Mathematical theory for the interface mode in an acoustic waveguide bifurcated from a Dirac point

Abstract. In this work, we prove the existence of a bound state in an acoustic waveguide that consists of two semi-infinite periodic media separated by an interface. The two periodic media under consideration are perturbed from the same periodic medium that attains a Dirac point in its spectrum, and they possess a common spectral gap enclosing the Dirac point. The bound state, which is an eigenmode of the partial differential operator for the acoustic waveguide, decays exponentially away from the interface and is thereby called an interface mode. Using layer potential techniques and asymptotic analysis, we first characterize the spectral gap opening for the two perturbed periodic media at the Dirac point and derive the asymptotics of the Bloch modes near the band gap edges. By reformulating the PDE eigenvalue problem for the acoustic waveguide as a boundary-integral equation over the interface of two semi-infinite media and analyzing the characteristic values of the associated boundary-integral operator, we prove the existence of the interface mode when the perturbation of two periodic media is small. It is shown that the spectral energy of the interface mode is located in the common band gap of two periodic media and can be viewed as a bifurcation from the Dirac point.