Relativistic many-body calculation of energies, lifetimes, polarizabilities, and hyperpolarizabilities in Li-like Be$^+$

Excitation energies of the $ns$, $np$, $nd$, and $nf$ ($n \leq$ 9) states in Li-like Be$^+$ are evaluated within the framework of relativistic many-body theory. All-order calculations of reduced matrix elements, oscillator strengths, transition rates, and lifetimes are given for levels up to $n$ = 9. Electric-dipole ($2s\ -np$), electric-quadrupole ($2s\ -nd$), and electric-octupole ($2s\ -nf$) matrix elements are evaluated in order to obtain the corresponding ground state multipole polarizabilities using the sum-over-states approach. Recommended values are provided for a large number of electric-dipole matrix elements. Scalar and tensor polarizabilities for the $ns$, $np_{1/2}$, $np_{3/2}$, $nd_{3/2}$, and $nd_{5/2}$ states with $n\leq 9$ are also calculated. Scalar hyperpolarizability for the ground $2s$ state is evaluated and compared with non-relativistic calculation. The uncertainties of our calculations are evaluated for most of the values listed in this work. These calculations provide recommended values critically evaluated for their accuracy for a number of Be$^+$ atomic properties useful for a variety of applications.