Solving the Spectral Problem via the Periodic Boundary Approximation in φ⁶ Theory

In φ⁶ theory, the resonance scattering structure is triggered by the so-calls delocalized modes trapped between the K̄K pair. The frequencies and configurations of such modes depend on the K̄K half-separation 2a, 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.