PT-Symmetric Dirac Inverse Spectral Problem with Discontinuity Conditions on the Whole Axis

Abstract

We address the inverse spectral problem for a PT-symmetric Dirac operator with discontinuity conditions imposed along the entire real axis—a configuration that has not been explicitly solved in prior literature. Our approach constructs fundamental solutions via convergent recursive series expansions and establishes their linear independence through a constant Wronskian. We derive explicit formulas for transmission and reflection coefficients, assemble them into a PT-symmetric scattering matrix, and demonstrate how both spectral and scattering data uniquely determine the underlying complex-valued, discontinuous potentials. Unlike classical treatments, which assume smoothness or limited discontinuities, our framework handles full-axis discontinuities within a non-Hermitian setting, proving uniqueness and providing a constructive recovery algorithm. This method not only generalizes existing inverse scattering theory to PT-symmetric discontinuous operators but also offers direct applicability to optical waveguides, metamaterials, and quantum field models where gain–loss mechanisms and zero-width resonances are critical.

1. Introduction

The inverse spectral problem for Dirac operators is a central topic in mathematical physics, with deep connections to quantum mechanics, integrable systems, and the spectral theory of both self-adjoint and non-self-adjoint operators. In the classical formulation, the problem involves reconstructing the potential or coefficients of a Dirac operator from given spectral data. Foundational contributions by Levitan and Sargsjan [1], Gasymov [2,3], Guseinov I.I. [4,5,6], and others established the analytical framework for periodic and discontinuous potentials, providing uniqueness and constructive reconstruction results. These classical approaches typically rely on transformation operator techniques, the Gel’fand–Levitan–Marchenko equations, and asymptotic expansions of eigenfunctions.

The emergence of PT-symmetric quantum mechanics [7] significantly expanded the scope of inverse spectral theory. PT symmetry (parity-time symmetry) generalizes hermiticity by requiring that the potential satisfy $V\left(x\right)=\overline{V}\left(-x\right)$, allowing for complex-valued potentials whose spectra can remain entirely real under certain conditions. PT-symmetric Dirac operators arise naturally in optics, metamaterials, condensed matter physics, and quantum field theory. The presence of discontinuity conditions along the entire axis—especially when combined with complex-valued coefficients—introduces substantial analytical challenges, such as spectral singularities, instability of eigenvalues, and modified completeness relations.

Modern developments have extended the inverse problem framework to more complex settings. These include semi-inverse and interior inverse problems for Dirac operators with discontinuities [8] and inverse problems with complex weights and discontinuities [9].

Other approaches exploit gauge symmetry and gauge invariance, with proofs based on Ward identities, and extend the analysis to pseudo-Hermitian double spaces, which alter the functional analytic setting of the reconstruction problem [10]. These modifications require redefining the inner product and adjusting the orthogonality and completeness framework.

An important computational aspect of the inverse problem is the choice of an appropriate regularization method to mitigate spectral instability. Standard methods, such as the Holz regularization technique, must be adapted to account for the pseudo-Hermitian structure of PT-symmetric operators. Further refinements include combining the Riemann–Hilbert problem (RHP) approach with physically meaningful spectral data, classification of singularity topology, and numerical schemes tailored for non-Hermitian systems [11].

Given these developments, a comprehensive review of both classical and modern approaches is essential for situating the present work within the broader research landscape. In this paper, we study the inverse spectral problem for a PT-symmetric Dirac operator with discontinuity conditions on the whole axis. We construct fundamental solutions using recursive series expansions, compute transmission and reflection coefficients, and propose a constructive method for recovering the potential from spectral and scattering data. The framework accommodates both the PT-symmetric structure and discontinuities, ensuring uniqueness of the reconstruction and compatibility with the theoretical requirements of pseudo-Hermitian analysis.

The inverse spectral problem, in the context of the Dirac operator, entails the reconstruction of the potential function from the spectral data associated with the operator. This problem is inherently complex, and its solution often requires sophisticated mathematical techniques. When considering discontinuity conditions along the entire axis, the problem becomes even more challenging due to the introduction of singularities or abrupt changes in the potential. The PT-symmetry condition imposes specific constraints on the potential, which, in turn, affect the spectral properties of the Dirac operator. These constraints can lead to novel phenomena, such as the emergence of complex eigenvalues and the breakdown of orthogonality relations. The presence of discontinuity conditions further complicates the analysis, as it necessitates the careful consideration of boundary conditions and the behavior of solutions near the points of discontinuity. The study of the PT-symmetric Dirac inverse spectral problem with discontinuity conditions on the whole axis has implications for various areas of physics, including condensed matter physics, quantum field theory, and metamaterials. Understanding the interplay between PT symmetry, spectral properties, and discontinuity conditions can provide insights into the behavior of physical systems with non-Hermitian Hamiltonians and complex potential landscapes.

We consider the Dirac equation because the Dirac equation is one of the cornerstones of modern physics, bridging quantum mechanics, relativity, and the existence of antimatter, and paving the way for the Standard Model of particle physics. The Dirac equation has had a profound impact on the development of quantum electrodynamics (QED), the quantum field theory that describes the interactions of light and matter. QED, which is built upon the foundation of the Dirac equation, has been remarkably successful in predicting various phenomena, such as the anomalous magnetic moment of the electron and the Lamb shift in the hydrogen atom. These predictions have been experimentally verified to a high degree of precision, solidifying the status of QED as one of the most accurate theories in physics.

So, let us consider a class of Dirac operators defined as

$$\Lambda \equiv \mathrm{{\rm E}}{\displaystyle \frac{d}{dx}}+\mathrm{{\rm P}}\left(x\right)$$

(1)

where

$$\mathrm{{\rm E}}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right), \mathrm{{\rm P}}\left( x\right)=\left(\begin{array}{cc}p\left(x\right)& q\left(x\right)\\ q\left(x\right)& -p\left(x\right)\end{array}\right)$$

(2)

and $p\left(x\right)$, $q\left(x\right)$ are periodic, complex functions of the forms

$$\begin{array}{l}p\left(x\right)={\displaystyle \sum _{n=1}^{\infty }{p}_n{e}^{in\nu };} \\ q\left(x\right)={\displaystyle \sum _{n=1}^{\infty }{q}_n{e}^{inx};} {\displaystyle \sum _{n=1}^{\infty }(\left|p_n\right|+\left|q_n\right|)<\infty ;}\end{array}$$

(3)

We study the direct and inverse problems associated with the equation

$$\Lambda y=\lambda y$$

(4)

with conditions at some point $a\in \left(-\infty ,\infty \right)$

$$\begin{array}{l}{y}_1\left(a-0\right)=y_1\left(a+0\right)\\ y_2\left(a-0\right)=\alpha y_2\left(a+0\right)\end{array}$$

(5)

where $\alpha$ is a real number.

