On the Frechet Root Kernel of Certain Wave Equations

TL;DR

This paper introduces the Fréchet root sensitivity kernel for complex-valued PDEs, unifies sensitivity analysis across systems, and links the kernel's structure to quantum mechanics' Born rule, offering new insights into wave-based systems.

Contribution

The paper extends the adjoint method to complex PDEs and introduces the Fréchet root kernel, revealing its structure and connection to quantum probability interpretation.

Findings

The Fréchet root kernel has a consistent structure across different PDEs.

The kernel propagates as a wave influenced by initial conditions.

For the Schrödinger equation, the kernel's integrand matches the Born rule.

Abstract

We extend the adjoint method to complex-valued PDEs and introduce the Fr\'echet root sensitivity kernel, as the most fundamental kernel from which all other material-sensitivity kernels can be derived. We apply this framework to four representative equations: two real-valued PDEs (the second-order wave equation and the Euler--Bernoulli beam equation) and two complex-valued PDEs (the complex transport equation and the Schroedinger equation with zero potential). We compute and analyze the Frechet root kernels for all four PDEs and show that, for constant material parameters, the kernel exhibits a consistent structure across systems, while its instantaneous form propagates as a wave whose shape depends on the initial conditions. For the Schroedinger equation, we find an especially notable result: the integrand of the Frechet root kernel coincides with the Born rule of quantum mechanics, suggesting that the probabilistic interpretation of the wavefunction may arise naturally from a general sensitivity-analysis framework rather than from an independent postulate. Our results establish a unified approach to sensitivity analysis for real- and complex-valued PDEs, provide a new perspective on the origin of the Born rule.