)={\int }_0^x(q^2\left(t\right)+p\left(t\right))dt.$$

(12)

Similarly to Lemma 1 in [25], we can prove the following asymptotics.

Lemma 1.

For $\left|\lambda \right|\to \infty ,$ the following asymptotics hold:

$$\begin{array}{cc}\hfill \phi (x,\lambda )=& \left(1-{\displaystyle \frac{q\left(x\right)-q\left(0\right)}{2\lambda }}\right)cos(\lambda x-Q\left(x\right))\hfill \\ \hfill & + {\displaystyle \frac{W\left(x\right)+h}{\lambda }}sin(\lambda x-Q\left(x\right))+{\displaystyle \frac{{\mathcal{A}}_{-,1}(x,\lambda )}{\lambda }},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\phi }^{\prime }(x,\lambda )=& - \lambda \left(1-{\displaystyle \frac{q\left(x\right)+q\left(0\right)}{2\lambda }}\right)sin(\lambda x-Q\left(x\right))\hfill \\ \hfill & +(W\left(x\right)+h)cos(\lambda x-Q\left(x\right))+{\mathcal{A}}_{-,2}(x,\lambda ),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \psi (x,\lambda )=& \left(1-{\displaystyle \frac{q\left(\pi \right)-q\left(x\right)}{2\lambda }}\right)cos(\lambda (\pi -x)-Q\left(\pi \right)+Q\left(x\right))\hfill \\ \hfill & + {\displaystyle \frac{W\left(\pi \right)-W\left(x\right)+H}{\lambda }}sin(\lambda (\pi -x)-Q\left(\pi \right)+Q\left(x\right))\hfill \\ \hfill & + {\displaystyle \frac{{\mathcal{A}}_{+,1}(x,\lambda )}{\lambda }},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\psi }^{\prime }(x,\lambda )=& - \lambda \left(1-{\displaystyle \frac{q\left(\pi \right)+q\left(x\right)}{2\lambda }}\right)sin(\lambda (\pi -x)-Q\left(\pi \right)+Q\left(x\right))\hfill \\ \hfill & - \left(W\right(\pi )-W(x)+H)cos\left(\lambda \right(\pi -x)-Q(\pi )+Q(x\left)\right)\hfill \\ \hfill & + {\mathcal{A}}_{+,2}(x,\lambda ).\hfill \end{array}$$

(16)

Here, ${\mathcal{A}}_{-,i}(x,\lambda )\in {\mathcal{L}}_x$ and ${\mathcal{A}}_{+,i}(x,\lambda )\in {\mathcal{L}}_{(\pi -x)}$ for $i=1,2.$

The ${\lambda }_n$ for $n\in \mathbb{Z}$ are the eigenvalues of the operator $\mathcal{H}$ if, and only if, each ${\lambda }_n$ is the zero of $\Delta \left(\lambda \right)$, which is defined by

$$\begin{array}{c}\hfill \Delta \left(\lambda \right)=⟨\phi (x,\lambda ),\psi (x,\lambda )⟩=\phi (x,\lambda ){\psi }^{\prime }(x,\lambda )-{\phi }^{\prime }(x,\lambda )\psi (x,\lambda ),\end{array}$$

(17)

where $\Delta \left(\lambda \right)$ is called the characteristic function of $\mathcal{H}.$ Note that $\Delta \left(\lambda \right)$ does not depend on x. The characteristic function has the following representation from (13) and (14):

$$\begin{array}{cc}\hfill \Delta \left(\lambda \right)=& \lambda \left(1-{\displaystyle \frac{q\left(\pi \right)+q\left(0\right)}{2\lambda }}\right)sin(\lambda \pi -Q\left(\pi \right))\hfill \\ \hfill & - \left(W\right(\pi )+h+H)cos(\lambda \pi -Q(\pi \left)\right)+\mathcal{A}\left(\lambda \right),\hfill \end{array}$$

