  In hypernuclear systems, interactions involving nucleons and hyperons are
intricately influenced by the surrounding particles, particularly by the
density and isospin feature of the nuclear medium. In this work, the
relativistic mean-field (RMF) theory is adopted to describe the structure of
several typical $ \Xi^{-} $ hypernuclei. New sets of $\Xi N$ effective
interactions, by taking a density-dependent meson-nucleon/hyperon coupling
perspective, are developed by fitting experimental data on the $\Xi^{-}$
hyperon $1s$ and $1p$ state separation energy of $^{15}_{\Xi^{-}}$C. It is
found that the density-dependent behavior of meson-hyperon coupling strengths
sensitively affects the description of hyperon single-particle levels, which
play a crucial role in the consistent description of the theoretical and
experimental separation energies of $\Xi^{-}$ $1s$ and $1p$ states in
$^{15}_{\Xi^{-}}$C. In fact, the density-dependent meson-baryon coupling
strengths introduce additional rearrangement contributions in the hyperon
self-energy. Correspondingly, detailed forms of density dependence in these
coupling strengths and different consideration of meson-baryon coupling
channels will impact the hyperon single-particle properties within hypernuclei.
Especially with the additional inclusion of isovector scalar $\delta$ meson,
the significant enhancement of rearrangement terms in the effective interaction
DD-ME$ \delta $ impacts the shape of the hyperon potential and alters the
characteristics of the isovector channel dynamics balance in the effective
nuclear force. Such effects may be crucial for describing hyperon separation
energies in specific $\Xi^{-}$ hypernuclei such as $^{12}_{\Xi^{-}}$Be and in
hypernuclei with higher mass numbers. Relevant research underscores the
importance of precisely accounting for medium effects in hyperon-nucleon
interactions.
