This is the starting point of theory of long-range scattering for nonlinear Schrödinger equations. The principle proposed in this paper is referred to as "Ozawa Method" in a recent series of papers of Ginibre and Velo.
This paper gives an affirmative answer to Veselic-Weidmann's conjecture (Math. Z., 156(1977), 93-104) on the borderline of the existence of the ordinary wave operators for Stark effect Hamiltonians.
Several improvements are given by A.Jensen and T. O. : Ann. Inst. Henri Poincaré, physique théorique, 54 (1991), 228-243; Rev. Math. Phys. 5 (1993), 601-629.
An explicit solution is given to the Davey-Stewartson systems.
A detailed description is made on a concentration phenomenon and scattering of the solution.
This paper apparently made Zakharov equations popular.
A technique here to recover "loss of derivatives" in derivative coupling in Zakharov equations is sometimes referred to as "Ozawa-Tsutsumi's Method" as in T. Colin and M. Colin, Differential and Integral Equations, 17(2004), 297-330, though its origin may be traced back to Shibata-Tsutsumi's paper.
A method of recovering "loss of derivatives" in derivative coupling in NLS is introduced by means of "gauge transformation."
This paper gives an affirmative answer to the question posed by Kenig, Ponce, and Vega, Ann. Inst. Henri Poincaré, analyse non linéaire 10(1993), 255-288, on smallness of the data for the local Cauchy problem for NLS.
This is the second example to which the long-range theory of nonlinear scattering is successfully applied.
A new inequality of Trudinger type is introduced. This paper gives a partial answer to Brezis' question in Collége de France Seminar (Pitman Research Notes in Math., 60(1982), 86-97). For applications, see M. Nakamura and T. O.: J. Funct. Anal., 155(1998), 364-380; Math. Z., 231(1999), 479-487; Publ. RIMS, Kyoto Univ., 37(2001), 255-293. Optimality of the inequalities in this paper is studied by S. Adachi and K. Tanaka, Proc. AMS, 128(2000), 2051-2057, and by H. Kozono, T. Sato, and H. Wadade, Indiana Univ. Math. J., 55(2006), 1949-1972.
A positive answer to the conjecture of Hörmander on the global existence of solutions to NLKG in two space dimensions in the famous Lund Lectures 1986-1987.
A description is made on how singularity formation for KGZ is effectively controlled in the energy space by means of nonresonance mechanism in phase space.
A definitive result on the problem of nonrelativistic limit of NLKG to NLS in the energy space.
A unified theory of small data scattering for NLS in a scaling invariant setting. Asymptotic completeness for the wave operators is proved in the Strauss critical case $p=(n+2+ sqrt{n^2 +12n+4}) /(2n)$ without smallness. Another limiting case associated with $p=(2-n+ sqrt{n^2 +4n+36})/4$ with $n geq 4$ in J. Ginibre, T. O., and Velo, Ann. Inst. Henri Poincaré, physique théorique, 60(1994), 211-239, is also shown to be included in the short-range scattering.
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