Global Fourier Integral Operators and Semiclassical Asymptotics

Abstract

In this paper we introduce a class of semiclassical Fourier integral operators with global complex phases approximating the fundamental solutions (propagators) for time-dependent Schrödinger equations. Our construction is elementary, it is inspired by the joint work of the first author with Yu. Safarov and D. Vasiliev. We consider several simple but basic examples.

GLOBAL $h$ FOURIER INTEGRAL OPERATORS WITH COMPLEX-VALUED PHASE FUNCTIONS

“…The results here are similar to those proved recently in [6]. However, there are two main differences between the approach taken here and that used in [6].…”

“…However, there are two main differences between the approach taken here and that used in [6]. Firstly, we find it convenient to define the notion of h FIO in a way that is more analogous to the definition of admissible h PDO given in [4].…”

Regularity and Decay of Solutions to the Stark Evolution Equation

“…For fixed t # R, exp(itH 0 ) u 0 may not be in the domain of (I+ |H 0 |) 1Â4 as a selfadjoint operator in L 2 (R n ). However, exp(itH 0 ) maps S(R n ) (the Schwartz space of rapidly decaying smooth functions) continuously into itself (see (1.3) as can be verified directly or derived from more general formulae [1,12]. However, the t-dependence of the right-hand side in (1.3) makes it impossible to derive the``1 2 -derivative gain'' based on that gain for the free group exp( &it2).…”

WKB Analysis of Bohmian Dynamics

“…This problem can be overcome by considering a class of Fourier integral operators whose Schwartz kernel furnishes an "-oscillatory integral with complex phase and quadratic imaginary part; see [40, theorem 2.1] for a precise definition. Using this, the authors of [40] construct a global-in-time approximation of U " .t / for potentials satisfying V 2 C 1 b .R d /, i.e., smooth and bounded together with all derivatives (see also [30,36] for closely related results with slightly different assumptions). By applying the stationary phase lemma to this type of (global) Fourier integral operator, one infers the following result as a slight generalization of [40, theorem 5.1]:…”