For a system of N identical quantum particles in which every particle feels one external potential and each pair interacts through a pair potential, the time-dependent Hartree-Fock approximation replaces the N-particle wave function by a Slater determinant of one-particle orbitals, turning the linear N-body Schrodinger equation into a coupled nonlinear system whose nonlinearity is a Hartree convolution term plus an exchange integral term. The inverse problem asks whether both potentials can be read back off the scattering operator, the map from a particle's free asymptotic state at minus infinity to its free asymptotic state at plus infinity. The high-velocity limit method pairs the scattering operator minus the identity with a fast-moving wave packet and expands in inverse powers of the velocity: the leading term yields the Fourier transform of the pair potential and the next term yields the X-ray transform of the external potential, which the Riesz-potential inversion formula inverts. Both potentials are thereby uniquely determined, with explicit reconstruction formulas, the pair potential coming from a first-kind integral equation solved through the Picard criterion on the singular system of a compact operator.
In a quantum system of N identical particles, every particle feels one external potential Vext and each pair interacts through a pair potential Vint. The time-dependent Hartree-Fock approximation rep…