(18)

where $\mathcal{A}\left(\lambda \right)\in {\mathcal{L}}_{\pi }.$ Let us mention that if $p\left(x\right),q\left(x\right)$ are known on $(0,\pi /2)$, then $\phi (\pi /2,\lambda ),{\phi }^{\prime }(\pi /2,\lambda )$ are obtained. The main aim of this paper is, by applying Theorem A, to uniquely reconstruct $\psi (\pi /2,\lambda ),{\psi }^{\prime }(\pi /2,\lambda )$. Following a well known method (see, for example, [24,32]), the eigenvalues ${\left\{{\lambda }_n\right\}}_{n\in \mathbb{Z}}$ behave asymptotically, as follows:

$${\lambda }_n=n+{\displaystyle \frac{Q\left(\pi \right)}{\pi }}+{\displaystyle \frac{W\left(\pi \right)+h+H}{n\pi }}+{\displaystyle \frac{c_{1,n}}{n}},$$

(19)

as $\left|n\right|\to \infty$, where ${\left\{c_{1,n}\right\}}_{n\in \mathbb{Z}}\in l^2.$ This yields

$$Q\left(\pi \right)=\underset{\left|n\right|\to \infty }{lim}({\lambda }_n\pi -n\pi ),$$

(20)

and

$$W\left(\pi \right)+H=\underset{\left|n\right|\to \infty }{lim}n({\lambda }_n\pi -n\pi -Q\left(\pi \right))-h.$$

(21)

Using Hadamard’s factorization theorem [28] (p. 74), one knows that $\Delta \left(\lambda \right)$ has the following representation:

$$\Delta \left(\lambda \right)=C\left(\lambda -{\lambda }_0\right)\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{\lambda }{{\lambda }_n}}\right)\left(1-{\displaystyle \frac{\lambda }{{\lambda }_{-n}}}\right).$$

(22)

Here, using (18), the constant C can be obtained by

$$C^{-1}=\underset{\tau \to +\infty }{lim}{\displaystyle \frac{2}{\tau [e^{-iQ\left(\pi \right)}-e^{\tau \pi }{e}^{iQ\left(\pi \right)}]}}(i\tau -{\lambda }_0)\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{i\tau }{{\lambda }_n}}\right)\left(1-{\displaystyle \frac{i\tau }{{\lambda }_{-n}}}\right),$$

(23)

where $i=\sqrt{-1}$. This implies that $\Delta \left(\lambda \right)$ is uniquely determined by $\sigma ={\left\{{\lambda }_n\right\}}_{n\in \mathbb{Z}}.$

3. The Reconstruction Method

In this section, we shall solve Problem 1, that is, provide the method for recovering H and the potential functions $p,q$ on $[\pi /2,\pi ]$ in terms of the spectral dataset

$$\mathcal{D}=\{h,(p,q)\mathrm{on}[0,{\displaystyle \frac{\pi }{2}}],\sigma ={\left\{{\lambda }_n\right\}}_{n\in \mathbb{Z}}\}.$$

(24)

Note that $\Delta \left(\lambda \right)$ can be written as

$$\Delta \left(\lambda \right)=\left|\begin{array}{cc}\phi ({\displaystyle \frac{\pi }{2}},\lambda )\hfill & \hfill \psi ({\displaystyle \frac{\pi }{2}},\lambda )\\ {\phi }^{\prime }({\displaystyle \frac{\pi }{2}},\lambda )\hfill & \hfill {\psi }^{\prime }({\displaystyle \frac{\pi }{2}},\lambda )\end{array}\right|.$$

(25)

If potential functions $p\left(x\right),q\left(x\right)$ on $(0,\pi /2)$ and h are given, then we can find $\phi ({\displaystyle \frac{\pi }{2}},\lambda )$ and its zeros ${\left\{v_n\right\}}_{n\in \mathbb{Z}}$ as well as ${\phi }^{\prime }({\displaystyle \frac{\pi }{2}},\lambda )$ and its zeros ${\left\{u_n\right\}}_{n\in \mathbb{Z}}$. Moreover, it is easy to know that the two sequences have the following asymptotics representations:

$$\begin{array}{cc}\hfill v_n& =2n+1+{\displaystyle \frac{2Q(\pi /2)}{\pi }}+{\displaystyle \frac{W(\pi /2)+h}{n\pi }}+{\displaystyle \frac{c_{2,n}}{n}},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill u_n& =2n+{\displaystyle \frac{2Q(\pi /2)}{\pi }}+{\displaystyle \frac{W(\pi /2)+h}{n\pi }}+{\displaystyle \frac{c_{3,n}}{n}},\hfill \end{array}$$

