Exact nonrelativistic polarizabilities of the hydrogen atom with the Lagrange-mesh method

Exact analytical expressions of the dipole polarizabilities of the nonrelativistic hydrogen atom in spherical coordinates are derived with the help of the Lagrange-mesh numerical method. This method can provide exact energies and wave functions for well-chosen conditions of calculation. Exact dipole polarizabilities are obtained after an unambiguous rounding up to at least principal quantum numbers around $n=30$. The scalar polarizability of any $nl$ level is given by $n^4[4n^2+14+7l(l+1)]/4$ and its tensor polarizability is given by $-n^4[3n^2-9+11l(l+1)]l/4(2l+3)$, which allows the calculation of the polarizability of any hydrogen state $nlm$.