Quantized Dissipation from the Inverse-Square Anomaly in a Non-Hermitian Klein–Gordon Field

Abstract

We construct an exactly solvable relativistic model that embeds the anomalous inverse-square interaction into a non-Hermitian Klein–Gordon field theory through a purely imaginary, scale-invariant scalar potential. The stationary field equation reduces to an inverse-square Schrödinger-type problem with a quadratic spectral parameter. By imposing a perfectly absorbing boundary condition at the singularity, the Hermitian inverse-square anomaly is naturally continued to an open dissipative system, where the geometric tower of bound states is reinterpreted as a hierarchy of resonant decay modes. We present an explicit analytical derivation showing that the resonance decay rates obey a universal log-periodic scaling relation determined solely by the anomalous scaling exponent and robust against generic subleading perturbations. The construction establishes a direct connection between self-adjoint boundary conditions and their non-Hermitian absorbing counterparts through analytic continuation of the boundary phase, demonstrating that the underlying discrete scale invariance survives in the presence of irreversible absorption. The resulting framework provides a minimal, exactly solvable laboratory for studying quantum scale anomalies, boundary-condition-induced non-Hermiticity, and quantized dissipation in relativistic open quantum systems.