On the WKB-theoretic structure of a Schrödinger operator with a merging pair of a simple pole and a simple turning point

A Schrödinger equation with a merging pair of a simple pole and a simple turning point (called MPPT equation for short) is studied from the viewpoint of exact Wentzel-Kramers-Brillouin (WKB) analysis. In a way parallel to the case of mergingturning-points (MTP) equations, we construct a WKB-theoretic transformation that brings anMPPTequation to its canonical form (the ∞-Whittaker equation in this case). Combining this transformation with the explicit description of the Voros coefficient for the Whittaker equation in terms of the Bernoulli numbers found by Koike, we discuss analytic properties of Borel-transformed WKB solutions of an MPPT equation. 0. Introduction The principal aim of this article is to form a basis for the exact WKB analysis of a Schrödinger equation (0.1) ( d dx2 − ηQ(x, η) ) ψ = 0 (η: a large parameter) with one simple turning point and with one simple pole in the potential Q. As [Ko1] and [Ko3] emphasize, the Borel transform of a WKB solution of (0.1) displays, near the simple pole singularity, behavior similar to that near a simple turning point. Hence it is natural to expect that such an equation plays an important role in exact WKB analysis in the large. Such an expectation has recently been enhanced by the discovery (see [KoT]) that the Voros coefficient of a WKB solution of (0.1) with (0.2) Q = 1 4 + α x + η−2 γ x2 (α, γ: fixed complex numbers) can be explicitly written down with the help of the Bernoulli numbers. The potential Q given by (0.2) plays an important role in Section 2; the Schrödinger Kyoto Journal of Mathematics, Vol. 50, No. 1 (2010), 101–164 DOI 10.1215/0023608X-2009-007, © 2010 by Kyoto University Received July 30, 2009. Revised October 2, 2009. Accepted October 9, 2009. Mathematics Subject Classification: Primary 34M60; Secondary 34E20, 34M35, 35A27, 35A30. Authors’ research supported in part by Japan Society for the Promotion of Science Grants-in-Aid 20340028, 21740098, and 21340029. 102 Kamimoto, Kawai, Koike, and Takei equation with the potential Q of the form (0.2), that is, the Whittaker equation with a large parameter η, gives us a WKB-theoretic canonical form of a Schrödinger equation with one simple turning point and with one simple pole in its potential. We note that the parameter α contained in the Whittaker equation in Section 2 is an infinite series α(η) = ∑ k≥0 αkη −k (αk: a constant), and we call such an equation the ∞-Whittaker equation when we want to emphasize that α is not a genuine constant but an infinite series as above. In order to make a semiglobal study of a Schrödinger equation with one simple turning point and with a simple pole in its potential, we let the simple pole singular point merge with the turning point and observe what kind of equation appears. For example, what if we let α tend to zero in (0.2) with γ being kept intact? Interestingly enough, the resulting equation is what we call a ghost equation (see [Ko2]); we have been wondering where we should place the class of ghost equations in regard to the whole WKB analysis. A ghost equation has no turning point by its definition (cf. Remark 1.1 in Section 1); still, a WKB solution of a ghost equation displays singularity similar to that which a WKB solution normally has near a turning point. The singularity is due to the singularities contained in the coefficients of η−k (k ≥ 1) in the potential Q (see [Ko2] for details; there a ghost (point) is tentatively called a “new” turning point). In view of the above observation, we regard a Schrödinger equation with one simple turning point and with one simple pole in its potential as an equation obtained through perturbation of a ghost equation by a simple pole term aq(x,a)/x, where a is a complex parameter and q(x,a) is a holomorphic function defined on a neighborhood of (x,a) = (0,0). An equation obtained by such a procedure is called an equation with a merging pair of a simple pole and a simple turning point, or, for short, an MPPT equation. Precisely speaking, we call a Schrödinger equation (0.1) an MPPT equation if its potential Q depends also on an auxiliary parameter a and has the form (0.3) Q = Q0(x,a) x + η−1 Q1(x,a) x + η−2 Q2(x,a) x2 , where Qj(x,a) (j = 0,1,2) are holomorphic near (x,a) = (0,0) and Q0(x,a) satisfies the following conditions (0.4) and (0.5): Q0(0, a) = 0 if a = 0, (0.4) Q0(x,0) = c (0) 0 x + O(x ) holds