Relativistic quantum motions of bosonic field under rainbow gravity's environment with point-like defect

In this paper, we focus on the relativistic quantum motions of spin-0 bosonic field in the background of a point-like global monopole taking into account the effects of background curvature. Moreover, we consider this quantum system in the presence of rainbow gravity's environment and analyze the influence on the behavior of scalar bosonic field. We solve the radial equation of the Klein-Gordon wave equation and present non-perturbative eigenvalue solutions for such a system by choosing well-known pair of rainbow functions, for examples, (i) $f\left(\chi \right)=1$, $h\left(\chi \right)=\sqrt{1-{\beta }_{0}\phantom{\rule{0.2em}{0ex}}\chi }$; (ii) $f\left(\chi \right)=1$, $h\left(\chi \right)=\sqrt{1-{\beta }_{0}\phantom{\rule{0.2em}{0ex}}{\chi }^{2}}$; (iii) $f\left(\chi \right)=\left[\mathrm{exp}\left({\beta }_{0}\phantom{\rule{0.2em}{0ex}}\chi \right)-1\right]/\left({\beta }_{0}\phantom{\rule{0.2em}{0ex}}\chi \right)$, $h\left(\chi \right)=1$, and (iv) $f\left(\chi \right)={\left(1-{\beta }_{0}\phantom{\rule{0.2em}{0ex}}\chi \right)}^{-1}=h\left(\chi \right)$, where $\chi =|E|/{E}_{p}$, and ${\beta }_{0}$ is the rainbow parameter. In fact, it is shown that the energy eigenvalues and the wave functions of bosonic fields are influenced by the rainbow parameter ${\beta }_{0}$. Furthermore, the eigenvalue solutions depend on the global monopole of the geometry characterized by the parameter α. The presence of a global monopole breaks the degeneracy of energy spectra and modifies the results compared to the flat space.