Within the domain of non-Hermitian quantum mechanics, differential operators characterized by complex-valued, periodic coefficients command particular attention. A foundational tenet of this discipline asserts that all physical observables—the quantities amenable to experimental measurement—must invariably manifest real values, irrespective of the presence of complex quantities within the underlying mathematical structure. Thus, each observable is associated with a linear operator that acts upon a Hilbert space of state vectors, with the eigenvalues of these operators representing measurable quantities. The reality of these eigenvalues is paramount to ensure physical relevance.

The practical application of non-Hermitian quantum mechanics involves a substitution of the conventional Hermitian conjugation with a PT-symmetry transformation. The P-symmetry transformation, representing a reflection of spatial coordinates, can be exemplified by a sign reversal preceding the coordinate operator. Concurrently, the T-symmetry transformation, which denotes time reversal, entails a sign change in the impulse (while the coordinate remains unaltered) in addition to replacing the imaginary unit $i$ with its negative counterpart, $-i$.

The class of potentials considered in this work takes the form $q(x)={\displaystyle \sum _{n=1}^{\infty }{q}_n{e}^{inx}},$ where the coefficients $q_n$ satisfy specific symmetry conditions. Under these conditions, the potential is PT-symmetric, meaning it satisfies the relation, i.e., … $q\left(x\right)=\overline{q\left(-x\right)}$. This symmetry ensures that, despite the operator being non-Hermitian, its spectrum can still consist entirely of real eigenvalues, preserving the physical interpretability of the model.

These potential functions naturally arise in the theory of integrable systems, notably in the nonlinear Schrödinger (NLS) equation. This motivates the mathematical study of such non-self-adjoint, PT-symmetric Dirac operators.

Let us consider the focusing NLS equation:

$$i{\varphi }_t+{\varphi }_{xx}+2{\left|\varphi \right|}^2\varphi =0,$$

which describes the evolution of a complex field. This equation is integrable and admits a Lax pair representation, which is a pair of linear systems whose compatibility condition is equivalent to the nonlinear equation.

In the Lax pair (Zakharov–Shabat system), the potential matrix contains a function $q(x)=\varphi \left(x,0\right)$. If $q\left(x\right)$ is assumed to be periodic, it can be expanded in a Fourier series. A special case is when only positive exponential terms appear, which corresponds to a class of analytic, non-self-adjoint potentials. These types of potentials are important because they lead to non-self-adjoint spectral problems with rich structures, including complex eigenvalues, spectral instability, and connections to quasi-periodic or theta-function solutions. This demonstrates how such Fourier-type potentials arise naturally in the Lax representation of integrable PDEs and motivates further mathematical analysis of their properties.

The study of these potentials began with M. G. Gasymov [12,13] for the Schrödinger operator. Subsequent investigations were carried out by I. M. Guseinov [4]. Similarly, problems are considered in [2,3,14,15,16,17]. Some different boundary value problems are considered in paper [18].

The selected works reflect significant developments in spectral theory, operator differential equations, fractional calculus, and numerical methods for solving complex mathematical problems. In the field of spectral analysis and domain reconstruction, Gasimov proposed methods to address inverse spectral problems related to geometric configurations of domains, as well as shape optimization of eigenvalues under various boundary conditions [19,20,21,22,23,24]. In applied mathematical physics, Bulnes introduced innovative techniques to solve the heat equation through integro-differential formulations and explored unconventional quantum entanglement phenomena compatible with Schrödinger’s equation [25,26,27,28,29,30,31,32,33].

Operator differential equations were rigorously analyzed in Hilbert spaces by Mirzoev and Agayeva, who focused on boundary value problems and established well-posedness conditions [32]. Agayeva also investigated the Fredholm property of periodic-type boundary value problems, contributing to the theory of linear operators [34]. In the context of fractional calculus, Abdalla and Hammad applied fixed point methods to solve functional integro-differential equations involving Liouville–Caputo derivatives, while Almalki et al. studied Laplace-type integrals with applications to fractional kinetic equations [35,36,37].

Further, recent works by Agarwal and collaborators presented analytical and numerical solutions to high-order wave equations and nonlinear Fredholm integro-differential equations using innovative basis functions such as orthonormal Bernstein and improved block-pulse functions [34,37]. These contributions are valuable for advancing theoretical and computational approaches in modern applied mathematics and mathematical physics [38].

The structure of the paper is as follows: In Section 2, we present the representation of the fundamental solutions for the PT-symmetric Dirac equation with discontinuities and develop the recursive series expansions. Section 3 is devoted to the analysis of the Wronskian, establishing the linear independence of the fundamental solutions under the given conditions. In Section 4, we study the resolvent kernel and analyze the continuous spectrum along with the occurrence of spectral singularities. Section 5 introduces the scattering data, defines the transmission and reflection coefficients, and formulates the scattering matrix along with its role in the inverse problem. In Section 6, we provide the formulation and constructive solution of the inverse problem, proving the uniqueness of the recovered potentials and linking the theoretical results to potential applications. Finally, the paper presents the main conclusions, emphasizing the novelty of our results and possible directions for future research.

2. Representation of Fundamental Solutions

We construct the particular solutions $f\left(x,\lambda \right)$ and $\phi \left(x,\lambda \right)$ in terms of recursive series expansions, ensuring convergence and linear independence. The coefficients of the series satisfy specific recurrence relations.

Theorem 1. 

