Tadayoshi Adachi

Asymptotic completeness for long-range many-particle systems with Stark effect. II

We prove the existence and the asymptotic completeness of the Dollard-type modified wave operators for many-particle Stark Hamiltonians with long-range potentials.

On multidimensional inverse scattering for Stark Hamiltonians

Based on the Enss-Weder [“The geometrical approach to multidimensional inverse scattering,” J. Math. Phys. 36, 3902–3921 (1995)] time-dependent method, we study one of multidimensional inverse scattering problems for Stark Hamiltonians. We first show that when the space dimension is greater than or equal to 2, the high velocity limit of the scattering operator determines uniquely the potential such as ∣x∣−γ with γ>1∕2 which is short range under the Stark effect. This is an improvement of previous results obtained by Nicoleau [“Inverse scattering for Stark Hamiltonians with short-range potentials,” Asymptotic Anal. 35, 349–359 (2003)] and Weder [“Multidimensional inverse scattering in an electric field,” J. Funct. Anal. 139, 441–465 (1996)]. Moreover, we prove that for a given long-range part of the potential under the Stark effect, the high velocity limit of the Dollard-type modified scattering operator determines uniquely the short-range part of the potential.

Long‐range scattering for three‐body Stark Hamiltonians

The existence and the asymptotic completeness of the Graf‐type modified wave operators for three‐body Stark Hamiltonians with long‐range potentials are proven under the condition that the electric field is sufficiently strong and the particles are accelerated with different acceleration.

On multidimensional inverse scattering in an external electric field asymptotically zero in time

Based on the Enss?Weder time-dependent method, we study one of the multidimensional inverse scattering problems for quantum systems in an external electric field asymptotically zero in time as E0(1 + |t|)?? with 0 1/(2 ? ?). Moreover, we prove that the high velocity limit of any one of the Dollard-type modified scattering operators determines uniquely the total potential.Dedicated to the memory of Professor Tetsuro Miyakawa.

Scattering in an external electric field asymptotically constant in time

We show the asymptotic completeness for two-body quantum systems in an external electric field asymptotically non-zero constant in time. One of the main ingredients of this paper is to give some propagation estimates for physical propagators generated by time-dependent Hamiltonians which govern the systems under consideration.

On multidimensional inverse scattering in time-dependent electric fields

We study one of the multidimensional inverse scattering problems for quantum systems in time-dependent electric fields E(t), which is represented as E0(1 + |t|)−μ with 0 ≤ μ 1/(2 − μ) of the potential belonging to the class rather wider than the one given by Adachi, Kamada, Kazuno and Toratani. Our method can also improve previous results in the case where E(t) is periodic in t with non-zero mean E0.

Scattering theory for two-body quantum systems with singular potentials in a time-periodic electric fielda)

Recently, the first author [Adachi, “Asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field,” Commun. Math. Phys. 275, 443 (2007)] proved the asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field whose mean in time is nonzero by obtaining propagation estimates for the physical propagator. However, in his work, it is needed that potentials under consideration are sufficiently smooth. In this paper, when N=2, we prove the asymptotic completeness of (modified) wave operators under the assumption that the potential has the local singularity of type |x|−1+ϵ when d≥3 and 0<ϵ<1, where d is the space dimension. We also discuss the modifiers in the position representation, which are used in the definition of the modified wave operators in the long-range case.

ON SPECTRAL AND SCATTERING THEORY FOR N-BODY, SCHRÖDINGER OPERATORS IN A CONSTANT MAGNETIC FIELD

We consider an N-body quantum system in a constant magnetic field which consists of just one charged and the other N - 1 neutral particles. We prove the existence of a conjugate operator for the Hamiltonian which governs the system, and show the asymptotic completeness of the system under short-range assumptions on the pair potentials.