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The in-medium feature of nuclear force which includes both nucleon-nucleon($NN$) and hyperon-nucleon ($\Lambda N$) interactions impacts the descriptionof single-$\Lambda$ hypernuclei. With the alternated mass number or isospin ofhypernuclei, such effects could be unveiled by analyzing systematical evolutionof the bulk and single-particle properties. From a density-dependentmeson-nucleon/hyperon coupling perspective, a new $\Lambda N$ effectiveinteraction in the covariant density functional (CDF) theory, namelyDD-LZ1-$\Lambda1$, is obtained by fitting the experimental data of $\Lambda$separation energies for several single-$\Lambda$ hypernuclei. It is thenadopted to study the structure and transition properties of single-$\Lambda$hypernuclei in Oxygen isotopes, comparing with several selected CDFLagrangians. Discrepancy is observed explicitly in the isospin evolution of$\Lambda1p$ spin-orbit splitting with various effective interactions, ascribedto their divergence of the meson-hyperon coupling strengths with increasingdensity. In particular, the density-dependent CDFs introduce an extracontribution to enhance the isospin dependence of the splitting, which isoriginated from the rearrangement terms of $\Lambda$ self-energies. Inaddition, the characteristics of hypernuclear radii are studied along theisotopic chain. Owing to the impurity effect of $\Lambda$ hyperon, a sizeshrinkage is observed in the matter radii of hypernuclei as compared to theircores of normal nuclei, while its magnitude is elucidated further to correlatewith the incompressibility of nuclear matter. Besides, there exists a sizablemodel-dependent trend that $\Lambda$ hyperon radii evolve with the neutronnumber, which is decided partly by the in-medium $NN$ interactions as well asthe core polarization effects.

Density-dependent relativistic mean field approach and its application to single-$\Lambda$ hypernuclei in Oxygen isotopes