Differential operators for systems of linear partial differential equations

The aim of this paper is the representation of solutions of systems of formally hyperbolic differential equations of second order. I. N.Vekua gave a representation of the solutions using the Riemann-matrix-function. Here we introduce special differential operators which map holomorphic functions into the set of solutions. An existence theorem for such operators is proved which gives a necessary and sufficient condition on the coefficients of a system. These operators are represented explicitly and several properties of them are investigated. We give different representations of the solutions of such systems and discuss the relation between the integral operator method and the method using differential operators which leads to an explicit representation of the Riemann-matrix-function by means of the differential operators. Two examples of special systems with differential operators are given.     

Constructive methods for certain Bergman operators

This paper is concerned with the class of linear partial differential equations of second order such that there exist Bergman operators with polynomial kernels (cf, [12]). In an earlier paper [ll] the authors have shown that these equations also admit differential operators as introduced by K. W. Bauer [I]. In the present paper, relations between different types of representations of solutions are investigated. These representations are of interest in developing a function theory of solutions; cf., for instance, K. W. Bauer [I] and S. Ruscheweyh [19]. They are also essential to global extensions of local results obtained by means of Bergman operators of the first kind. The inversion problem for those operators is solved, and it is shown that all solutions of equations of that class which are holomorphic in a domain of C² can be represented by operators with polynomial kernels. Furthermore, a construction principle for deriving the equations investigated by K. W. Bauer [2] is obtained; this yields corresponding representations of solutions by differential and integral operators in a systematic fashion     

The Fourth-order Bessel–type Differential Equation

The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane.

There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation.

When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coefficients. Embedded in this form of the equation is a spectral parameter; this combination leads to the generation of self-adjoint operators in a weighted Hilbert function space. In the second-order case one of these associated operators has an eigenfunction expansion that leads to the Hankel integral transform.

This article is devoted to a study of the spectral theory of the Bessel-type differential equation of order four; considered on the positive real axis this equation has singularities at both end-points. In the associated Hilbert function space these singular end-points are classified, the minimal and maximal operators are defined and all associated self-adjoint operators are determined, including the Friedrichs self-adjoint operator. The spectral properties of these self-adjoint operators are given in explicit form.

From the properties of the domain of the maximal operator, in the associated Hilbert function space, it is possible to obtain a virial theorem for the fourth-order Bessel-type differential equation.

There are two solutions of this fourth-order equation that can be expressed in terms of classical Bessel functions of order zero and order one. However it appears that additional, independent solutions essentially involve new special functions not yet defined. The spectral properties of the self-adjoint operators suggest that there is an eigenfunction expansion similar to the Hankel transform, but details await a further study of the solutions of the differential equation.     

Bergman-Operatoren mit Polynomen als Erzeugenden

The Bergman integral operators considered in this paper have particularly simple generating functions. This property is important in the study of solutions of partial differential equations by function theoretical methods. It is shown that those operators can be represented in a form free of integrals and that the corresponding class of equations includes equations which are related to the wave and Laplace equations and have been investigated in detail by K. W. Bauer and E. Peschl.     

Differential Operator Solutions of the Bauer-Peschl Equation

In this paper new solutions of the Bauer-Peschl equation represented by differential operators are derived. A relation to the solutions of Bauer is given. Furthermore, it will be said in which new way Bauer's solutions can be obtained. Also three other ways for obtaining the solutions of the Bauer-Peschl equation are sketched.     

Polynomial modules and a generalized cauchy problem for systems of holomorphic partial differential operators

In this paper, we study holomorphic solutions of overdetermined systems of partial differential equatoins with constant coefficients. We parametrize the solution space, using an ideal of holomorphic functions generated by certain polynomials associated to the principal symbols of the operators, by proving a Cauchy-Kowalevsky type theorem for these systems. In this theorem, we introduce a non-characteristicity condition for systems which is a direct generalization of a condition for single equations introduced by the author and H.S. Shapiro in a previous paper.

As a tool for studying the systems mentioned above, certain estimates, which may be of independent interest, for elements of analytic modules generated by polynomials are obtained.     

