Quasinormal modes and bound states of massive scalar fields in wormhole spacetimes

In this work we explore the propagation of massive scalar fields on some wormhole backgrounds. On one side, we consider the Bronnikov-Ellis wormhole solution and wormhole geometries with a non-constant redshift function by introducing a gravitational mass $M$, which goes over into the Bronnikov-Ellis wormhole when the gravitational mass parameter vanishes. We employ the continued fraction method to calculate accurately the quasinormal frequencies of massive scalar fields, particularly focusing on low values of the angular number, and we show an anomalous behaviour of the decay rate of the quasinormal frequencies, for $n \geq \ell$. Also, we show that for a massive scalar field and $M \neq 0$ the effective potential allows potential wells for some values of the parameters which support bound states, which are obtained using the continued fraction method and they are characterized by having only a frequency of oscillation and they do not decay; however, for the Bronnikov-Ellis wormhole the effective potential do not support bound states. On the other side, we consider a wormhole geometry which is an exact solution of $f(R)$ modified gravity. For this geometry the quasinormal frequencies can be obtained analytically, being the longest-lived modes the ones with lowest angular number $\ell$. So, in this wormhole background the anomalous behaviour is avoided.