(27)

where the sequences ${\left\{c_{2,n}\right\}}_{n\in \mathbb{Z}},{\left\{c_{3,n}\right\}}_{n\in \mathbb{Z}}\in l^2.$ Substituting $\lambda =v_n$ into (25), we obtain

$$\psi \left({\displaystyle \frac{\pi }{2}},v_n\right)=-{\displaystyle \frac{\Delta \left(v_n\right)}{{\phi }^{\prime }\left(\frac{\pi }{2},v_n\right)}}.$$

(28)

This, together with (15), yields that

$$\begin{array}{cc}\hfill {\mathcal{A}}_{+,1}({\displaystyle \frac{\pi }{2}},v_n)=& - {\displaystyle \frac{v_n\Delta \left(v_n\right)}{{\phi }^{\prime }(\frac{\pi }{2},v_n)}}-v_{n}cos({\displaystyle \frac{v_n\pi }{2}}-Q\left(\pi \right)+Q\left({\displaystyle \frac{\pi }{2}}\right))\left(1-{\displaystyle \frac{q\left(\pi \right)-q(\pi /2)}{2v_n}}\right)\hfill \\ \hfill & - (W\left(\pi \right)-W\left({\displaystyle \frac{\pi }{2}}\right)+H)sin({\displaystyle \frac{v_n\pi }{2}}-Q\left(\pi \right)+Q\left({\displaystyle \frac{\pi }{2}}\right))\hfill \\ \hfill =:& {\alpha }_n.\hfill \end{array}$$

(29)

Note that

$$\begin{array}{cc}\hfill & sin(v_n\pi -Q\left(\pi \right))=-sin(2Q\left({\displaystyle \frac{\pi }{2}}\right)-Q\left(\pi \right))+{\alpha }_{n,1},\hfill \\ \hfill & cos(v_n\pi -Q\left(\pi \right))=-cos(2Q\left({\displaystyle \frac{\pi }{2}}\right)-Q\left(\pi \right))+{\alpha }_{n,2},\hfill \\ \hfill & sin({\displaystyle \frac{v_n\pi }{2}}-Q\left(\pi \right)+Q\left({\displaystyle \frac{\pi }{2}}\right))={(-1)}^{n}cos(2Q\left({\displaystyle \frac{\pi }{2}}\right)-Q\left(\pi \right))+{\alpha }_{n,3},\hfill \\ \hfill & cos({\displaystyle \frac{v_n\pi }{2}}-Q\left(\pi \right)+Q\left({\displaystyle \frac{\pi }{2}}\right))={(-1)}^{n+1}sin(2Q\left({\displaystyle \frac{\pi }{2}}\right)-Q\left(\pi \right))+{\alpha }_{n,4},\hfill \end{array}$$

with ${\left\{{\alpha }_{n,i}\right\}}_{n\in \mathbb{Z}}\in l^2$ for $i=1,2,3,4.$

Substituting the above results into (29), we can deduce that ${\left\{{\alpha }_n\right\}}_{n\in \mathbb{Z}}\in l^2.$ Thus, applying Theorem 1, we obtain

$${\mathcal{A}}_{+,1}({\displaystyle \frac{\pi }{2}},\lambda )=\phi ({\displaystyle \frac{\pi }{2}},\lambda )\sum _{n=-\infty }^{+\infty }{\displaystyle \frac{{\alpha }_n}{\frac{d\phi (\frac{\pi }{2},\lambda )}{d\lambda }{|}_{\lambda =v_n}(\lambda -v_n)}}.$$

(30)

The series on the right-hand side of (30) converges uniformly to a function which belongs to ${\mathcal{L}}_{{\displaystyle \frac{\pi }{2}}}$. Then, $\psi ({\displaystyle \frac{\pi }{2}},\lambda )$ can be constructed from (15).