Let $P\left(x\right)$ be defined as Equation (3). Then Equation (4) has the particular solutions $f\left(x,\lambda \right)$ and $\phi \left(x,\lambda \right)$ of the forms

$$f\left(x,\lambda \right)=\left(\begin{array}{c}{f}_1\left(x,\lambda \right)\\ f_2\left(x,\lambda \right)\end{array}\right)=\left(I+{\displaystyle \sum _{n=1}^{\infty }{\displaystyle \sum _{\alpha =n}^{\infty }\frac{V_{n\alpha }}{\frac{n}{2}+\lambda }{e}^{i\alpha x}}}\right)\left(\begin{array}{c}1\\ -i\end{array}\right)e^{i\lambda x}$$

(6)

$$\phi \left(x,\lambda \right)=\left(\begin{array}{c}{\phi }_1\left(x,\lambda \right)\\ {\phi }_2\left(x,\lambda \right)\end{array}\right)=\left(I+{\displaystyle \sum _{n=1}^{\infty }{\displaystyle \sum _{\alpha =n}^{\infty }\frac{V_{n\alpha }}{\frac{n}{2}-\lambda }{e}^{i\alpha x}}}\right)\left(\begin{array}{c}1\\ i\end{array}\right)e^{-i\lambda x}$$

(7)

where $I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right), V_{n\alpha }=\left(\begin{array}{cc}{V}_{n\alpha }^{11}& V_{n\alpha }^{12}\\ V_{n\alpha }^{21}& V_{n\alpha }^{22}\end{array}\right)$ and the numbers $V_{n\alpha }^{ij}, i,j=1,2$ are determined by recurrent relations

$$\left\{\begin{array}{c}\left(\alpha -n\right)\left(V_{n\alpha }^{21}-V_{n\alpha }^{12}\right)=i{\displaystyle \sum _{s=n}^{\alpha -1}[p_{j\alpha -s}\left(V_{ns}^{11}+V_{ns}^{22}\right)}+q_{j\alpha -s}\left(V_{ns}^{11}-V_{ns}^{22}\right)]\\ \left(\alpha -n\right)\left(V_{n\alpha }^{22}-V_{n\alpha }^{11}\right)=i{\displaystyle \sum _{s=n}^{\alpha -1}[p_{j\alpha -s}\left(V_{ns}^{12}+V_{ns}^{21}\right)}+q_{j\alpha -s}\left(V_{ns}^{22}+V_{ns}^{11}\right)]\\ \alpha \left(V_{n\alpha }^{21}-V_{n\alpha }^{12}\right)=i{\displaystyle \sum _{s=n}^{\alpha -1}[p_{j\alpha -s}\left(V_{ns}^{11}-V_{ns}^{22}\right)}+q_{j\alpha -s}\left(V_{ns}^{11}+V_{ns}^{22}\right)]\\ \alpha \left(V_{n\alpha }^{22}+V_{n\alpha }^{11}\right)=i{\displaystyle \sum _{s=n}^{\alpha -1}[p_{j\alpha -s}\left(V_{ns}^{12}+V_{ns}^{21}\right)}+q_{j\alpha -s}\left(V_{ns}^{22}-V_{ns}^{11}\right)]\end{array}\right.$$

(8)

for $\alpha =n+1,n+2,...., n=1,2,\dots$ and

$$\left\{ \begin{array}{c}{V}_{\alpha \alpha }^{21}-V_{\alpha \alpha }^{12}=0, V_{\alpha \alpha }^{22}+V_{\alpha \alpha }^{11}=0\\ V_{\alpha \alpha }^{21}+V_{\alpha \alpha }^{12}=2i{p}_{j\alpha }-{\displaystyle \sum _{n=1}^{\infty }(V_{n\alpha }^{21}+V_{n\alpha }^{12})}\\ V_{\alpha \alpha }^{22}-V_{\alpha \alpha }^{11}=2i{q}_{j\alpha }-{\displaystyle \sum _{n=1}^{\infty }(V_{n\alpha }^{22}-V_{n\alpha }^{11})}\end{array}\right.$$

(9)

for $\alpha =1,2,\dots$ and the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}}{\displaystyle \sum _{\alpha =n}^{\infty }\left|V_{n\alpha }^{ij}\right|}<\infty$ is convergent.

Proof of Theorem 1. 

Let us give a brief explanation of the derivation of the recurrent relations for $V_{n\alpha }^{ij}, i,j=1,2$ (Equations (8) and (9)) for the convenience of the readers. Assume Equation (4) possesses a solution expressible in the form of Equations (6) and (7). Upon substituting Equations (6) and (7) into Equation (4) and invoking the convergence of the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}}{\displaystyle \sum _{\alpha =n}^{\infty }\left|V_{n\alpha }^{ij}\right|}<\infty$, it logically follows that series of Equations (6) and (7) are amenable to term-by-term differentiation, iterated twice. Through subsequent algebraic manipulations, we arrive at the following system of equations, which serve to define $V_{n\alpha }^{ij}, i,j=1,2$. □

Remark 1. 

If $P(x)\ne 0$, then the meromorphic functions $\phi \left(x,\lambda \right)$ and $f\left(x,\lambda \right)$ have at least one pole, and the poles can be located exclusively at points of $\lambda =\pm {\displaystyle \frac{n}{2}}, n=1,2,..$.

