PARAMETRICES FOR A CLASS OF SCHRODINGER-EQUATIONS

The article describes the construction of a parametrix, global in space, in the initial value problem for the Schrodinger equation t partial derivative u/partial derivative t = Delta u - Q(x)u - V(x)u, with Q a real quadratic form and V a real C-infinity potential in R(n). The hypothesis on V is that \V(x)\ less than or equal to const. \x\, and \V-(alpha)(x)\ less than or equal to const. \x\(1-\alpha\)( as \x\ --> +infinity). By conjugating the equation with the unitary group exp(itH)(H = -Delta + Q(x)) the problem is reduced to a hyperbolic pseudodifferential operator to which a global version of the geometric optics method applies. The symbolic calculus used by the construction is based on weights that reflect the role of the propagation set. (C) 1995 John Wiley & Sons, Inc.