The 4503 rovibronic term values belonging to the mutually perturbed $A^1Sigma^+_u$ and $b^3Pi_u$ states of Cs$_2$ were extracted from laser induced fluorescence (LIF) $Asim brightarrow X^1Sigma^+_g$ Fourier transform spectra with the 0.01 cm$^{-1}$ uncertainty. The experimental term values of the $A^1Sigma^+_usim b^3Pi_u$ complex covering the rotational levels $Jin [4,395]$ in the excitation energy range $[9655,13630]$ cm$^{-1}$ were involved into coupled-channel (CC) deperturbation analysis. The deperturbation model takes explicitly into account spin-orbit coupling of the $A^1Sigma^+_u(A0^+_u)$ and $b^3Pi^+_{0_u}(b0^+_u)$ states as well as spin-rotational interaction between the $Omega=0$, $1$ and $2$ components of the $b^3Pi^+_{Omega_u}$ state. The emph{ab initio} relativistic calculations on the low-lying electronic states of Cs$_2$ were accomplished in the framework of Fock space relativistic coupled cluster (FSRCC) approach to provide the interatomic potentials of the interacting $A0^+_u$ and $b0^+_u$ states as well as the relevant $Asim b$ spin-orbit coupling function. To validate the present CC deperturbation analysis solely obtained by energy-based data, the $Asim b to X(v^{primeprime}_X)$ LIF intensity distributions were measured and compared with their theoretical counterparts obtained by means of the non-adiabatic vibrational wave functions of the $Asim b$ complex and the FSRCC $Asim b to X$ transition dipole moments calculated by the finite-field method.
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