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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

It is shown that the singularity spectrum of the phase space integral associated with any partially ordered Landau diagram is confined to a variety defined by a set of modified Landau equations. These equations are similar to the ordinary Landau equations but involve limiting procedures. The variety defined by the modified equations coincides with the variety defined by the ordinary equations except at points calledu=0 points. Next the causal parts of the sets defined by the modified Landau equations are defined, in a natural way, and it is conjectured that the singularity spectrum of theS-matrix is confined to the union of the causal parts of the singularity spectra of the phase space integrals. An analogous conjecture on general bubble diagram functions asserts that the singularity spectrum of each of these functions is confined to sets defined by the modified Landau equations augmented by appropriate positive-α and negative-α conditions. Generalized Landau equations are introduced. These equations do not involve limiting procedures, but provide a useful partial characterization of the sets defined by the modified Landau equations augmented by these positive-α and negative-α conditions.

The purpose of this paper is to present a way of explicit construction of local elementary solutions for Cauchy problems for a class of hyperbolic linear differential operators with multiple characteristics. We also investigate their singularity structure microlocally. The class we deal with in this article is a subclass dealt with by Kashiwara-Kawai [1] and our results can be derived from the very general and precise results given in §6 of Kashiwara-Kawai [1]. (See Miwa [1] for related topics.) However, the argument of Kashiwara -Kawai [1] is highly transcendental in its nature, and? hence3 we cannot expect that their results could be a guidepost for the "Structure Theorem" for general systems of micro-differential ( = pseudodifferential) equations in the multiple characteristic case, which is originally obtained in Sato-Kawai-Ka shiwara [1](*} in the simple-characteristic case(**}. To the contrary, our results in this article are, though restricted, tied up with the recent results of Nakamura [1] and Hamada-Nakamura [1] whose arguments are constructive in the sense that the coefficients of the expansion used there are determined successively in an explicit manner. Thus, we hope our argument and results in this article will serve as a starting point for the investigation of "Structure Theorem" for general systems with multiple

The previously proved results that every analytically renormalized Feynman integral is a regular holonomic function suggests that theS-matrix should be locally expressible as an infinite sum of regular holonomic functions. A regularity propertyR is formulated that expresses the condition that theS-matrix be locally expressible near each physical pointp as a convergent sum of regular holonomic functions, with each term enjoying some of the regularity properties of a corresponding Feynman integral. This propertyR holds at every physical pointp that has yet been analyzed by the methods of axiomatic field theory orS-matrix theory. Some analyticity properties of unitarity-type integrals are then examined under the assumption that theS-matrix satisfies propertyR and a weak integrability condition. These results rest heavily on some recently proved properties of regular holonomic functions.

A case of ischemic colitis following abdominal aortic aneurysmectomy in a 74-year-old male is reported. The patient was admitted to the hospital with a complaint of severe abdominal pain. A ruptured abdominal aortic aneurysm was suspected, and emergency laparotomy was performed. At the operation a massive hematoma was encountered in the retroperitoneal space caused by rupture of an abdominal aortic aneurysm. Aneurysmectomy was performed uneventfully. The left colon appeared to be viable at the end of the procedure. Three days after the operation he had diarrhea, and the symptoms of bowel ischemia progressed. Colonoscopy on the 7th day after the first operation revealed ischemic colitis. He was treated conservatively for 2 weeks without significant improvement. On the 19th postoperative day uncontrollable massive melena occurred. Emergency laparotomy was carried out again, and resection of the left colon and transverse colostomy were done. The postoperative course after the second operation was uneventful, and the patient is doing well now. Although clinically significant ischemic colitis following aortic reconstruction is rare, colonoscopy after the operation is useful for early recognition of ischemic changes before clinical manifestations.

The purpose of this paper is to prove some theorems on the extensibility of hyperfunction solutions and real analytic solutions of systems of linear differential equations with real analytic coefficients. It is Ehrenpreis [2] that revealed the intimate relations between the algebraic character of "overdetermin ed" systems of linear differential equations and Hartogs' theorem on the removable singularity of holomorphic functions of several complex variables. In fact, the essential part of his results can be summarized as follows: if a (generalized) function satisfies an "overdetermin ed" system of linear differential equations with constant coefficients outside a compact set in Rn, then it must be extended uniquely all over Rn as a solution of the same system. Note that the notion of "overdetermin ed" system is purely algebraic and that the Cauchy-Riemann equation, which holomorphic functions of n complex

Let U be an open subset of R", and let P(x, Dx) be a linear differential operator with analytic coefficients defined on a neighborhood of the closure [£/] of U. Suppose that the boundary dU of U is smooth (i.e., non-singular and analytic) and that dU is non-characteristic with respect to P at each point in dU. Then it is well-known ([7], [11]) that the boundary value of a hyperfunction solution of the equation Pu = 0 on U is a well-defined hyperfunction. However, little is known about the characterization of a solution whose boundary value determines a hyperfunction near a characteristic boundary point. The purpose of this article is to discuss this problem for one n-l special case, i.e., the pair of the heat operator d/dt — A = d/dt — J^ d/dxj def j=i and the domain {(£, x)eR"; t > 0}. Our main result (Theorem 1 below) asserts that, (i) if a C°°-solution u(t, x) does not behave too wildly as t|0, and (ii) if u(t, x) uniformly tends to zero outside a compact set K a R"' as *iO, then we can assign a compactly supported hyperfunction g(x) to u(t, x) so that the vanishing of g(x) entails the vanishing of u(t, x) itself. Furthermore we can find such a tame solution u(t, x) of the heat equation for any compactly supported hyperfunction g(x). [See Theorem 1 for the precise statement. Note also that a hyperfunction supported by a compact set, say L, is an analytic functional with the real carrier L.] Let us note the following two facts: First, if u(t, x) tends to infinity too rapidly as t j 0, then our procedure will not assign a hyperfunction g(x). (Cf. §2(i)) Second, we know (see [3], for example) that there exists a hyperfunction e(t, x) (xeR) supported by {(t, x)eR; t = 0, x ^ 0} satisfying the equation

We prove that the micro-local holonomic structure controls the local monodromy structure of functions involved. This result plays an essential role in investigating “hierarchical principle” in perturbation theory.