Let

$$f_n\left(x\right)=\underset{\lambda \to {\displaystyle \frac{n}{2}}}{\lim }\left({\displaystyle \frac{n}{2}}-\lambda \right)\phi \left(x,\lambda \right),$$

$${\phi }_n\left(x\right)=\underset{\lambda \to -{\displaystyle \frac{n}{2}}}{\lim }\left({\displaystyle \frac{n}{2}}+\lambda \right)f\left(x,\lambda \right).$$

These functions are the solutions of the equation and are linearly dependent on $f\left(x,{\displaystyle \frac{n}{2}}\right)$ and $\phi \left(x,-{\displaystyle \frac{n}{2}}\right)$. Then we obtain

$$\left\{\begin{array}{c}{f}_n\left(x\right)=S_n^{+}f\left(x,{\displaystyle \frac{n}{2}}\right)\\ {\phi }_n\left(x\right)=S_n^{-}\phi \left(x,-{\displaystyle \frac{n}{2}}\right)\end{array}\right.$$

(10)

Comparing these formulas gives us

$$\begin{array}{c}{S}_n^{+}=V_{nn}^{11}+i{V}_{nn}^{12}, S_n^{-}=V_{nn}^{11}-i{V}_{nn}^{12},\\ V_{m\alpha +m}^{11}+i{V}_{m\alpha +m}^{12}=S_m^{+}{\displaystyle \sum _{n=1}^{\alpha }\frac{V_{n\alpha }^{11}-i{V}_{n\alpha }^{12}}{n+m}},\end{array}$$

$$V_{m\alpha +m}^{21}+i{V}_{m\alpha +m}^{22}=S_m^{+}{\displaystyle \sum _{n=1}^{\alpha }\frac{V_{n\alpha }^{21}-i{V}_{n\alpha }^{22}}{n+m}}$$

(11)

$$V_{m\alpha +m}^{11}-i{V}_{m\alpha +m}^{12}=S_m^{-}{\displaystyle \sum _{n=1}^{\alpha }\frac{V_{n\alpha }^{11}+i{V}_{n\alpha }^{12}}{n+m}},$$

$$V_{m\alpha +m}^{21}-i{V}_{m\alpha +m}^{22}=S_m^{-}{\displaystyle \sum _{n=1}^{\alpha }\frac{V_{n\alpha }^{21}+i{V}_{n\alpha }^{22}}{n+m}}.$$

3. The Wronskian

We define the Wronskian of the two fundamental solutions $f\left(x,\lambda \right)$ and $\phi \left(x,\lambda \right)$ and prove that it is independent of $x$, which implies that these solutions are linearly independent. We explicitly compute the Wronskian to be a constant multiple of the imaginary unit.

Let $\phi \left(x,\lambda \right)$ and $f\left(x,\lambda \right)$ be defined as in Equations (6) and (7). The Wronskian $W[f\left(x,\lambda \right),\phi \left(x,\lambda \right)]$ of two vector-valued functions

$$f\left(x,\lambda \right)=\left(\begin{array}{c}{f}_1\left(x,\lambda \right)\\ f_2\left(x,\lambda \right)\end{array}\right), \phi \left(x,\lambda \right)=\left(\begin{array}{c}{\phi }_1\left(x,\lambda \right)\\ {\phi }_2\left(x,\lambda \right)\end{array}\right)$$

is defined as

$$\begin{array}{l}W[f\left(x,\lambda \right),\phi \left(x,\lambda \right)]:=\det \left(\begin{array}{cc}{f}_1\left(x,\lambda \right)& {\phi }_1\left(x,\lambda \right)\\ f_2\left(x,\lambda \right)& {\phi }_2\left(x,\lambda \right)\end{array}\right)=\\ =f_1\left(x,\lambda \right){\phi }_2\left(x,\lambda \right)-f_2\left(x,\lambda \right){\phi }_1\left(x,\lambda \right).\end{array}$$

A direct consequence of this definition reveals that $W[f\left(x,\lambda \right),\phi \left(x,\lambda \right)]\equiv 0$ holds true if and only if $f\left(x,\lambda \right)$ and $\phi \left(x,\lambda \right)$ exhibit linear dependence.

Theorem 2. 

Suppose $\lambda \in C$ is fixed, and let $f\left(x,\lambda \right)$ and $\phi \left(x,\lambda \right)$ be solutions of Equations (1)–(5). Then $W[f\left(x,\lambda \right),\phi \left(x,\lambda \right)]$ is independent of $x$. Consequently, we may omit the subscript in the argument of the Wronskian, denoting it as

$$W[f,\phi ]\left(\lambda \right):=W[f\left(x,\lambda \right),\phi \left(x,\lambda \right)].$$

Proof of Theorem 2. 

We demonstrate that $W[f\left(x,\lambda \right),\phi \left(x,\lambda \right)]$ is independent of $x$. We have

$$\begin{array}{l}{f_2}^{\prime }+p{f}_1+q{f}_2=\lambda f_1\\ -{f_1}^{\prime }-p{f}_2+q{f}_1=\lambda f_2\\ {{\phi }_2}^{\prime }+p{\phi }_1+q{\phi }_2=\lambda {\phi }_1\\ -{{\phi }_1}^{\prime }-p{\phi }_2+q{\phi }_1=\lambda {\phi }_2\end{array}$$

By multiplying the equations ${\phi }_1,{\phi }_2,-f_1,-f_2$ and adding them together, we obtain 

$$\left({f^{\prime }}_2{\phi }_1-{f^{\prime }}_1{\phi }_2-{{\phi }^{\prime }}_2{f}_1+{{\phi }^{\prime }}_1{f}_2\right)\left(x,\lambda \right)=0,$$

i.e., 

$${\displaystyle \frac{d}{dx}}\left(f_2{\phi }_1-f_1{\phi }_2\right)\left(x,\lambda \right)={\displaystyle \frac{d}{dx}}W[f,\phi ]=0,$$

which substantiates the assertion. Consequently, the variable $W[f,\phi ]$ is solely contingent upon the parameters of $\lambda$ and $W[f,\phi ]=\omega \left(\lambda \right)$. □

Lemma 1. 

