Applying the sum rule approach, we investigate the energy of a soft dipole motion in $Lambda$ hypernuclei, which results from a dipole oscillation of a $Lambda$ hyperon against the core nucleus. To this end, we systematically study single-$Lambda$ hy
pernuclei, from $^{16}_{;,Lambda}$O to $^{208}_{;;;Lambda}$Pb, for which the ground state wave function is obtained in the framework of Hartree-Fock method with several Skyrme-type $Lambda N$ interactions. Our results indicate that the excitation energy of the soft dipole $Lambda$ mode, $E_{sdLambda}$, decreases as the mass number increases. We find that the excitation energy is well parametrized as $E_{sdLambda}=26.6A^{-1/3}+11.2A^{-2/3}$ MeV as a function of mass number $A$.
Effect of the tensor force on $beta$?-decay is studied in the framework of the proton-neutron random-phase-approximation (RPA) with the Skyrme force. The investigation is performed for even-even semi-magic and magic nuclei, $^{34}$Si, $^{68}$, $^{78}
$Ni and $^{132}$Sn. The tensor correlation induces strong impact on low-lying Gamow-Teller state. In particular, it improves the ?$beta$-decay half-lives. $Q$ and $ft$ values are also investigated and compared with experimental data.
Using the Hartree-Fock plus random-phase-approximation (HF+RPA), we study the impurity effect of $Lambda$ hyperon on the collective vibrational excitations of double-$Lambda$ hypernuclei. To this end, we employ a Skyrme-type $Lambda N$ and $LambdaLam
bda$ interactions for the HF calculations, and the residual interactions for RPA derived with the same interactions. We find that inclusion of two $Lambda$ hyperons in $^{16}$O shifts the energy of the collective states towards higher energies. In particular, the energy of the giant monopole resonance of $^{,,18}_{LambdaLambda}$O, as well as that of $^{210}_{LambdaLambda}$Pb, becomes larger. This implies that the effective incompressibility modulus increases due to the impurity effect of $Lambda$ particle, if the $beta$-stability condition is not imposed.
Fission-related phenomena of heavy $Lambda$ hypernuclei are discussed with the constraint Skyrme-Hartree-Fock+BCS (SHF+BCS) method, in which a similar Skyrme-type interaction is employed also for the interaction between a $Lambda$ particle and a nucl
eon. Assuming that the $Lambda$ particle adiabatically follows the fission motion, we discuss the fission barrier height of $^{239}_{Lambda}$U. We find that the fission barrier height increases slightly when the $Lambda$ particle occupies the lowest level. In this case, the $Lambda$ particle is always attached to the heavier fission fragment. This indicates that one may produce heavy neutron-rich $Lambda$ hypernuclei through fission, whose weak decay is helpful for the nuclear transmutation of long-lived fission products. We also discuss cases where the $Lambda$ particle occupies a higher single-particle level.