Let P(#, D} be a linear partial differential operator with real analytic coefficients in a domain V in Rn+1 and let SC^ be a real analytic hypersurface non-characteristic with respect to P(#, D). The purpose of this paper is to show that every hyperfunction solution u of P(x, D)u = Q on one side of V\S has boundary values on S which are hyperfunctions of n variables on 5. This fact has been proved by H. Komatsu [_6~] and P. Schapira Q8] in the case where P(x, J9) is elliptic. Their method applies with minor modifications to the general operators. In §1 we show that the Cauchy-Kowalevsky theorem for the dual equation with the initial values on 5 is equivalent to a theorem of division of hyperfunction s with supports in 5 by the differential operator P(x, D). We define the boundary values in § 2 and prove the uniqueness of hyperfunction solutions of the Cauchy problems.

In this paper we construct local elementary solutions for linear differential operators P(x9 Dx) whose principal symbols are real and of simple characteristics and investigate their regularity properties using Sato's theory of the sheaf ^ (Sato DC^CAl). Throughout this paper we assume that the coefficients of differential operators are analytic. In §1 we prepare some theorems which extend the classical existence theorem of Cauchy-Kowalevsky in complex domain to cases of singular initial data. (Cf. Hamada [1].) In §2 we employ the results of §1 to construct local elementary solutions for Cauchy problems for (/-) hyperbolic operators. As an application of the method employed there, we construct a singular solution u(x) of P(x, Dx)u = Q whose singular support is "very small". (See Theorem 2.8 for the precise meaning of this statement.) In §3 we construct a local elementary solution for a linear partial differential operator of real principal symbols with simple characteristics and decide where its singularities locate to conclude that the result of §2 on the existence of singular solutions with small singularities is the best possible one. Throughout this paper we denote by M an ra-dimensional real analytic manifold which we identify with a domain in R containing the origin.

On demontre un resultat de Hartogs sur la localisation des singularites des fonctions holomorphes a l'aide de la theorie des systemes surdetermines d'equations differentielles lineaires

Publisher Summary Although it is at present commonly accepted that microdifferential operators of infinite order are important in studying the structure of microdifferential equations of finite order, the importance of microdifferential equations of infinite order has just been being recognized. In view of this situation, this chapter discusses the way microdifferential equations of infinite order arise in analysis, analysis of microdifferential equations, and the kind of novelties so far observed. The chapter highlights the way microdifferential equations of infinite order arise in analysis by exploring the classical elliptic theta function. This chapter explains the way linear differential equations of infinite order come into the study of elliptic theta functions. So far the known way of characterizing the theta-zero-values was based upon their automorphic properties such as Jacobi's imaginary transformation; the chapter presents an approach that is designed to deduce a global property of the theta-zerovalue from its local property, that is, from the fact that it solves a system of linear differential equations.

Publisher Summary This chapter discusses the microlocal analysis of infrared singularities. Probably the most secure feature of relativistic quantum theory is the physical-region analytic structure: the singularities of any scattering function are confined to certain surfaces, called Landau-Nakanishi surfaces or Landau surfaces for short, and in a neighborhood of any regular point of such a surface the scattering function is, normally, the boundary value of a function holomorphic on a certain specified kind of domain. The physical-region singularity structure is the momentum-space form of an asymptotic spacetime structure that corresponds to classical physics in just the way demanded by the correspondence principle—the physical-region momentum-space singularity structure corresponds to the classical property that a stable particle, left undisturbed, moves with constant velocity, and neither disappears nor generates clones of itself, in the course of an undisturbed motion. The chapter presents a rigorous treatment of the problems associated with the singular character of the photon propagator at k = 0, in the context of the simplest nontrivial example.

On demontre la propriete de domaine ferme d'un operateur differentiel lineaire P agissant sur l'espace A(K) des fonctions analytiques reelles sur un sous ensemble compact K de R n sous la condition de P-convexite uniforme de K

On montre que l'analyse microlocale permet de deduire des theoremes d'existence pour des solutions hyperfonctions et des solutions analytiques reelles a partir du theoreme d'existence semiglobal pour des solutions analytiques reelles de Kiro

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 Sato's conjecture.

Publisher Summary This chapter presents some global existence theorems for real analytic solutions on a compact set K satisfying some convexity condition with respect to the system in question. It is assumed that the linear differential operators in question are with analytic coefficients. Various theorems are proven, and several examples are also presented in the chapter.

This report is intended to give an intuitive explanation of the theory of pseudo-differential operators in hyperfunction theory. The reader is refer~ed to Sato-Kawai-Kashiwara [i] for further details of the theory and its applications. Hereafter Sato-Kawai-Kashiwara [I] will be quoted as S-K-K for short. In this exposition the present speaker wants to lay great emphasis on the following two points.