Introduction and main resultsWe consider the unique global existence of solutions in a weaker class than H^1 (R^n) and some properties of the solution operator for the following nonlinear Schrodinger equation: where t₀in R and lambdain R. By a (n) we denote infty if n=1 or n=2 and (n+2) / (n-2) if ngeqq 3. There are many papers concerning the global existence of solutions for Problem (1. 1)? (1. 2) (see, e. g. , 1-2, 4-7 and 9). Ill 1 Baillon, Cazenave and Figueira show that if 1leqq nleqq 3, 10, Problem (1. 1)? (1. 2) has a unique global strong solution u (t) in C (R;H^2 (R^n) ) cap C^1 (R;L^2 (R^n) ) for any u₀in H^2 (R^n). In -2 Ginibre and Velo show that if 10 or if 10 and p>1, Problem (1. 1)? (1. 2) has at least one global weak solution u (t) in L^infty (R;H^1 (R^n) cap L^p+1 (R^n) ) for any u₀in H^1 (R^n) cap L^p+1 (R^n) (see also 5). In 10 M. Tsutsumi and N. Hayashi mathrmdmathrmimathrmsmathrmcmathrmu₃₎ₓ₌₀ₓ₇ₑ₌{₎}mathrms the unique global existence of classical solutions for (1. 1)? (1. 2) with lambda>0 (see also Pecher and von Wahl 4). Furthermore, in 9M. Tsutsumi discusses the unique global solution in ovalboxttsmall REJECT (R^n) or in the weighted Sobolev space for (1. 1)? (1. 2) with lambda>0. In almost all of previous papers the solution of (1. 1)? (1. 2) has been constructed in a space not larger than the energy space, that is, H^1 (R^n), because the proofs in almost all of previous papers are based on the energy conservation law. However, in 7 Strauss constructs the wave operators from L^2 (R^n) to L^2 (R^n) for the equation (1. 1) with p=1+4/n (see 7, Theorem 5). His results are almost equivalent to the construction in L^2 (R^n) of unique local solutions for (1. 1)? (1. 2) with p=1+4/n. In this paper we prove that when 1<p<1+4/n, we can construct the unique global solution of (1. 1)? (1. 2) for u₀ in L^2 (R^n) (but possibly not in H^1 (R^n) ). Such a solution is
Yoshio Tsutsumi (Wed,) studied this question.