Takahiro Kawai

The Bender-Wu Analysis and the Voros Theory

In their pioneering work [BW], Bender and Wu presented the secular equation for anharmonic oscillators ([BW], (F.56)~(F.58) in p. 1257), which was supported by their ingenious WKB analysis. As we shall discuss in our forthcoming article, we can validate their conjecture by Voros’ epoch-making article [V]. At the same time several ideas contained in [BW] can be effectively employed to understand the Voros theory from the viewpoint of (micro)differential operators, and this is what we report here.

Second-microlocalization and asymptotic expansions

The asymptotic expansions of (holonomic) microfunctions is neatly dealt with by the second micro-localization.

A study of Feynman integrals by micro-differential equations

We use the holonomic character of Feynman integrals to describe their singularity structure explicitly in some simple cases. The results in §1 show that under moderate conditions Feynman amplitudes can be locally expressed essentially in terms of Legendre functions near the points where two positive-α Landau-Nakanishi surfaces meet. Related topics such as hierarchical principle in perturbation theory are also discussed in terms of holonomic systems involved. In §4 we use the concrete expressions for Feynman amplitudes obtained in §1 to discuss the validity of Satos conjecture.

Microlocal Analysis of Fixed Singularities of WKB Solutions of a Schrödinger Equation with a Merging Triplet of Two Simple Poles and a Simple Turning Point

We first show that the WKB-theoretic canonical form of an M2P1T (merging two poles and one turning point) Schrodinger equation is given by the algebraic Mathieu equation. We further show that, in analyzing the structure of WKB solutions of a Mathieu equation near fixed singular points relevant to simple poles of the equation, we can focus our attention on the pole part of the equation so that we may reduce it to the Legendre equation. The Borel transformation of WKB-theoretic transformations thus obtained gives rise to microdifferential relations, which lead to the microlocal analysis of the Borel transformed WKB solutions of an M2P1T equation near their fixed singular points. The fully detailed account of the results will be given in Kamimoto et al. (Exact WKB analysis of a Schrodinger equation with a merging triplet of two simple poles and one simple turning point—its relevance to the Mathieu equation and the Legendre equation, 2011).

Application to the Noumi-Yamada System with a Large Parameter

It is known that a traditional Painleve equation (of the variable t) is obtained by the compatibility condition of a system of second order linear differential equations of the variables x and t. Here, when we focus upon the underlying linear system, the latter variable t is often called a deformation parameter. We can consider, with the appropriate introduction of a large parameter (eta ) into these systems, the Stokes geometry for both the linear and non-linear systems in the same way as that described in the previous chapter.

Definition and Basic Properties of Virtual Turning Points

As we mentioned in Preface, global asymptotic analysis of higher order differential equations was thought impossible to construct in 1980s among specialists in asymptotic analysis. In hindsight we find it reasonable, because neither new Stokes curves nor virtual turning points were not in their toolboxes. At that time, physicists H.L. Berk et al. observed in [BNR] that the totality of ordinary Stokes curves was insufficient to describe the Stokes phenomena for WKB solutions of higher order equations and that “new” Stokes curves were needed. Unfortunately it seems that the importance of their observation was not properly appreciated by mathematical specialists in asymptotic analysis as the publication date of the survey article [F] indicates. We believe the reason was that they were not familiar with the Borel transformation depending on parameters, whereas such notion is crucially important in formulating the notion of virtual turning points. Thus we begin our discussion by recalling the core part of “WKB analysis based upon the Borel transformation with parameters” of the Schrodinger equation, a typical second order differential equation. Such analysis is usually referred to as the “exact WKB analysis”; here the adjective “exact” is used in contrast to “asymptotic”.

Exact WKB Analysis of Non-adiabatic Transition Problems for 3-Levels

As we have observed so far, virtual turning points and (new) Stokes curves emanating from them play a crucially important role in discussing the Stokes geometry of a higher order ordinary differential equation and/or a system of ordinary differential equations of size greater than two. Once all the non-redundant virtual turning points are provided, then we can explicitly calculate the analytic continuation of solutions of an ordinary differential equation in view of its complete Stokes geometry and connection formulas discussed in Sect. 1.4. Adopting this approach, we consider the non-adiabatic transition problem for three levels and compute transition probabilities of solutions in this chapter. This is a good application of the exact WKB analysis for a higher order ordinary differential equation to a physical problem, illuminating the role of virtual turning points in the calculation of analytic continuation of solutions.

Virtual turning points — A gift of microlocal analysis to the exact WKB analysis

Several aspects of the notion of virtual turning points are discussed; its background, its relevance to the bifurcation phenomena of a Stokes curve, its importance in the analysis of the Noumi-Yamada system (a particular higher order Painleve equation) and a concrete recipe for locating them. Examples given here make it manifest that virtual turning points are indispensable in WKB analysis of higher order linear ordinary differential equations with a large parameter.

On infrared singularities

The traditional separation of infrared divergent part of the S-matrix from a finite remainder ([YFS], [GY]) is effective only at points where the S-matrix is non-singular, as was pointed out in [S2]. This limitation is due primarily to the approximation n n