A Linear Scale‑Mixing Operator with Power‑Law Eigenfunctions and a Continuous Flow

Overview

This paper introduces and analyzes a linear scale-mixing operator acting on the Banach space of bounded continuous functions over the positive real axis. The framework formalizes the dynamics of scale-dependent mixing by combining local evaluation with scaled evaluation to describe how information or fields evolve across scales. 

Key Mathematical Results

• Operator Formalism: Defines the operator \bm{(\mathbb{R}A)(x) = \lambda A(x) + \mu A(sx)} and proves its boundedness on \bm{C_b(\mathbb{R}_+)}. 

• Exact Spectral Decomposition: Derives the eigenvalue map \bm{\Lambda(\beta) = \lambda + \mu s^\beta} for power-law eigenfunctions \bm{A(x) = x^\beta}. 

• Neutral Stability: Identifies a unique scale-invariant exponent \bm{\alpha} where the eigenfunction remains stable under iteration, providing a "stable ground" for scale-mixing trajectories. 

• Continuous Scaling Limit: Establishes a rigorous transition from discrete iterations to a first-order transport-type partial differential equation: \bm{\partial_t A = -\kappa A + \mu_0 x \partial_x A}. 

• Nonlinear Dynamics: Extends the continuous flow to include a quadratic saturation term, identifying the trivial vacuum and a nontrivial equilibrium \bm{A = \kappa/\gamma}. 

Significance

The model serves as an analytically tractable dynamical system for studying infinitesimal scale evolution. It bridges the gap between discrete scaling operations and continuous transport-type PDEs, offering a foundation for generalizations into nonlinear, stochastic, or higher-dimensional mixing processes.