New approach to differential equations with countable impulses

This paper provides a new approach to study the solutions of a class of generalized Jacobi equations associated with the linearization of certain singular flows on Riemannian manifolds with dimension n + 1.A new class of generalized differential operators is defined.We investigate the kernel of the corresponding maximal operators by applying operator theory.It is shown that all nontrivial solutions to the generalized Jacobi equation are hyperbolic,in which there are n dimension solutions with exponential...     

On the Relation between Linear Difference and Differential Equations with Polynomial Coefficients

This paper represents the third part of a contribution to the “dictionary” of homogeneous linear differential equations with polynomial coefficients on one hand and corresponding difference equations on the other. In the first part (cf. [4]) we studied the case that the differential equation (D) has at most regular singularities at O and at ∞, and arbitrary singularities in the rest of the complex plane. We constructed fundamental systems of solutions of a corresponding difference equation (A), using integral transforms of microsolutions of (D) at its singular points in ?. In the second part ([5]) we considered differential equations having at most a regular singularity at O and an irregular one at O. We used integral transforms of asymptotically flat solutions of (D) to define it fundamental system of solutions of (Δ), holomorphic in a right half plane, and integral transforms of sections of the sheaf of solutions of (D) modulo solutions with moderate growth as t → 0 in some sector, to define a fundamental system of (Δ), holomorphic in a left half plane. In this final part we combine the techniques and results of the preceding papers to deal with the general case.     

First Order Differential Operators Associated to the Cauchy—Riemann Operator in the Plane

The article deals with initial value problems of type δw/δt = Fw, w(0, ·) = φ where t is the time and F is a linear first order operator acting in the z = x ? iy-plane. In view of the classical Cauchy-Kovalevkaya Theorem, the initial value problem is solvable provided F has holomorphic coefficients and the initial function is holomorphic. On the other hand, the Lewy example [H. Lewy (1957). An example of a smooth linear partial differential equation without solution. Ann. of Math., 66, 155–158.] shows that there are equations of the above form with infinitely differentiable coefficients not having any solutions. The article in hand constructs, conversely, all linear operators F for which the initial value problem with an arbitrary holomorphic initial function is always solvable. In particular, we shall see that there are equations of that type whose coefficients are only continuous.     

Asymptotic existence theorems for formal power series whose coefficients satisfy certain partial differential recursions

We study power series whose coefficients are holomorphic functions of another complex variable and a nonnegative real parameter s, and are given by a differential recursion equation. For positive integer s, series of this form naturally occur as formal solutions of some partial differential equations with constant coefficients, while for s=0 they satisfy certain perturbed linear ordinary differential equations. For arbitrary s?0, these series solve a differential-integral equation. Such power series, in general, are not multisummable. However, we shall prove existence of solutions of the same differential-integral equation that in sectors of, in general, maximal opening have the formal series as their asymptotic expansion. Furthermore, we shall indicate that the solutions so obtained can be related to one another in a fairly explicit manner, thus exhibiting a Stokes phenomenon.     

Global Representatives of Colombeau Holomorphic Generalized Functions

It is proved that a holomorphic generalized function in the sense of Colombeau has a representative consisting of a net of holomorphic functions. More generally, the existence of global representatives of Colombeau solutions to elliptic partial differential equations is proved. Further, it is shown that a generalized holomorphic function on Ω, which is equal to zero in an open set L ì WL\subset{\rm{\Omega}} , is equal to zero.     

Quadratic fermionic dynamics with dissipation

Gaussian solutions of the Cauchy problem for the GKS-L equation (in the Schrödinger picture) with quadratic fermionic generators are obtained. These Gaussian solutions are represented both as exponentials of quadratic forms in fermionic creation-annihilation operators and by their normal symbols. The coefficients of these forms are represented as algebraic functions of matrices.

     

Über die Koerzitivität gewöhnlicher Differenzenoperatoren und die Konvergenz von Mehrschrittverfahren

Summary For linear ordinary difference operators under general boundary conditions a so called coerciveness inequality is proved. As an application the solutions and their difference quotients up to the order of the approximated differential equation of stable and consistent multistep difference methods are shown to converge.

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Herrn Prof. Dr. Dr. h.c. L. Collatz zum 60. Geburtstag gewidmet     

Differential symmetry breaking operators: I. General theory and F-method

We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case, symmetry breaking operators are characterized by differential equations of second order via the F-method.     

