Scalar perturbation of gravitating double-kink solutions

In this letter, a two-dimensional (2D) gravity-scalar model is studied. This model supports interesting double-kink solutions, and the corresponding metric solutions can be derived analytically. Depending on a tunable parameter $c$, the metric can be symmetric or asymmetric. The Schr\"odinger-like equation for normal modes of the physical linear perturbation is derived. As $c$ varies, the effective potential can have one or two singular barriers. If $c$ is larger than a critical value, the zero mode will be normalizable, despite of the appearance of a strong repulsive singularity. The double-kink solution is always stable against linear perturbations.