Moreover, substituting $\lambda =u_n$ into Equation (25), we obtain

$${\psi }^{\prime }\left({\displaystyle \frac{\pi }{2}},u_n\right)={\displaystyle \frac{\Delta \left(u_n\right)}{\phi \left(\frac{\pi }{2},u_n\right)}},$$

which, in combination with (16), yields

$$\begin{array}{cc}\hfill {\mathcal{A}}_{+,2}({\displaystyle \frac{\pi }{2}},u_n)=& {\displaystyle \frac{\Delta \left(u_n\right)}{{\phi }_1\left(\frac{\pi }{2},u_n\right)}}-u_{n}sin({\displaystyle \frac{v_n\pi }{2}}-Q\left(\pi \right)+Q\left({\displaystyle \frac{\pi }{2}}\right))\left(1-{\displaystyle \frac{q\left(\pi \right)+q(\pi /2)}{2v_n}}\right)\hfill \\ \hfill & +(W\left(\pi \right)-W\left({\displaystyle \frac{\pi }{2}}\right)+H)cos({\displaystyle \frac{v_n\pi }{2}}-Q\left(\pi \right)+Q\left({\displaystyle \frac{\pi }{2}}\right))\hfill \\ \hfill =:& {\beta }_n.\hfill \end{array}$$

(31)

Note that

$$\begin{array}{cc}\hfill & sin(u_n\pi -Q\left(\pi \right))=sin(2Q\left({\displaystyle \frac{\pi }{2}}\right)-Q\left(\pi \right))+{\beta }_{n,1},\hfill \\ \hfill & cos(u_n\pi -Q\left(\pi \right))=cos(2Q\left({\displaystyle \frac{\pi }{2}}\right)-Q\left(\pi \right))+{\beta }_{n,2},\hfill \\ \hfill & sin({\displaystyle \frac{u_n\pi }{2}}-Q\left(\pi \right)+Q\left({\displaystyle \frac{\pi }{2}}\right))={(-1)}^{n}sin(2Q\left({\displaystyle \frac{\pi }{2}}\right)-Q\left(\pi \right))+{\beta }_{n,3},\hfill \\ \hfill & cos({\displaystyle \frac{u_n\pi }{2}}-Q\left(\pi \right)+Q\left({\displaystyle \frac{\pi }{2}}\right))={(-1)}^{n}cos(2Q\left({\displaystyle \frac{\pi }{2}}\right)-Q\left(\pi \right))+{\beta }_{n,4},\hfill \end{array}$$

with ${\left\{{\beta }_{n,1}\right\}}_{n\in \mathcal{Z}}\in l^2$ for $i=1,2,3,4.$ Substituting the above results into (31), and by a simple calculation, we know that ${\left\{{\beta }_n\right\}}_{n\in \mathbb{Z}}\in l^2.$ So, applying Theorem 1, we achieve

$${\mathcal{A}}_{+,2}({\displaystyle \frac{\pi }{2}},\lambda )={\phi }^{\prime }({\displaystyle \frac{\pi }{2}},\lambda )\sum _{n=-\infty }^{+\infty }{\displaystyle \frac{{\beta }_n}{\frac{d{\phi }^{\prime }(\frac{\pi }{2},\lambda )}{d\lambda }{|}_{\lambda =u_n}(\lambda -u_n)}}.$$

(32)

Then, ${\psi }^{\prime }({\displaystyle \frac{\pi }{2}},\lambda )$ can be reconstructed from (16).

Based on the above discussion, the algorithm for solving Problem 1 is shown below.

Remark 1.

In fact, for Algorithm 1, we only need the basis property of ${\left\{{\lambda }_n\right\}}_{n\in \mathbb{Z}}$ and with a few modifications, the method we propose above can be adapted to the case of the Dirichlet boundary condition at right point $x=\pi$, corresponding to the value $H=\infty$.

Algorithm 1 Let the input dataset $\Omega$ be given.

(1) Find $Q\left(\pi \right)$ and $W\left(\pi \right)+H$ from (20) and (21), respectively. (2) Construct $\Delta \left(\lambda \right)$ via (22). (3) Compute the solutions $\phi \left(x,\lambda \right)$ and ${\phi }^{\prime }\left(x,\lambda \right)$ of (1) for $x\in (0,\pi /2]$, then obtain the zeros ${\left\{v_n\right\}}_{n\in \mathbb{Z}}$ and ${\left\{u_n\right\}}_{n\in \mathbb{Z}}$ of $\phi \left({\displaystyle \frac{\pi }{2}},\lambda \right)$ and ${\phi }^{\prime }\left({\displaystyle \frac{\pi }{2}},\lambda \right)$, respectively. (4) Construct ${\mathcal{A}}_{+,1}({\displaystyle \frac{\pi }{2}},\lambda )$ via (30) together with (29) and then obtain $\psi ({\displaystyle \frac{\pi }{2}},\lambda )$ by (15); Construct ${\mathcal{A}}_{+,2}({\displaystyle \frac{\pi }{2}},\lambda )$ via (32) together with (31) and then obtain ${\psi }^{\prime }({\displaystyle \frac{\pi }{2}},\lambda )$ by (16). (5) Construct the functions $p,q$ on $(\pi /2,\pi ]$ by the well known procedure described in [20]. (6) Obtain $W\left(\pi \right)$ in virtue of (12) and then obtain H by the value of $W\left(\pi \right)+H$ in step (1).

