Solving the spectral problem via the boundary approximation in theory

In theory, the resonance scattering structure is triggered by the so-called delocalized modes trapped between the pair. The frequencies and configurations of such modes depend on the half-separation a, which can be derived from the Schrödinger-like equation. We propose to use the boundary conditions to connect the half-localized and delocalized modes, and use boundary approximation (BA) 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 Schrödinger-like equation, and obtain astonishing agreement between them at the long separation regime.