with c 0 being (0.5) a constant different from 0. Clearly we find a ghost equation at a = 0; furthermore, the implicit function theorem together with the assumption (0.5) guarantees the existence of a unique holomorphic function x(a) that satisfies (0.6) Q0 ( x(a), a ) = 0. Assumption (0.4) entails (0.7) x(a) = 0 if a = 0, On the WKB-theoretic structure of an MPPT operator 103 and the assumption (0.5) guarantees that, for a sufficiently small a( = 0), x = x(a) is a simple turning point of the operator in question. As the term “an MPPT equation” indicates, it is a counterpart of an MTP equation in our context. An MTP equation, that is, a merging-turning-points equation introduced in [AKT4] contains, by definition, two simple turning points that merge into one double turning point as the parameter t tends to zero; whereas, in an MPPT equation, a simple pole and a simple turning point merge into a ghost point where neither zero nor singularity is observed in the highest degree (i.e., degree zero) in η part of the potential. The parallelism of these two notions is not a superficial one. The reduction of an MPPT equation to a canonical one is achieved in Sections 1 and 2 in a way parallel to that used in the reduction of MTP equation to a canonical one. First, in Section 1 we construct a WKB-theoretic transformation that brings an MPPT equation with the parameter a being zero to a particular ∞-Whittaker equation, that is, the ∞Whittaker equation with the top degree part of the parameter α(η) being zero (i.e., α(η) = ∑ k≥1 αkη −k), and then in Section 2 we construct the transformation of a generic (i.e., a = 0) MPPT equation to the ∞-Whittaker equation in the form of a perturbation series in a, starting with the transformation constructed in Section 1. In Sections 1 and 2 we focus our attention on the formal aspect of the problem, and the estimation of the growth order of the coefficients that appear in several formal series is given separately in Appendices A and B. One important implication of the estimates given in Appendix B is that they endow the formal transformation with an analytic meaning as a microdifferential operator through the Borel transformation. Furthermore, as is shown in Theorems 1.7 and 2.7, the action of the resulting microdifferential operator upon multivalued analytic functions such as Borel-transformed WKB solutions is described in terms of an integro-differential operator of particular type; its kernel function contains a differential operator of infinite order in x-variable. Thus it is of local character in x-variable, whereas it is suited for the global study related to the resurgence phenomena in y-variable (see, e.g., [SKK], [K] for the notion of a differential operator of infinite order; see also [AKT4], which first used a differential operator of infinite order in exact WKB analysis). As the domain of definition of the integrodifferential operator may be chosen to be uniform with respect to the parameter a (see Remark 2.3), our results in Section 2 are of semiglobal character, as is noted in Remark 4.1. This uniformity is one of the most important advantages in introducing the notion of an MPPT operator. It is worth emphasizing that the uniformity becomes clearly visible through the Borel transformation. In order to use the results obtained in Section 2 for the detailed study of the structure of Borel-transformed WKB solutions of an MPPT equation, we first study in Section 3 analytic properties of Borel-transformed WKB solutions of the Whittaker equation, and then in Section 4 we analyze Borel-transformed WKB solutions of the ∞-Whittaker equation using the results obtained in Section 3. The basis of the study in Section 3 is a recent result of Koike [KoT], and the analysis in Section 4 makes essential use of the estimate (B.3) of the coefficients {αk(a)}k≥0 104 Kamimoto, Kawai, Koike, and Takei of the parameter α(a, η) = ∑ k≥0 αk(a)η −k; the effect of this infinite series that appears in the ∞-Whittaker equation is grasped as a microdifferential operator acting on Borel-transformed WKB solutions of the Whittaker equation. Combining all the results obtained in Sections 2 and 4, we summarize in Section 5 basic properties of Borel-transformed WKB solutions of an MPPT equation with a = 0. 1. Construction of the transformation to the canonical form, I: The case where a = 0 The purpose of this section is to show how to construct the Borel-transformable series (1.1) x(x̃, η) = ∑ k≥0 x (0) k (x̃)η −k