4. The Existence Problem

In this section, we consider the existence of Problem 1 according to the reconstruction counterparts, as discussed above.

In the dataset

$$\mathcal{D}=\{h,(p_{-}\left(x\right),q_{-}\left(x\right))\mathrm{on}[0,{\displaystyle \frac{\pi }{2}}],\sigma ={\left\{{\lambda }_n\right\}}_{n\in \mathbb{Z}}\},$$

(33)

where $h\in \mathbb{R}$, $q_{-}\left(x\right)\in W_2^1[0,\pi /2]$ and $p_{-}\left(x\right)\in L_2[0,\pi /2]$ are real-valued functions and the real sequences ${\left\{{\lambda }_n\right\}}_{n\in \mathbb{Z}}$ satisfy the following asymptotics:

$$\begin{array}{c}\hfill {\lambda }_n=n+A+{\displaystyle \frac{B}{n}}+{\displaystyle \frac{c_n}{n}},\end{array}$$

(34)

for $\left|n\right|\to \infty$, where $A,B\in \mathbb{R}$ and ${\left\{c_n\right\}}_{n\in \mathbb{Z}}\in l^2.$

Let $\Delta \left(\lambda \right)$ be defined by

$$\Delta \left(\lambda \right)=C(\lambda -{\lambda }_0)\prod _{n\in {\mathbb{Z}}_0}\left(1-{\displaystyle \frac{\lambda }{{\lambda }_n}}\right),$$

(35)

where the constant C is given by

$$C^{-1}=\underset{\tau \to +\infty }{lim}{\displaystyle \frac{2}{\tau [e^{-iA\pi }-e^{\tau \pi }{e}^{iA\pi }]}}(i\tau -{\lambda }_0)\prod _{n=1}^{\infty }\left(1-{\displaystyle \frac{i\tau }{{\lambda }_n}}\right)\left(1-{\displaystyle \frac{i\tau }{{\lambda }_{-n}}}\right).$$

(36)

Let $\phi (x,\lambda )$ be the solution of Equation (1), defined on the interval $x\in [0,\pi /2]$, satisfying the initial conditions $\phi (0,\lambda )=1$ and ${\phi }^{\prime }(0,\lambda )=h$. Because the potential functions ${p,q|}_{x\in [0,\pi /2]}$ are given a priori, we can find that the solutions $\phi (x,\lambda )$ and $\phi (\pi /2,\lambda ),{\phi }^{\prime }(\pi /2,\lambda )$ satisfy the following asymptotics

$$\begin{array}{cc}\hfill \phi ({\displaystyle \frac{\pi }{2}},\lambda )=& \left(1-{\displaystyle \frac{q\left(\frac{\pi }{2}\right)-q\left(0\right)}{2\lambda }}\right)cos({\displaystyle \frac{\pi }{2}}\lambda -Q\left({\displaystyle \frac{\pi }{2}}\right))\hfill \\ \hfill & +{\displaystyle \frac{W\left(\frac{\pi }{2}\right)+h}{\lambda }}sin({\displaystyle \frac{\pi }{2}}\lambda -Q\left({\displaystyle \frac{\pi }{2}}\right))+{\displaystyle \frac{{\mathcal{A}}_{-,1}(\frac{\pi }{2},\lambda )}{\lambda }},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\phi }^{\prime }({\displaystyle \frac{\pi }{2}},\lambda )=& - \lambda \left(1-{\displaystyle \frac{q\left(\frac{\pi }{2}\right)+q\left(0\right)}{2\lambda }}\right)sin({\displaystyle \frac{\pi }{2}}\lambda -Q\left({\displaystyle \frac{\pi }{2}}\right))\hfill \\ \hfill & +(W\left({\displaystyle \frac{\pi }{2}}\right)+h)cos({\displaystyle \frac{\pi }{2}}\lambda -Q\left({\displaystyle \frac{\pi }{2}}\right))+{\mathcal{A}}_{-,2}({\displaystyle \frac{\pi }{2}},\lambda ).\hfill \end{array}$$

