Solving the Spectral Problem via the Periodic Boundary Approximation in $\phi^6$ Theory

TL;DR

This paper introduces a periodic boundary approximation method to analytically solve the spectral problem in $$ theory, effectively connecting localized and delocalized modes and accurately predicting their frequencies and configurations.

Contribution

The paper presents a novel analytical approach using periodic boundary conditions to solve the spectral problem in $$ theory, providing explicit formulas and confirming accuracy with numerical simulations.

Findings

Analytical formulas for frequencies and configurations of delocalized modes.

Excellent agreement between analytical predictions and numerical simulations at long separation regimes.

Identification of spectral wall locations for delocalized modes.

Abstract

In $\phi^6$ theory, the resonance scattering structure is triggered by the so-calls delocalized modes trapped between the $\bar{K}K$ pair. The frequencies and configurations of such modes depend on the $\bar{K}K$ half-separation 2$a$, can be derived from the Schr\"{o}dinger-like equation. We propose to use the periodic boundary conditions to connect the localized and delocalized modes, and use periodic boundary approximation (PBA) to solve the spectrum analytically. In detail, we derive the explicit form of frequencies, configurations and spectral wall locations of the delocalized modes. We test the analytical prediction with the numerical simulation of the Schr\"{o}dinger-like equation, and obtain astonishing agreement between them at the long separation regime.