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Inspired by the symbol calculus of linear differential operators of infinite order applied to the Borel transformed WKB solutions of simple-pole type equation [Kamimoto et al. (RIMS Kokyuroku Bessatsu B 52:127–146, 2014)], which is summarized in Section 1, we introduce in Section 2 the space of simple resurgent functions depending on a parameter with an infra-exponential type growth order, and then we define the assigning operator \({\mathscr{A}}\) which acts on the space and produces resurgent functions with essential singularities. In Section 3, we apply the operator \({\mathscr{A}}\) to the Borel transforms of the Voros coefficient and its exponentiation for the Whittaker equation with a large parameter so that we may find the Borel transforms of the Voros coefficient and its exponentiation for the boosted Whittaker equation with a large parameter. In Section 4, we use these results to find the explicit form of the alien derivatives of the Borel transformed WKB solutions of the boosted Whittaker equation with a large parameter. The results in this paper manifest the importance of resurgent functions with essential singularities in developing the exact WKB analysis, the WKB analysis based on the resurgent function theory. It is also worth emphasizing that the concrete form of essential singularities we encounter is expressed by the linear differential operators of infinite order.

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 introduce an appropriate class of higher order linear ordinary differential operators with a large parameter and with simple poles in their coefficients, and study their structure from the viewpoint of exact WKB analysis, i.e., the WKB analysis based on the Borel resummation. Our main result is that an operator in the class is expressed as a product of two operators Q and R, with Q being irrelevant to the Stokes geometry and with R being a second order simple-pole type operator studied by Koike ([Ko1], [Ko2], [Ko3]). This decomposition theorem gives us a connection formula for WKB solutions near the simple pole in question, and it is used to explain the background mechanism of some intriguing phenomenon we encounter in the study of a particular third order operator. Some discussions on the scope of the future development of the theory are included.

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.

Nous introduisons la notion d'operateur microdifferentiel du type BKW et developpons une analyse BKW exacte pour de tels operateurs en utilisant l'analyse microlocale : nous donnons une methode recursive pour construire une solution BKW d'un tel operateur au moyen du calcul symbolique des operateurs microdifferentiels. Nous etudions leur structure locale au voisinage des points de transition par un theoreme de division du type Weierstrass. Nous detaillons l'equation de Berk-Book en appendice.

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.

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.

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”.

The Schrodinger equation idψ/dt = ηHψ with a large parameter η and an appropriate 3×3 matrix H is studied by the exact WKB method, i.e. WKB analysis based on the Borel resummation. Transition probabilities of solutions of the equation are explicitly calculated with the help of the connection formula for WKB solutions. Since a 3×3 system is considered, addition of new Stokes curves is necessary to obtain the correct connection formula (Berk-Nevins-Roberts' observation), and we present a practically useful procedure for finding required new Stokes curves.

In order to determine the region where the Borel sum of a Wentzel–Kramers–Brillouin (WKB) solution ψ of an mth order (m⩾3) ordinary differential equation Pψ=0 with polynomial coefficients is well defined, we apply the steepest descent method to the Laplace integral of a WKB solution ψ of Pψ=0, with P denoting the Laplace transform of P. As a counterpart of the connection formula for ψ, we introduce a new notion of the exact steepest descent path, which is the union of bifurcated steepest descent paths. Both theoretical and computer-assisted experimental studies are given to show the importance of this notion.

n-th order with n 2 has not yet been well established; its local aspect is clari ed under the assumption that the discriminant of the characteristic equation is of simple zeros ([AKT, Theorem 1.4 and Theorem 1.6]), but its global aspect is far from complete understanding ([BNR], [AKT, Section 2]). The purpose of this paper is to examine the validity of our Ansatz concerning the Stokes geometry ([AKT, Section 2]) through the study of particular di erential equations whose solutions can be explicitly given in the form of integrals.

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).

This report shows how effective microlocal analysis is in alien calculus of WKB solutions. Concretely speaking, we describe how to analyze the structure of fixed singularities of Borel transformed WKB solutions of an MPPT (= a merging pair of a simple pole and a simple turning point) Schrodinger equation by finding out its microlocal canonical form. The microlocal canonical form of the Borel transformed MPPT equation is the Borel transformed Whittaker equation (Theorem 3 and Theorem 5), whose Borel transformed Voros coefficient can be concretely given (Theorem 2). These results give a tangible description of the alien derivative of WKB solutions of an MPPT equation (Theorem 6). The details are given in [5] and [11].

