A WKB approach to semiclassical parametrices

Abstract

In this paper, we examine the relationship between WKB approximations and semiclassical parametrices. The fact that the two are related is a point of mathematical folk law in the field of semiclassical analysis. Here, we will set the relationship out precisely and illustrate it with a couple of examples. We will see that semiclassical parametrices for the Schrödinger propagator can be constructed directly from WKB solutions and that this approach has the advantage of allowing us to consider parametrices accurate only to highest order (in $h$) rather than all orders. Such parametrices are often sufficient to suit the technical needs of semiclassical analysis and the WKB method of finding approximate solutions is particularly suitable for obtaining useful explicit expressions for the top-order terms.