Superoscillations and the Klein-Gordon equation via the Fourier method

TL;DR

The paper analyzes the time evolution of 1D Klein-Gordon dynamics with superoscillatory initial data and Dirac-type sources, using the Fourier method to derive explicit solutions and to reveal the role of infinite-order differential operators. It connects superoscillations to the Segal-Bargmann (Fock-space) framework, providing integral representations via the Segal-Bargmann transform and illustrating how the initial superoscillatory data propagate and persist under relativistic dynamics. A key finding is an additional integral term in the inhomogeneous evolution that links to covariance-like kernels of fractional Brownian motion, suggesting a bridge between superoscillatory phenomena in quantum field settings and stochastic processes. Overall, the work embeds superoscillations within the Segal-Bargmann formalism, yielding both explicit evolution formulas and novel integral representations with potential applications in quantum dynamics and stochastic modeling.

Abstract

We investigate the time-evolution problem associated with the Klein-Gordon equation, using superoscillations as initial data. Additionally, the Segal-Bargmann transform is used to derive integral representations of the resulting solutions.

Figures (2)

• Figure 1: Plot of solution $u_n(x,t)$ of Theorem \ref{['Theorem: time evolution P1']} for $m=3$ and $n=10$. The solution is plotted for different values of $a$ as a function of $x$, for a fixed $t=0$.
• Figure 2: Two-dimensional plot of the evolution in time $u_n(x,t)$, for the example of $n=10, a=1.5, m=3$.

Theorems & Definitions (51)

• Definition 2.1: Supershift Property
• Remark 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Remark 2.5
• Example 2.6
• Theorem 2.7
• Remark 2.8
• Remark 2.9