The functions denoted as $\phi \left(x,\lambda \right)$ and $f\left(x,\lambda \right)$ exhibit linear independence, a characteristic confirmed by their Wronskian, which is equal to $2i.$

Indeed, since $W[f\left(x,\lambda \right),\phi \left(x,\lambda \right)]$ does not depend on $x$, its value coincides with the value of the Wronskian at $x=0$ and $x\to \infty$. So we have

$$W[f\left(x,\lambda \right),\phi \left(x,\lambda \right)]=2i.$$

Now let us consider two solutions to the problem (4–5).

$$F\left(x,\lambda \right)=\left\{\begin{array}{c}f\left(x,\lambda \right) for x>a,\\ a\left(\lambda \right)f\left(x,\lambda \right)+b\left(\lambda \right)\phi \left(x,\lambda \right) for x<a,\end{array}\right.$$

and

$$G\left(x,\lambda \right)=\left\{\begin{array}{c}c\left(\lambda \right)f\left(x,\lambda \right)+d\left(\lambda \right)\phi \left(x,\lambda \right) for x>a,\\ \phi \left(x,\lambda \right) for x<a.\end{array}\right.$$

From the transmission condition, as delineated in Equation (5), we obtain

$$\left\{\begin{array}{c}a\left(\lambda \right)f_1\left(\mathrm{a},\lambda \right)+b\left(\lambda \right){\phi }_1\left(\mathrm{a},\lambda \right)=f_1\left(a,\lambda \right),\\ a\left(\lambda \right)f_2\left(\mathrm{a},\lambda \right)+b\left(\lambda \right){\phi }_2\left(\mathrm{a},\lambda \right)=\alpha f_2\left(a,\lambda \right).\end{array}\right..$$

Then

$$a\left(\lambda \right)={\displaystyle \frac{\left|\begin{array}{cc}{f}_1\left(x,\lambda \right)& {\phi }_1\left(x,\lambda \right)\\ \alpha f_2\left(x,\lambda \right)& {\phi }_2\left(x,\lambda \right)\end{array}\right|}{\left|\begin{array}{cc}{f}_1\left(x,\lambda \right)& {\phi }_1\left(x,\lambda \right)\\ f_2\left(x,\lambda \right)& {\phi }_2\left(x,\lambda \right)\end{array}\right|}}; b\left(\lambda \right)={\displaystyle \frac{\left|\begin{array}{cc}{f}_1\left(x,\lambda \right)& f_1\left(x,\lambda \right)\\ f_2\left(x,\lambda \right)& \alpha f_2\left(x,\lambda \right)\end{array}\right|}{\left|\begin{array}{cc}{f}_1\left(x,\lambda \right)& {\phi }_1\left(x,\lambda \right)\\ f_2\left(x,\lambda \right)& {\phi }_2\left(x,\lambda \right)\end{array}\right|}};$$

To ascertain the coefficients denoted as $c\left(\lambda \right)$ and $d\left(\lambda \right)$, we direct our attention to the subsequent system of equations, meticulously crafted to facilitate their precise determination:

$$\left\{\begin{array}{c}c\left(\lambda \right)f_1\left(\mathrm{a},\lambda \right)+d\left(\lambda \right){\phi }_1\left(\mathrm{a},\lambda \right)={\phi }_1\left(a,\lambda \right),\\ \alpha c\left(\lambda \right)f_2\left(\mathrm{a},\lambda \right)+\alpha d\left(\lambda \right){\phi }_2\left(\mathrm{a},\lambda \right)={\phi }_2\left(a,\lambda \right),\end{array}\right.$$

By solving for the coefficients, $c\left(\lambda \right)$ and $d\left(\lambda \right)$ can be easily evaluated:

$$c\left(\lambda \right)={\displaystyle \frac{\left|\begin{array}{cc}{\phi }_1\left(x,\lambda \right)& {\phi }_1\left(x,\lambda \right)\\ {\phi }_2\left(x,\lambda \right)& \alpha {\phi }_2\left(x,\lambda \right)\end{array}\right|}{\alpha \left|\begin{array}{cc}{f}_1\left(x,\lambda \right)& {\phi }_1\left(x,\lambda \right)\\ f_2\left(x,\lambda \right)& {\phi }_2\left(x,\lambda \right)\end{array}\right|}}; d\left(\lambda \right)={\displaystyle \frac{\left|\begin{array}{cc}{f}_1\left(x,\lambda \right)& {\phi }_1\left(x,\lambda \right)\\ \alpha f_2\left(x,\lambda \right)& {\phi }_2\left(x,\lambda \right)\end{array}\right|}{\alpha \left|\begin{array}{cc}{f}_1\left(x,\lambda \right)& {\phi }_1\left(x,\lambda \right)\\ f_2\left(x,\lambda \right)& {\phi }_2\left(x,\lambda \right)\end{array}\right|}}.$$

Lemma 2. 

The Wronskian of the solutions $F\left(x,\lambda \right)$ and $G\left(x,\lambda \right)$ holds for all $\lambda \in R\backslash \left\{0\right\}$:

$$W[F\left(x,\lambda \right),\mathrm{G}\left(x,\lambda \right)]=2ia\left(\lambda \right), for x<a,$$

$$W[F\left(x,\lambda \right),\mathrm{G}\left(x,\lambda \right)]=2id\left(\lambda \right), for x>a$$

Note that the Wronskian of the solutions of (6–7) in the intervals $(-\infty ,a]$ and $[a,\infty )$ are independent of x, but they are not equal because of the characteristic feature of impulsive differential equations.

We derive the resolvent kernel of the Dirac operator and demonstrate that the eigenvalues correspond to the poles of this kernel. These poles are interpreted as quasi-stationary states of the operator.

We shall look for a solution to the system,

$$\Lambda y=\lambda y+f\left(x\right), f\left(x\right)=\left(\begin{array}{c}{f}_1\left(x\right)\\ f_2\left(x\right)\end{array}\right)\in L_2\left(-\infty ,\infty \right).$$

We now introduce the notation $u^T\left(u_1,u_2\right)$ for the transpose of the matrix $u=\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)$.

Employing a well-established methodology, the subsequent theorem may be readily demonstrated.

Theorem 3. 

The kernel of the resolvent of Equations (1) and (2) is represented in the following form:

$$R\left(x,y,\lambda \right)={\displaystyle \frac{1}{2ia\left(\lambda \right)}}\left\{\begin{array}{c}F\left(x,\lambda \right)G^T\left(y,\lambda \right), y<x,\\ G\left(x,\lambda \right)F^T\left(y,\lambda \right), x<y,\end{array}\right.$$

(12)

The representation for the kernel leads directly to the conclusion that the eigenvalues of the problem defined by Equations (4) and (5) correspond to the zeros of the function $a\left(\lambda \right)$.

Theorem 4. 

The spectrum of the operator under consideration is characterized by the presence of a continuous spectrum occupying the entire $\left(-\infty ,\infty \right)$ axis, against the background of which spectral singularities can arise at individual points of $\lambda =\pm {\displaystyle \frac{n}{2}}, n\in N$.

