Cayley-transform analysis and numerical validation of the convergent Born series for the Helmholtz equation

Abstract: We develop an operator-theoretic framework for the Convergent Born Series (CBS) method applied to the Lippmann--Schwinger equation for high-frequency Helmholtz problems. In contrast to the Fourier-based analysis of Osnabrugge et al., our approach expresses the preconditioned Lippmann--Schwinger iteration entirely in terms of the resolvent of a self-adjoint background operator. This leads to a unitary Cayley-transform representation of the CBS iteration operator, from which we derive basis-independent bounds on its numerical range and a general convergence criterion valid on arbitrary bounded domains and for complex-valued wave numbers. Because the analysis does not rely on an explicit Green's function in the Fourier domain, the Cayley-transform framework extends naturally to a broader class of frequency-domain wave and diffusion equations whose fundamental solutions are not available in closed form. We further incorporate smoothly tapered complex-wavenumber absorbing layers that preserve the self-adjoint structure of the reference operator and enhance the contractivity of the iteration without modifying the differential operator. In addition to this theoretical generalization, we present a detailed numerical validation in which CBS solutions are benchmarked against PML-based finite-difference wavefield simulations. These experiments demonstrate that the operator-theoretic CBS formulation delivers accurate and stable results across a broad range of contrasts and frequencies, thereby significantly extending the applicability and theoretical foundation of the CBS method beyond previously analyzed settings.