Scalars on asymptotically locally AdS wormholes with $\mathcal{R}^{2}$ terms

In this paper we study the propagation of a probe scalar on an asymptotically locally AdS wormhole geometry, that is a solution of General Relativity with a negative cosmological constant and a Gauss-Bonnet term in five dimensions. The metric is characterized by a single integration constant $\rho _{0}$ and the wormhole throat is located at $\rho=0$. In the region $0<\rho <\rho _{0}$, both the gravitational pull as well as the centrifugal contributions to the geodesic motion point in the same direction and therefore they cannot balance. We explore the consequences of the existence of this region on the propagation of a scalar probe. The cases with $\rho _{0}=0$ as well as the limit $\rho _{0}\rightarrow +\infty $ (also possessing a traversable throat) lead to an exactly solvable differential eigenvalue problem, in a shape-invariant potential of the Rosen-Morse and Scarf family, respectively. Here, by numerical methods, we compute the normal modes of a scalar field when $\rho_0\neq 0$, with reflecting boundary conditions at both asymptotic regions. We also explore the effect of a non-minimal coupling between the scalar curvature and the scalar field. Remarkably, there is a particular value of the non-minimal coupling that leads to fully resonant spectra in the limit on vanishing $\rho_0$ as well as when $\rho_0\rightarrow+\infty$, for purely radial modes.