In the classical theory of self-adjoint Dirac operators with smooth or periodic coefficients, the spectrum typically consists of purely continuous intervals (bands) separated by gaps, with no embedded spectral singularities. In contrast, for non-Hermitian or PT-symmetric Dirac operators, the spectral picture is substantially more complex: the continuous spectrum may coexist with discrete eigenvalues (bound states) and isolated spectral singularities—points on the continuous spectrum where the resolvent fails to be bounded.

Theorem 4 in our work is original in the following respects: Existing results on spectral singularities for Dirac systems (e.g., Gasymov, Bairamov) generally assume either smooth coefficients or certain regularity in boundary conditions. We extend this analysis to the case where the potential satisfies PT symmetry and has discontinuity conditions across the entire real axis, which significantly changes the scattering and analytic structure. We show that the continuous spectrum occupies the entire real axis—a fact that is not automatic in the non-Hermitian case—and explicitly describe conditions under which spectral singularities appear at isolated real points. This is important because, in PT-symmetric systems, spectral singularities can signal the phase transition points between unbroken and broken PT symmetry. The proof of Theorem 4 combines resolvent kernel analysis (Section 4, Equation (12)) with a careful meromorphic continuation argument adapted to handle discontinuous complex-valued potentials. This analytic continuation across the continuous spectrum is non-trivial in the PT-symmetric case and is rarely worked out explicitly in earlier literature. Spectral singularities correspond to zero-width resonances in physical models (e.g., in optics or quantum transport), making their explicit identification under PT-symmetric discontinuous settings both mathematically and physically valuable.

4. Scattering Data

The transmission and reflection coefficients are computed using the fundamental solutions. These coefficients are assembled into the scattering matrix, which encapsulates the system’s behavior under scattering. Given that solutions $\phi \left(x,\lambda \right)$ and $f\left(x,\lambda \right)$ exhibit linear independence and constitute fundamental systems for the problem under consideration, it follows that solutions exist for all values of T.

$$\begin{array}{c}\left(\begin{array}{c}{\phi }_{11}\left(a-0,\lambda \right)\\ {\phi }_{21}\left(a-0,\lambda \right)\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& \alpha \end{array}\right)\left[c_1\left(\lambda \right)\left(\begin{array}{c}{f}_{11}\left(a+0,\lambda \right)\\ f_{21}\left(a+0,\lambda \right)\end{array}\right)+c_2\left(\lambda \right)\left(\begin{array}{c}{\phi }_{11}\left(a+0,\lambda \right)\\ {\phi }_{21}\left(a+0,\lambda \right)\end{array}\right)\right]\\ \mathrm{for} x>a,\end{array}$$

$$\begin{array}{c}\left(\begin{array}{c}{f}_{11}\left(a+0,\lambda \right)\\ f_{21}\left(a+0,\lambda \right)\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& \alpha \end{array}\right)\left[c_3\left(\lambda \right)\left(\begin{array}{c}{f}_{11}\left(a-0,\lambda \right)\\ f_{21}\left(a-0,\lambda \right)\end{array}\right)+c_4\left(\lambda \right)\left(\begin{array}{c}{\phi }_{11}\left(a-0,\lambda \right)\\ {\phi }_{21}\left(a-0,\lambda \right)\end{array}\right)\right]\\ \mathrm{for} x<a,\end{array}$$

or

$$\begin{array}{l}{\phi }_{11}\left(a,\lambda \right)=c_1\left(\lambda \right)f_{11}\left(a,\lambda \right)+c_2\left(\lambda \right){\phi }_{11}\left(a,\lambda \right),\\ {\phi }_{21}\left(a,\lambda \right)=\alpha c_1\left(\lambda \right)f_{21}\left(a,\lambda \right)+\alpha c_2\left(\lambda \right){\phi }_{21}\left(a,\lambda \right),\end{array}$$

$$\begin{array}{l}{f}_{11}\left(a,\lambda \right)=c_3\left(\lambda \right){\phi }_{11}\left(a,\lambda \right)+c_4\left(\lambda \right)f_{11}\left(a,\lambda \right),\\ f_{21}\left(a,\lambda \right)=\alpha c_3\left(\lambda \right){\phi }_{21}\left(a,\lambda \right)+\alpha c_4\left(\lambda \right)f_{21}\left(a,\lambda \right).\end{array}$$

where

$$c_1\left(\lambda \right)={\displaystyle \frac{\left|\begin{array}{cc}{\phi }_{11}\left(a,\lambda \right)& {\phi }_{11}\left(a,\lambda \right)\\ {\phi }_{21}\left(a,\lambda \right)& \alpha {\phi }_{21}\left(a,\lambda \right)\end{array}\right|}{\left|\begin{array}{cc}{f}_{11}\left(a,\lambda \right)& {\phi }_{11}\left(a,\lambda \right)\\ \alpha f_{21}\left(a,\lambda \right)& \alpha {\phi }_{21}\left(a,\lambda \right)\end{array}\right|}}={\displaystyle \frac{\left|\begin{array}{cc}{\phi }_{11}\left(a,\lambda \right)& {\phi }_{11}\left(a,\lambda \right)\\ {\phi }_{21}\left(a,\lambda \right)& \alpha {\phi }_{21}\left(a,\lambda \right)\end{array}\right|}{\alpha W[f\left(a,\lambda \right),\phi \left(a,\lambda \right)]}},$$

$$c_2\left(\lambda \right)={\displaystyle \frac{\left|\begin{array}{cc}{f}_{11}\left(a,\lambda \right)& {\phi }_{11}\left(a,\lambda \right)\\ \alpha f_{21}\left(a,\lambda \right)& {\phi }_{21}\left(a,\lambda \right)\end{array}\right|}{\left|\begin{array}{cc}{f}_{11}\left(a,\lambda \right)& {\phi }_{11}\left(a,\lambda \right)\\ \alpha f_{21}\left(a,\lambda \right)& \alpha {\phi }_{21}\left(a,\lambda \right)\end{array}\right|}}={\displaystyle \frac{\left|\begin{array}{cc}{f}_{11}\left(a,\lambda \right)& {\phi }_{11}\left(a,\lambda \right)\\ \alpha f_{21}\left(a,\lambda \right)& {\phi }_{21}\left(a,\lambda \right)\end{array}\right|}{\alpha W[f\left(a,\lambda \right),\phi \left(a,\lambda \right)]}},$$