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 merging-turning-points (MTP) equations, we construct a WKB-theoretic transformation that brings an MPPT equation to its canonical form (the $\infty$ -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.

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.

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

Support is adduced for two related conjectures of simplicity of the analytic structure of the S-matrix and related function; namely, Sato's conjecture that the S-matrix is a solution of a maximally over-determined system of pseudo-differential equations, and our conjecture that the singularity spectrum of any bubble diagram function has the conormal structure with respect to a canonical decomposition of the solutions of the relevant Landau equations. This latter conjecture eliminates the open sets of allowed singularities that existing procedures permit.

We announce a generalization of the reduction theorem for 0-parameter solutions of the traditional (i.e., second order) Painlevé equations with a large parameter to those of some higher order Painlevé equations, i.e., each member of the Painlevé hierarchies $(P_J)$ ($J = \mbox{I}$, II-1 and II-2) discussed in [KKNT]. Thus the scope of applicability of the reduction theorem ([KT1, KT2]) has been substantially enlarged; only six equations were covered by our previous result, while the result reported here applies to infinitely many equations.

Introducing a new notion of the exact steepest descent path, we develop a new steepest descent method applicable to general ordinary differential equations with polynomial coefficients. Its application to the connection problem for solutions is also discussed.

We prove the local extensibility across a k-convex hypersurface of solutions of a system of linear differential equations with constant coefficients which satisfies some algebraic conditions determined by k. A global extensibility is also proved in the case of completely k-convex set. The purpose of this note is to prove under some algebraic conditions on a system L of linear differential equations with constant coefficients the (local) extensibility of hyperfunction solutions of system L across a hypersurface which enjoys some "convexity" property, whose precise meaning will be given in Definition 4. Our results make a contrast to the original Hartogs-type theorems of Ehrenpreis on removable singularities of solutions of linear differential equations with constant coefficients in the sense that our extension is achieved out of a "convex" open set, not into it. In fact, what we need is the underdetermined character of system L involved, not the overdetermined character, which the original result of Ehrenpreis requires. One important point in our arguments is that the flabbiness of the sheaf of hyperfunctions (Sato [8]) enables us to extend solutions even locally in spite of the fact that such an extension is not unique. See Ehrenpreis [1], Palamodov [7], Malgrange [6], Komatsu [5] and references cited there for Hartogs-type extension theorems and related results for linear differential equations with constant coefficients. See Kawai [3] for the case of linear differential equations with variable coefficients. The main tool in our proof is the vanishing theorem for cohomology groups on a completely (n k)-convex set which have as their coefficients the hyperfunction solution sheaf of a system of linear differential equations with constant coefficients (Kawai [2]). In formulating the algebraic conditions on L we also use modules Ei(L) (i = 0, 1, 2, . . . ) introduced and fully investigated by Palamodov [7, Part II, Chapter VIII, ? 13]. The author would like to express his heartiest thanks to Professor Eric Bedford of Courant Institute, New York University, for calling the author's Received by the editors August 20, 1975 and, in revised form, December 12, 1975. AMS (MOS) subject classifications (1970). Primary 35E10; Secondary 46F15.

In this report we summalize some recent results and make a conjecture on the micro-analytic structure of the S-matrix and related functions. The details of this report shall appear elsewhere. We refer the reader to Kawai-Stapp [4], [5] for the definition of the symbols and terminologies used here. In order to determine the micro-analytic structure of the 5-matrix and other bubble diagram functions we first examine the singularity spectrum of the phase space integral ID(p) = ^n(Pi-, "°,Pn) associated with a (stable particle) Landau diagram D with n external lines, N internal lines and n' vertices. A formal definition of the phase space integral ID(J>} is given as follows