Zur Konstruktion gewisser Integraloperatoren für partielle Differentialgleichungen Teil II

Using results of Part I of this paper, we shall now develop two methods of constructing linear partial differential equations which admit Bergman operators with polynomial kernels; these equations will be obtained explicitly. Those methods will also yield general representations of solutions of such an equation which are holomorphic in some domain of complex two-space. For generating all those solutions, one needs a pair of Bergman operators. Whereas in Part I of this paper we required at least one of the two operators to have a polynomial kernel, we now impose the condition that both operators be of that kind. This entails further basic results about the existence, construction, and uniqueness of solutions.     

Expansion of the boundaries of the solutions of the linear Volterra integral equations with convolution kernel

We solve a certain differential equation and system of integral equations. As applications, we characterize holomorphic symbols of commuting Toeplitz operators on the pluriharmonic Bergman space. In addition, pluriharmonic symbols of normal Toeplitz operators are characterized. Also, zero semi-commutators for certain classes of Toeplitz operators are characterized.This research is partially supported by KOSEF(98-0701-03-01-5).     

Asymptotic regularity of solutions to Hartree–Fock equations with Coulomb potential

We study the asymptotic regularity of solutions to Hartree–Fock (HF) equations for Coulomb systems. To deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo‐differential operators on conical manifolds. First, the non‐self‐consistent‐field case is considered, which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator, which guarantee ellipticity for the local part of the Fock operator on the open stretched cone ?₊ × S². This proves the existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed‐point approach based on Cancès and Le Bris analysis of the level‐shifting algorithm, we show via another bootstrap argument that the corresponding self‐consistent‐field solutions to the HF equation have the same type of asymptotic regularity. Copyright © 2008 John Wiley & Sons, Ltd.     

Homogeneous differential-operator equations in locally convex spaces

We describe a general method that allows us to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is not interpreted as the triviality of the right-hand side of an equation. It is understood in the sense that the left-hand side of an equation is a homogeneous function with respect to operators appearing in that equation. Solutions are represented as functional vector-valued series which are uniformly convergent and generated by solutions to a kth order ordinary differential equation, by the roots of the characteristic polynomial and by elements of a locally convex space. We find sufficient conditions for the continuous dependence of the solution on a generating set. We also solve the Cauchy problem for the considered equations and specify conditions for the existence and the uniqueness of the solution. Moreover, under certain hypotheses we find the general solution to the considered equations. It is a function which yields any particular solution. The investigation is realized by means of characteristics of operators such as the order and the type of an operator, as well as operator characteristics of vectors, namely, the operator order and the operator type of a vector relative to an operator. We also use a convergence of operator series with respect to an equicontinuous bornology.     

Properties of the solutions of the fourth-order Bessel-type differential equation

The structured Bessel-type functions of arbitrary even-order were introduced by Everitt and Markett in 1994; these functions satisfy linear ordinary differential equations of the same even-order. The differential equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity. They are all natural extensions of the classical second-order Bessel differential equation. Further these differential equations have real-valued coefficients on the positive real half-line of the plane, and can be written in Lagrange symmetric (formally self-adjoint) form. In the fourth-order case, the Lagrange symmetric differential expression generates self-adjoint unbounded operators in certain Hilbert function spaces. These results are recorded in many of the papers here given as references. It is shown in the original paper of 1994 that in this fourth-order case one solution exists which can be represented in terms of the classical Bessel functions of order 0 and 1. The existence of this solution, further aided by computer programs in Maple, led to the existence of a linearly independent basis of solutions of the differential equation. In this paper a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane. The methods lead to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane.     

Bergmansche Polynom-Erzeugende Erster Art

This paper is concerned with special Bergman operators for linear partial differential equations. E. Kreyszig introduced a class of equations which possess polynomials as generating functions for Bergman operators. These generating functions permit an integral-free representation of the solutions which can also be obtained by K. W. Bauer's and E. Peschl's differential operators, as was shown by M. Kracht and E. Kreyszig. For an extension of Kreyszig's class by W. Watzlawek we state conditions that are necessary and sufficient for the existence of a polynomial generating function of the first kind. We give an explicit characterization of the corresponding subclass and a representation of those generating functions, which are important for developing a function theory of the equations considered.