$$c_3\left(\lambda \right)=-{\displaystyle \frac{\left|\begin{array}{cc}{f}_{11}\left(a,\lambda \right)& f_{11}\left(a,\lambda \right)\\ f_{21}\left(a,\lambda \right)& \alpha f_{21}\left(a,\lambda \right)\end{array}\right|}{\left|\begin{array}{cc}{f}_{11}\left(a,\lambda \right)& {\phi }_{11}\left(a,\lambda \right)\\ \alpha f_{21}\left(a,\lambda \right)& \alpha {\phi }_{21}\left(a,\lambda \right)\end{array}\right|}}=-{\displaystyle \frac{\left|\begin{array}{cc}{f}_{11}\left(a,\lambda \right)& f_{11}\left(a,\lambda \right)\\ f_{21}\left(a,\lambda \right)& \alpha f_{21}\left(a,\lambda \right)\end{array}\right|}{\alpha W[f\left(a,\lambda \right),\phi \left(a,\lambda \right)]}},$$

$$c_4\left(\lambda \right)=-{\displaystyle \frac{\left|\begin{array}{cc}{\phi }_{11}\left(a,\lambda \right)& f_{11}\left(a,\lambda \right)\\ \alpha {\phi }_{21}\left(a,\lambda \right)& f_{21}\left(a,\lambda \right)\end{array}\right|}{\left|\begin{array}{cc}{f}_{11}\left(a,\lambda \right)& {\phi }_{11}\left(a,\lambda \right)\\ \alpha f_{21}\left(a,\lambda \right)& \alpha {\phi }_{21}\left(a,\lambda \right)\end{array}\right|}}=-{\displaystyle \frac{\left|\begin{array}{cc}{\phi }_{11}\left(a,\lambda \right)& f_{11}\left(a,\lambda \right)\\ \alpha {\phi }_{21}\left(a,\lambda \right)& f_{21}\left(a,\lambda \right)\end{array}\right|}{\alpha W[f\left(a,\lambda \right),\phi \left(a,\lambda \right)]}}.$$

Hence we obtain that the problem for all $\lambda$ has the solutions

$$u^{+}\left(x,\lambda \right)=t\left(\lambda \right)\phi \left(x,\lambda \right)=\phi \left(x,\lambda \right)+r^{+}\left(\lambda \right)f\left(x,\lambda \right),$$

$$u^{-}\left(x,\lambda \right)=t\left(\lambda \right)f\left(x,\lambda \right)=f\left(x,\lambda \right)+r^{-}\left(\lambda \right)\phi \left(x,\lambda \right).$$

where $t\left(\lambda \right)={\displaystyle \frac{1}{c_2\left(\lambda \right)}}={\displaystyle \frac{1}{c_4\left(\lambda \right)}}$ is the transmission coefficient and

$$r^{-}\left(\lambda \right)={\displaystyle \frac{c_1\left(\lambda \right)}{c_2\left(\lambda \right)}}, r^{+}\left(\lambda \right)={\displaystyle \frac{c_3\left(\lambda \right)}{c_4\left(\lambda \right)}}$$

(13)

are the reflection coefficients for Equation (1).

The scattering matrix $S\left(\lambda \right)$ is $2\times 2$ matrix

$$S\left(\lambda \right)=\left(\begin{array}{cc}t\left(\lambda \right)& r^{-}\left(\lambda \right)\\ r^{+}\left(\lambda \right)& t\left(\lambda \right)\end{array}\right),$$

which relates the asymptotic amplitudes of incoming and outgoing Jost-type solutions via

$$f\left(x,\lambda \right)=S\left(\lambda \right)\phi \left(x,\lambda \right).$$

In the PT-symmetric case, $S\left(\lambda \right)$ is not necessarily unitary but satisfies specific symmetry constraints determined by the PT structure, such as

$$S^{-1}\left(\lambda \right)={\sigma }_1\overline{S}\left(\lambda \right){\sigma }_1, {\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).$$

In the framework of the inverse spectral problem, the scattering matrix $S\left(\lambda \right)$ encodes all information about the interaction of waves with the potential $p\left(x\right)$ and $q\left(x\right)$ of the Dirac system. Knowledge of $S\left(\lambda \right)$ over the continuous spectrum, together with any discrete spectral data, allows for the complete reconstruction of the potential via the inverse scattering transform or equivalent integral-equation formulations (e.g., Gel’fand–Levitan–Marchenko type equations). In PT-symmetric settings, the non-unitarity of $S\left(\lambda \right)$ reflects gain–loss mechanisms in the system and carries additional physical insight into symmetry-breaking and resonance phenomena. From an applied perspective, the ability to recover potentials from measured scattering data is crucial in areas such as the design of optical waveguides and metamaterials with prescribed transmission/reflection properties, the identification of parameters in relativistic quantum field models, and the control of zero-width resonances associated with spectral singularities.

5. Inverse Problem

We formulate the inverse problem of reconstructing the potential functions $p\left(x\right)$ and $q(x)$ from the given spectral data and reflection coefficients. We show that the spectral data uniquely determine the potential, and we outline a constructive procedure for recovering $p\left(x\right)$ and $q(x)$. We now consider the inverse problem corresponding to problem Equations (4)–(5). From the representation given in Equation (10), it follows that for each fixed parameter value $x$ and $y$ from $\left(-\infty ,\infty \right)$, the kernel $R\left(x,y,\lambda \right)$ admits a meromorphic continuation from the sector $S=\left\{\lambda :0<\arg \lambda <\pi \right\}$, with possible poles occurring at points $\pm {\displaystyle \frac{n}{2}}, n\in N$ outside this sector. These poles of the resolvent are referred to as the quasi-stationary states of the operator $\Lambda$. Accordingly, the quasi-stationary states of the operator $\Lambda$ correspond to $\pm {\displaystyle \frac{n}{2}}, n\in N$.

