Quantum inverse scattering for time-dependent repulsive Hamiltonians of quadratic type
Atsuhide Ishida
TL;DR
This work analyzes inverse scattering for a multidimensional quantum system with a time-dependent repulsive quadratic Hamiltonian $H_0(t)$, where the coefficient balances between free-like and repulsive-quadratic behavior as $k(t)=\omega^2$ for $|t|\le1$ and $k(t)=\sigma/t^2$ for $|t|>1$. Using the Enss–Weder time-dependent method and a Mehler-based propagator decomposition, the authors derive a reconstructing limit that ties high-velocity scattering data to integrals involving the singular and regular parts of the short-range potential $V=V^{\rm sing}+V^{\rm reg}$, under distinct decay regimes: $\sigma\le2$ (sufficient decay with $\rho>1/2$) and $\sigma>2$ (necessitating Graf-type modified wave operators to recover decay). The key result is a uniqueness theorem: if two potentials yield the same scattering operator $S$, then they must coincide, even in the presence of Coulomb-like singularities. The approach combines propagator decompositions, decay estimates, and Radon-transform arguments to ensure that the potential is recoverable from the velocity limit of the scattering data, highlighting the mathematical richness of time-dependent repulsive quadratic models.
Abstract
We study a multidimensional inverse scattering problem under the time-dependent repulsive Hamiltonians of quadratic type. The time-dependent coefficient on the repulsive term decays as the inverse square of time, which is the threshold between the standard free Schroedinger operator and the time-independent repulsive Hamiltonians of quadratic type. Applying the Enss-Weder time-dependent method, we can determine uniquely the short-range potential functions with Coulomb-like singularities from the velocity limit of the scattering operator.