Partial Jost determinants and spectral density of the resolvent with respect to the momentum transfer

Integral representations for partial lost determinants and lost operator on the unit sphere are derived in terms of the spectral density in the momentum transfer of the elliptic operator resolvent in $R_N$. The connection is found between the spectral densities for the potential $V(r)$ in different dimensions $r_N$, $N\geqslant2$. New integral representations for the solutions of the Coulomb problem are obtained.