Yes, the ideas presented in this work can be applied to other types of partial differential equations (PDEs) beyond the Helmholtz equation, including the Schrödinger equation and Maxwell's equations. The fundamental principles of analytic continuation and complexification are not unique to the Helmholtz equation; they are applicable to a broad class of linear PDEs that exhibit similar wave-like behavior.
For the Schrödinger equation, which describes quantum mechanical wave functions, the complexification approach can be utilized to analyze scattering problems and bound states in potential wells. The analytic continuation of the wave function into the complex plane can provide insights into the behavior of quantum states and their interactions with potentials.
In the case of Maxwell's equations, which govern electromagnetic wave propagation, the complexification scheme can be adapted to handle boundary conditions at interfaces between different media. By employing the same principles of analytic continuation and boundary integral formulations, one can derive solutions for electromagnetic fields in complex geometries, including layered media and waveguides.
Overall, the versatility of the complexification approach makes it a valuable tool for tackling a variety of PDEs, enabling researchers to explore new applications in fields such as acoustics, optics, and quantum mechanics.