We have performed shell-model calculations for the two one valence-neutron isotones $^{135}$Te and $^{137}$Xe and the two one valence-proton isotopes $^{135,137}$Sb. The main aim of our study has been to investigate the evolution of single-particle states with increasing nucleon number. To this end, we have focused attention on the spectroscopic factors and the effective single-particle energies. In our calculations, we have employed a realistic low-momentum two-body effective interaction derived from the CD-Bonn nucleon-nucleon potential that has already proved quite successful in describing the spectroscopic properties of nuclei in the $^{132}$Sn region. Comparison shows that our results reproduce very well the available experimental data. This gives confidence in the evolution of the single-particle states 4 figures predicted by the present study.
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