The purpose of this report is to show that the (generalized) Feynman integral should satisfy a holonomic system***' of linear differential equations. We also discuss the analyticity of the defining function of the Feynman amplitude in complex domain as a corollary of this result. Our main result, i.e. the existence of holonomic system, gives an affirmative answer to the conjecture of Regge [15], who first understood and emphasized the importance of the role of systems of differential equations in the investigation of Feynman integrals. In his report a homological approach to this problem is suggested. It is very illuminating but seems to be accompanied with many technical difficulties, as Professor Regge himself points out in the report. This important property of the Feynman integral has also been conjectured and proved in simple cases by Sato [16] independently and in a little different context See also BarucchiPonzano [1], Kawai-Stapp [11], [12] and references cited there. Note that Kawai-Stapp ([11], [12]) discusses the ^-matrix itself, not the individual Feynman integral, as Sato [16] originally proposed. We also give the diagramatic description of the characteristic variety of the system involved. It enjoys a nice physical interpretation as is shown by Kashi

It is shown that the singularity spectrum of (generalized) position-space Feynman amplitude AD (x; /I) is confined to the extended Landau variety (introduced by KashiwaraKawai [8], [9] and Kashiwara-Kawai-Stapp [11]) associated with the Feynman diagram .D(O), which is obtained by replacing all the masses associated with the internal lines of D with zero, at least under the assumption that the diagram D is external. One recent trend in the investigation of analytic properties of the functions important in physics is to investigate the singularity structure of the functions in question from the view point of micro-local analysis, namely, to analyze their singularity structure on the cotangent bundle. (See Pham [15], Sato [16], lagolnizer [4], [5], [6], Kawai-Stapp [12], [13], K-K-S [11], Sato et al. [18] and references cited there. See also Chandler-Stapp [2] and lagolnitzer-Stapp [7], whose analysis is essentially of the micro-local character.) In the articles quoted above, the quantity in question is studied in the momentum-space and the celebrated LaridauNakanishi variety is naturally the starting point of their analysis. During the coruse of the precise argument on the unitarity integrals and so on, thanks to the harmony between mathematics and physics, K-K-S [11] and Kashiwara-Kawai [8], [9] have been naturally led to the notion of the extended Landau variety which describes the geometric feature of the singularity structure of phase space integrals, the (oil-shell) 5-matrix and Feynman amplitudes (K-K-S [11], Kashiwara-Kawai [8], [9]). Now, Received September 16, 1977. * Supported in part by NSF GP 36269. Department of Mathematics, Harvard University and Research Institute for Mathematical Sciences, Kyoto University. Current address: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606 Japan. (0) Hereafter referred to K-K-S [11] for short.

Inspired by some physical problems, Sato has recently proposed to investigate the deformation of systems of (micro-) differential equations with the emphasis on the in variance of global character of solutions. (Sato et al. [4]) Especially, he has observed that the classical result of Schlesinger [5] is most neatly explained from this point of view. The purpose of this note is to show that the monodromy structure of solutions of holonomic systems of linear differential equations is invariant under the deformation of the system specified below (Definition 1) on the condition that related topological structure does not change. (See Corollary 1 for the precise statement.) We first list up some notations used in this note. We refer the reader to S-K-K [3] for notations not listed here. X: a complex maniford. A point in X shall be denoted by x. 7t: the projection map from T*X to X. U: a connected Stein open set in CN. A point in CN shall be denoted by *=(*i, —,**)• Xc={c}xXc:UxX for c^U. The natural injection from Xc to UxX shall be denoted by cc. We sometimes identify Xc with X.

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

Poisson's summation formula tells us that the Fourier transform of the sum of Dirac's (J-functions supported by the integral points is again the sum of Dirac's (^-functions supported by the integral points. In this paper we first consider the converse problem, that is, we characterize a distribution which is a sum of distributions supported by the integral points and whose Fourier transform is again of the same form. Using this result we give another proof of the classical theorem of Hamburger on the characterization of the zata function of Riemann. We also show a generalization of the result to the zeta function associated to the imaginary quadratic field Q(^ — 1) -. In what follows, we use the notation 2 (f) or f to denote the Fourier transform of f normalized in the form I f (x) exp(27Tv — 1 ) dx. We denote by Z>£ (resp., xa} (a= (al9 •--, am) eZJ) the differential operator d™/da%* — da%? (resp., the monomial x"l~>x°tf), where Z+ m denotes the set of ra-tuple of non-negative integers and \a\ = XI #/• j=i We denote Dirac's d-f unction supported at x — n by 8(x — 7i) and its derivative Dax8(x-n) by 8(a}(x-n). Now the first result is stated as follows

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