LET US CONSIDER THE initial-value problem FOR THE NONLINEAR SCHRODINGER EQUATION (FORMOLA) WHERE D = ¶2X1 + ... + ¶2XD IS THE LAPLACIAN IN D DIMENSIONS, V (X): RD ® R IS A BOUNDED POTENTIAL, AND ¦(|U|2): R+ ® R IS A REAL ANALYTIC FUNCTION. THE MAIN QUESTION IS: IF U0 Î HS(RD) FOR SOME S ³ 0, DOES THERE EXIST A UNIQUE LOCAL SOLUTION U(T) TO THE initial-value problem (1), WHICH REMAINS IN HS(RD) FOR ALL T Î [0,T] AND, IF IT IS SO, DOES IT EXTEND GLOBALLY FOR ALL TIMES T Î R+? THE CLASSICAL THEORY OF PARTIAL DIFFERENTIAL EQUATIONS CONSIDERS SMOOTH SOLUTIONS FOR LARGE valueS OF S ³ 0. IT WAS ONLY RECENTLY THIS THEORY WAS EXTENDED TO SOLUTIONS OF LOW REGULARITY (FOR SMALLER valueS OF S). THE FOLLOWING THEOREM GIVES LOCAL WELL-POSEDNESS OF initial-value problem (1) IN HS(RD) FOR S > D/2 .