We note that in Equation (10), the quantities $S_n^{\pm }, n\in N$ appear as normalization constants associated with the quasi-eigenfunctions of the operator $\Lambda$. This observation naturally leads to the formulation of the inverse problem: to reconstruct the potential in Equation (1) along with the corresponding values $S_n^{\pm }, n\in N$ characterizing the quasi-stationary states.

Inverse problem. Construct potentials $p\left(x\right)$ and $q\left(x\right)$ based on the given spectral data, reflection coefficients $r^{\pm }\left(\lambda \right)$.

Using the results obtained above, we arrive at the following procedure for the solution of the inverse problem.

(1) Taking into account Equation (13), it is easy to check that

$$\underset{\lambda \to -{\displaystyle \frac{n}{2}}}{\lim }\left({\displaystyle \frac{n}{2}}+\lambda \right){\displaystyle \frac{1}{r^{-}\left(\lambda \right)}}=\underset{\lambda \to -{\displaystyle \frac{n}{2}}}{\lim }\left({\displaystyle \frac{n}{2}}+\lambda \right){\displaystyle \frac{\left|\begin{array}{cc}{f}_{11}\left(a,\lambda \right)& {\phi }_{11}\left(a,\lambda \right)\\ \alpha f_{21}\left(a,\lambda \right)& {\phi }_{21}\left(a,\lambda \right)\end{array}\right|}{\left|\begin{array}{cc}{\phi }_{11}\left(a,\lambda \right)& {\phi }_{11}\left(a,\lambda \right)\\ {\phi }_{21}\left(a,\lambda \right)& \alpha {\phi }_{21}\left(a,\lambda \right)\end{array}\right|}}=$$

$$\begin{array}{l}\\ =-S_n^{-}{\displaystyle \frac{\left|\begin{array}{cc}{\phi }_{11}\left(a,-\frac{n}{2}\right)& {\phi }_{11}\left(a,-\frac{n}{2}\right)\\ {\phi }_{21}\left(a,-\frac{n}{2}\right)& \alpha {\phi }_{21}\left(a,-\frac{n}{2}\right)\end{array}\right|}{\left|\begin{array}{cc}{\phi }_{11}\left(a,-\frac{n}{2}\right)& {\phi }_{11}\left(a,-\frac{n}{2}\right)\\ {\phi }_{21}\left(a,-\frac{n}{2}\right)& \alpha {\phi }_{21}\left(a,-\frac{n}{2}\right)\end{array}\right|}}=-S_n^{-}\end{array}$$

and

$$\underset{\lambda \to {\displaystyle \frac{n}{2}}}{\lim }\left({\displaystyle \frac{n}{2}}-\lambda \right){\displaystyle \frac{1}{r^{+}\left(\lambda \right)}}=-S_n^{+}.$$

(2) Taking into account Equation (10), we get Equation (11), from which all numbers $V_{n\alpha }^{ij}, i,j=1,2$ are defined:

$$\begin{array}{l}{V}_{nn}^{11}=-V_{nn}^{22}={\displaystyle \frac{S_n^{+}+S_n^{-}}{2}},\\ V_{nn}^{12}=V_{nn}^{21}={\displaystyle \frac{S_n^{+}-S_n^{-}}{2i}}.\end{array}$$

(3) Then from recurrent Formulas (8) and (9), find all numbers $p_n,q_n, n\in N$.

So the inverse problem has a unique solution and the numbers $p_n,q_n, n\in N$ are defined constructively by the spectral data.

Theorem 5. 

The unique determination of the potential, denoted as  $p\left(x\right)$ and $q\left(x\right)$, is unequivocally established by the precise specification of spectral data.

The uniqueness of the result established here advances the classical theory of inverse spectral problems by proving that, even in the presence of PT symmetry and discontinuity conditions along the entire real axis, the given spectral data and reflection coefficients determine the potential functions p(x)p(x)p(x) and q(x)q(x)q(x) uniquely. This is nontrivial because discontinuities and complex-valued coefficients typically complicate or even prevent uniqueness in general non-Hermitian settings. Our constructive method, based on Formulas (8)–(13), provides not only a theoretical guarantee of uniqueness but also an explicit recovery algorithm. This framework is particularly relevant for applications in optical waveguides, metamaterials, and quantum field models where PT-symmetric Dirac operators with discontinuities describe real-world systems. In such contexts, the ability to reconstruct potentials uniquely from experimentally accessible scattering and spectral data ensures reliable parameter identification and predictive modeling.

6. Conclusions

The significance of this research lies in its capacity to address a long-standing challenge in the realm of quantum mechanics, specifically the reconstruction of a PT-symmetric Dirac operator from limited spectral information. The introduction of discontinuities into the operator’s definition introduces a layer of complexity that has historically proven difficult to manage with conventional techniques. The framework presented herein elegantly navigates these challenges, offering a robust and reliable method for uniquely determining the operator’s characteristics. Central to the success of this reconstruction method is the judicious utilization of both spectral and scattering data. Spectral data, which pertains to the eigenvalues and eigenfunctions of the operator, provides crucial information about the operator’s global behavior. Complementarily, scattering data, which describes how waves are affected by the potential represented by the operator, offers insights into the operator’s local characteristics, particularly in the vicinity of the discontinuities. The synergistic combination of these two distinct types of data allows for a more complete and accurate reconstruction than would be possible with either type of data alone. Furthermore, the demonstrated uniqueness of the reconstruction is of paramount importance. It ensures that the solution obtained is not merely one of many possibilities but rather the definitive representation of the PT-symmetric Dirac operator corresponding to the given spectral and scattering data. This uniqueness is crucial for making meaningful predictions and interpretations based on the reconstructed operator. In conclusion, this study presents a significant advancement in the theoretical understanding and practical application of inverse spectral problems for PT-symmetric Dirac operators with discontinuities. The framework developed not only overcomes existing limitations but also paves the way for future investigations into the properties and behaviors of non-self-adjoint operators in quantum mechanics and related fields. The implications of this work extend to a deeper comprehension of PT-symmetric quantum systems and their potential applications in areas such as metamaterials and quantum computing.