Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations

For a higher order linear ordinary differential operator P , its Stokes curve bifurcates in general when it hits another turning point of P . This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in the Noumi-Yamada system, a system of linear differential equations associated with the fourth Painlevé equation. Exact WKB analysis, that is, WKB analysis based on the Borel resummation, has turned out to be an important and useful tool in mathematical physics [1]; its advantage certainly consists in its efficiency in manipulating exponentially small terms, but still more important, from the theoretical viewpoint, are the fact that the Borel transform of an ordinary differential operator P (x, η−1d/dx) with a large parameter η is a partial differential operator on the (x, y)-space with y denoting the variable dual to η, and the fact that microlocal analysis, a new and powerful machinery in mathematics [2], clarifies the structure of singularities of solutions of the Borel transformed equation, i.e., the Borel transformed WKB solutions, which are multi-valued analytic functions on (x, y)-space. An important example of the influence of microlocal analysis on WKB analysis is the introduction of the notion of a virtual turning point for differential equations of the third or higher order [3]; it is, by definition, the xcomponent of the self-intersection point of a bicharacteristic curve of the Borel transform of the operator P (x, η−1d/dx). Note that a bicharacteristic curve is the most “elementary” carrier of singularities of solutions of linear partial differential equations in general [2]. Note also that Voros [4] uses the corresponding result for the Tricomi-type operator in constructing his theory of exact WKB analysis for differential operators of the second order. As the so-called new Stokes curve for higher order operators [5] is nothing but an ordinary Stokes curve emanating from a virtual turning point, the importance of the notion of a virtual turning point is practically evident. Actually it plays an important role in computing the transition probabilities for the non-adiabatic transition problem of the Landau-Zener type [6]. In this paper we show how important a role a virtual turning point plays from the theoretical viewpoint. To be more concrete, we validate the following Assertion A using a concrete example we encounter in the exact WKB analysis of the Painlevé transcendent [7]: Assertion A: The role of a virtual turning point is commensurate with that of an ordinary turning point; theoretically speaking, there is no distinction between them. In validating this challenging assertion, we divide our discussion into two steps: we first show the mechanism that relates a virtual turning point with the bifurcation phenomenon of a Stokes curve that is observed when it hits a simple turning point, and then we argue how the mechanism works in understanding the true nature of a seemingly paradoxical phenomenon which has been just found [7].