(38)

Here,

$$Q\left(x\right)={\int }_0^x{q}_{-}\left(t\right)dt,W\left(x\right)={\int }_0^x(q_{-}^2\left(t\right)+p_{-}\left(t\right))dt.$$

Now we consider the following functional equation:

$$\left|\begin{array}{cc}\phi ({\displaystyle \frac{\pi }{2}},\lambda )\hfill & \hfill \psi \left(\lambda \right)\\ {\phi }^{\prime }({\displaystyle \frac{\pi }{2}},\lambda )\hfill & \hfill \widehat{\psi }\left(\lambda \right)\end{array}\right|=\Delta \left(\lambda \right).$$

(39)

Here, $\Delta \left(\lambda \right)$ is defined by (35). When considering the existence problems of Problem 1, the key is to check under what conditions that the functional function (39) has the unique solution pair $(\psi \left(\lambda \right),\widehat{\psi }\left(\lambda \right)).$

Theorem 2.

Let $\mathcal{D}$ be given by (33), for which the sequence $\sigma ={\left\{{\lambda }_n\right\}}_{n\in \mathbb{Z}}$ satisfies (34). Let the function pair $(\psi \left(\lambda \right),\widehat{\psi }\left(\lambda \right))$ be the solution pair of functional Equation (39).

If the function $\psi /\widehat{\psi }\left(\lambda \right)$ belongs to the Nevanlinna class, then there uniquely exist two real-valued functions, $q_{+}\left(x\right)\in W_2^1[\pi /2,\pi ]$ and $p_{+}\left(x\right)\in L_2[\pi /2,\pi ]$, such that $\sigma ={\left\{{\lambda }_n\right\}}_{n\in \mathbb{Z}}$ is the spectrum of the quadratic pencil of differential operators, defined by (1)–(3), with potential functions $p\left(x\right)=p_{-}\left(x\right),q\left(x\right)=q_{-}\left(x\right)$ a.e. on $(0,\pi /2)$ and $p\left(x\right)=p_{+}\left(x\right),q\left(x\right)=q_{+}\left(x\right)$ a.e. on $(\pi /2,\pi ).$

Proof. 

From the above discussion in Section 3 and using the zeros of $\phi ({\displaystyle \frac{\pi }{2}},\lambda )$ and ${\phi }^{\prime }({\displaystyle \frac{\pi }{2}},\lambda )$ which satisfy the asymptotics (37)–(38), we can construct the functions $\psi \left(\lambda \right)$ and $\widehat{\psi }\left(\lambda \right)$, which obey the asymptotics

$$\begin{array}{cc}\hfill \psi \left(\lambda \right)=& \left(1-{\displaystyle \frac{c}{2\lambda }}\right)cos({\displaystyle \frac{\pi }{2}}\lambda -A\pi +Q\left({\displaystyle \frac{\pi }{2}}\right))\hfill \\ \hfill & +{\displaystyle \frac{B-h-W\left(\frac{\pi }{2}\right)}{\lambda }}sin({\displaystyle \frac{\pi }{2}}\lambda -A\pi +Q\left({\displaystyle \frac{\pi }{2}}\right))+{\displaystyle \frac{{\mathcal{A}}_{+,1}\left(\lambda \right)}{\lambda }},\hfill \\ \hfill \widehat{\psi }\left(\lambda \right)=& -\lambda \left(1-{\displaystyle \frac{\widehat{c}}{2\lambda }}\right)sin({\displaystyle \frac{\pi }{2}}\lambda -A\pi +Q\left({\displaystyle \frac{\pi }{2}}\right))\hfill \\ \hfill & +(B-h-W\left({\displaystyle \frac{\pi }{2}}\right))cos({\displaystyle \frac{\pi }{2}}\lambda -A\pi +Q\left({\displaystyle \frac{\pi }{2}}\right))+{\mathcal{A}}_{+,2}\left(\lambda \right).\hfill \end{array}$$

