And Schrodinger Operators

In this paper we consider smooth analogues of operators studied in connection with pointwise convergence of the solution of the free Schrr odinger equation to the given initial data. Such operators are interesting examples of oscillatory integral operators with degenerate phase functions, and we obtain sharp L 2 ! L 2 bounds. 0. Introduction. We begin this paper by giving motivation for the objects we will study, placing them in their proper context. Consider the initial value problem for the Schrr odinger equation with no potential, (i@ t u(x; t) + x u(x; t) = 0 (x; t) 2 R n R u(x; 0) = f(x) 2 L 2 (R n): (1) Then u(x; t) = (2) ?n Z R n e ix e itjj 2 b f() dd = (e itj j 2 b f())(x) (2) deenes a (weak) solution of (1) such that lim t!0 u(x; t) = f(x) in the L 2 sense. When the integral in (2) is absolutely convergent the limit is a point-wise limit. However, if f is an arbitrary L 2 function the integral in (2) may not be absolutely convergent, and we must take the right hand side of (2) as the deenition of u(x; t). It is not self-evident that u converges pointwise to the initial data in this case, and in fact it sometimes does not. The question of what extra smoothness conditions on f will guarantee the existence pointwise a.e. of lim t!0 u(x; t) arises. For a given s 0 let H s (R n) denote th e L 2-Sobolev space,