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We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrodinger equation. Our emphasis is put on the analysis of the singularity structure of its Borel transformed WKB solutions near fixed singular points relevant to the two simple poles contained in the potential of the equation. In Part I, we focus our attention on the construction and analytic properties of a WKB-theoretic transformation that transforms an M2P1T equation to an algebraic Mathieu equation. That transformation plays an important role in Part II ( [7] ) when we discuss the singularity structure of Borel transformed WKB solutions of an M2P1T equation.

We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrodinger equation. In Part II, using a WKB-theoretic transformation to the algebraic Mathieu equation constructed in Part I, we calculate the alien derivative of its Borel transformed WKB solutions at each fixed singular point relevant to the simple poles through the analysis of Borel transformed WKB solutions of the Legendre equations. In the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic Mathieu equation and the Legendre equation.

Higher order simple-pole type operators, that is, higher order linear ordinary differential operators with a large parameter η whose coefficients have simple poles at the origin, are discussed from the viewpoint of the exact WKB analysis. Making use of the technique of microdifferential operators, we clarify the singularity structure of the Borel transform of their WKB solutions.

We generalize the reduction theorem for 0-parameter solutions of a traditional (i.e., second order) Painleve equation with a large parameter to those of some higher order Painleve equation, that is, each member of the Painleve hierarchies ( P J ) ( J = I , II - 1 or II-2). Thus the scope of applicability of the reduction theorem in [KT1] has been substantially enlarged; only six equations were covered by our previous result, while Theorem 3.2.1 of this paper applies to infinitely many equations.

To analyze differential operators whose WKB solutions admit infinitely many phases, we introduce a class of differential operators of WKB type and analyze their exact WKB theoretic structure near their turning points. Our analysis makes full use of techniques and ideas in microlocal analysis; we use a quantized contact transformation to construct a WKB solution of a differential equation of WKB type, and we use a Spath-type division theorem for a differential operator of WKB type to study its structure near turning points. As an application, we show a connection formula for WKB solutions near a simple turning point.

The recent progress of the theory of microdifferential operators of infinite order [l-4] and their usefulness in several branches of analysis [IS, 11, 16-241 have enhanced the interest in such operators. However, our knowledge on such operators is still quite limited, particularly compared with our knowledge on operators of finite order. For example, we do not know much about such a basic problem as characterizing a linear differen- tial operator of infinite order which is of closed range as a map from germs of holomorphic functions to germs of holomorphic functions. To the best of our knowledge, Ishimura’s recent work [7] is the only result that has some general applicability, although it is still far from the final goal. The purpose of this paper is to prove some theorems which we hope to contribute toward the progress of the study of such operators. To be con- crete, we present some conditions on a linear differential operator P which guarantees the finiteness of the dimensions of the kernel and the cokernel of the map P: Ccn,o -+ O,,,,, where Ocn,O denotes germ at origin of sheaf 0,. of holomorphic functions on C:“. In particular, under those con- ditions the map has finite index, and it is of closed range (Theorem 4 and Theorem 5). Our results are a natural generalization of Bony and Schapira [S, Thtoreme 5.11 to the infinite-order case. It is also quite akin to Kashiwara, Kawai, and Sjostrand [12] in its philosophy in the sense that some ellipticity condition for a suitably chosen tangential system lies 155

The purpose of this article is to show that some finiteness theorem (= finite dimensionality of the space of solutions) holds for a class of systems of linear differential equations of infinite order. Although finiteness theorems for holonomic systems of (micro-)differential equations of finite order have recently become quite popular, the character of the theorems which we present here is different from the results for equations of finite order. Hence, in this introduction, we discuss a simple and instructive example so that it may help the reader’s understanding of the character of the results in this article. As the example will indicate, our results have close connection with the celebrated result of Hamburger on the characterization of the c-function of Riemann, although we deal with theta functions (Hamburger [2], Hecke [3], and Weil [8]; see also Ehrenpreis and Kawai [ 11). This connection was pointed out to one of us (T.K.) by Professor L. Ehrenpreis. Concerning the basic properties of linear differential operators of infinite order, we refer the reader to Sato-Kawai-Kashiwara [6, Chap. II]’ (hereafter referred to as S-K-K). Here we only emphasize that a linear differential operator of infinite order acts upon the sheaf of holomorphic functions as a sheaf homomorphism. Hence our main result (Theorem 2.14 in Section 2) is of local character. This forms a striking contrast to the hitherto known way of characterizing theta functions through their automorphic properties. Now, in order to provide an example of our results, let us show how the theta zero-value (Nullwerte) is related to a system of linear differential equations of infinite order. In order to fix the notations, let us consider