Here, $c,\widehat{c}\in \mathbb{R}$ and ${\mathcal{A}}_{+,1}\left(\lambda \right),{\mathcal{A}}_{+,2}\left(\lambda \right)\in {\mathcal{L}}_{\pi /2}.$ It is easy to check that their zeros, denoted by ${\left\{{\alpha }_{n,D}\right\}}_{k\in \mathbb{Z}}$ and ${\left\{{\alpha }_{n,N}\right\}}_{k\in \mathbb{Z}}$, satisfy the following two conditions:

$$\begin{array}{cc}\hfill {\alpha }_{n,D}& =2n+1+2A-{\displaystyle \frac{2Q(\pi /2)}{\pi }}+{\displaystyle \frac{B-h-W\left(\frac{\pi }{2}\right)}{n\pi }}+{\beta }_n,{\left\{{\beta }_n\right\}}_{n\in \mathbb{Z}}\in l^2,\hfill \\ \hfill {\alpha }_{n,N}& =2n+2A-{\displaystyle \frac{2Q(\pi /2)}{\pi }}+{\displaystyle \frac{B-h-W\left(\frac{\pi }{2}\right)}{n\pi }}+{\widehat{\beta }}_n,{\left\{{\widehat{\beta }}_n\right\}}_{n\in \mathbb{Z}}\in l^2.\hfill \end{array}$$

It should be noted that ${\left\{{\alpha }_{n,D}\right\}}_{k\in \mathbb{Z}}$ are the zeros of the meromorphic function $\psi /\widehat{\psi }\left(\lambda \right)$, while ${\left\{{\alpha }_{n,N}\right\}}_{k\in \mathbb{Z}}$ are its poles. If $\psi /\widehat{\psi }\left(\lambda \right)$ belongs to the Nevanlinna class, then the two sets ${\left\{{\alpha }_{n,D}\right\}}_{n\in \mathbb{Z}}$ and ${\left\{{\alpha }_{n,N}\right\}}_{n\in \mathbb{Z}}$ are interlacing:

$$\dots {\alpha }_{-2,D}<{\alpha }_{-1,N}<{\alpha }_{-1,D}<{\alpha }_{0,N}<{\alpha }_{0,D}<{\alpha }_{1,N}<{\alpha }_{1,D}<{\alpha }_{2,N}\dots .$$

Thus, the sets ${\left\{{\alpha }_{n,D}\right\}}_{k\in \mathbb{Z}}$ and ${\left\{{\alpha }_{n,N}\right\}}_{k\in \mathbb{Z}}$ satisfy the conditions of the Borg-type two-spectra theorem in [18], and there exist $q_{+}\left(x\right)\in W_2^1[\pi /2,\pi ]$ and $p_{+}\left(x\right)\in L_2[\pi /2,\pi ]$, which generates a Dirichlet–Dirichlet problem with the characteristic function $\psi \left(\lambda \right)$ and a Dirichlet–Neumann problem with the characteristic function $\widehat{\psi }\left(\lambda \right)$ on $[\pi /2,\pi ]$, such that ${\left\{{\alpha }_{n,D}\right\}}_{k\in \mathbb{Z}}$ and ${\left\{{\alpha }_{n,N}\right\}}_{k\in \mathbb{Z}}$ are exactly the Dirichlet–Dirichlet spectrum and Dirichlet–Neumann spectrum of the pencil defined on $[\pi /2,\pi ]$, respectively.

In virtue of the reconstruction procedure of the function pair $(\psi \left(\lambda \right),\widehat{\psi }\left(\lambda \right))$, one knows that it is the solution pair of functional Equation (39), which implies that $\Delta \left(\lambda \right)$ is the characteristic function of the quadratic pencil of differential operators on $[0,\pi ]$, defined by (1)–(3), with potential functions $p=p_{-},q=q_{-}$ on $(0,\pi /2)$ a.e. and $p=p_{+},q=q_{+}$ on $(\pi /2,\pi ).$ Thus, $\sigma ={\left\{{\lambda }_n\right\}}_{n\in \mathbb{Z}}$ is its spectrum. This completes the proof. □

5. Conclusions

This paper serves as a direct sequel to [24], in which the authors establish the uniqueness of the potential for a half-inverse problem. Using the same spectral data, we consider the existence and uniqueness of a solution for the inverse spectral problem to reconstruct the potential. We provide an algorithm to reconstruct the potential function and specify the necessary and sufficient conditions under which the existence of the differential pencil is unique, filling a gap in the the literature on this topic.