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

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.

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