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An inverse scattering problem for a quantized scalar field interacting ith a classical source J is considered. Assuming that J is of he form J(t, x) = JT (t)JX(x), (t, x) 2 R×R3, we represent JX (resp. T ) in terms of JT (resp. JX) and the asymptotic fields of and its onjugate field.

We consider the scattering problem for the nonlinear Klein-Gordon equation. The nonlinear term of the equation behaves like a power term. We show that we can define the scattering operator on a suitable 0-neighborhood of the weighted Sobolev space, which improves the known results in some sense. Our proof is based on the Strichartz type estimates.

We study the global well-posedness (GWP) and small data scattering of radial solutions of the relativistic Hartree type equations with nonlocal nonlinearity F(u) = ¸(j ¢ j¡° ¤ juj2)u, ¸ 2 R n f0g, 0 < ° < n, n ¸ 3. We establish a weighted L2 Strichartz estimate applicable to non-radial functions and some fractional integral estimates for radial functions.

Some properties of distributions f satisfying x ¢ rf 2 Lp(Rn), 1 · p < 1, are studied. The operator x ¢ r is the generator of a semi-group of dilations. We first give Sobolev type inequalities with respect to the operator x ¢r. Using the inequalities, we also show that if f 2 Lp loc(Rn), x ¢rf 2 Lp(Rn) and jxjn=pjf(x)j vanishes at infinity, then f belongs to Lp(Rn). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L2(Rn).

We study the inverse scattering problem for the nonlinear Schr\"odinger equation and for the nonlinear Klein-Gordon equation with the generalized Hartree type nonlinearity. We reconstruct the nonlinearity from knowledge of the scattering operator, which improves the known results.

We consider the scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity. We give some estimates for the nonlinearity, and prove the existence of the scattering operator, which improves the known results in some sense. Our proof is based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation.

We shall consider the inverse scattering problem for time dependent version of Hartree-Fock equation and nonlinear Klein-Gordon equation.The uniqueness theorem on identifying the cubic convolution nonlinearity from the knowledge of the scattering operator will be shown.