W. N. Everitt

Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators

Introduction: Fundamental algebraic and geometric concepts applied to the theory of self-adjoint boundary value problems Maximal and minimal operators for quasi-differential expressions, and GKN-theory Symplectic geometry and boundary value problems Regular boundary value problems Singular boundary value problems Appendix A. Constructions for quasi-differential operators Appendix B. Complexification of real symplectic spaces, and the real GKN-theorem for real operators References.

Complex symplectic geometry with applications to ordinary differential operators

Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems. 1. Fundamental definitions for complex symplectic spaces, and three motivating illustrations Complex symplectic spaces, as defined below, are non-trivial generalizations of the real symplectic spaces of Lagrangian classical dynamics [AM], [MA]. Further, these complex spaces provide important algebraic structures clarifying the theory of boundary value problems of linear ordinary differential equations, and the theory of the associated self-adjoint linear operators on Hilbert spaces [AG], [DS], [NA]. These fundamental concepts are introduced in connection with three examples or motivating discussions in this first introductory section, with further technical details and applications presented in the Appendix at the end of this paper. The new algebraic results are given in the second and main section of this paper, which developes the principal theorems of the algebra of finite dimensional complex symplectic spaces and their Lagrangian subspaces. A preliminary treatment of these subjects, with full attention to the theory of self-adjoint operators, can be found in the earlier monograph of these authors [EM]. Definition 1. A complex symplectic space S is a complex linear space, with a prescribed symplectic form [:], namely a sesquilinear form (i) u, v → [u : v], S × S → C, so [c1u + c2v : w] = c1[u : w] + c2[v : w], (1.1) which is skew-Hermitian, (ii) [u : v] = −[v : u], so [u : c1v + c2w] = c̄1[u : v] + c̄2[u : w] Received by the editors August 19, 1997. 1991 Mathematics Subject Classification. Primary 34B05, 34L05; Secondary 47B25, 58F05.

On the representation of holomorphic functions by integrals

Sufficient conditions are given for the holomorphy of functions defined by parametric integrals in Lebesgue integration spaces

Complex symplectic spaces and boundary value problems

This paper presents a review and summary of recent research on the boundary value problems for linear ordinary and partial differential equations, with special attention to the investigations of the current authors emphasizing the applications of complex symplectic spaces. In the first part of the previous century, Stone and von Neumann formulated the theory of self-adjoint extensions of symmetric linear operators on a Hilbert space; in this connection Stone developed the properties of self-adjoint differential operators generated by boundary value problems for linear ordinary differential equations. Later, in diverse papers, Glazman, Krein and Naimark introduced certain algebraic techniques for the treatment of appropriate generalized boundary conditions. During the past dozen years, in a number of monographs and memoirs, the current authors of this expository summary have developed an extensive algebraic structure, complex symplectic spaces, with applications to both ordinary and partial linear boundary value problems. As a consequence of the use of complex symplectic spaces, the results offer new insights into the theory and use of indefinite inner product spaces, particularly Krein spaces, from an algebraic viewpoint. For instance, detailed information is obtained concerning the separation and coupling of the boundary conditions at the endpoints of the intervals for ordinary differential operators (see the Balanced Intersection Principle), and the introduction of the generalized boundary conditions over the region for some elliptic partial differential operators (see the Harmonic operator).

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.

An unusual self-adjoint linear partial differential operator

hhIn an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region. This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the Harmonic operator. The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions. In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty. This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty. Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent. In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.

Fourth-order Bessel equation: eigenpackets and a generalized Hankel transform †

In connection with the fourth-order Bessel-type differential equation two expansion theorems are established, the convergence being pointwise or in an L 2-setting. If the positive parameter M tends to zero, these two expansion theorems reduce to the classical Hankel transform of order zero. In a previous article, the authors have proved that in one of the introduced Lebesgue–Stieltjes Hilbert function spaces, the differential expression x −1 L M gives rise to exactly one self-adjoint operator S M . In this article, it is proved, together with the corresponding expansion theorems, that S M has a complete eigenpacket. The orthogonality property of this eigenpacket is reflected in a distributional orthogonality on which the expansion theorems are based. †This paper is dedicated to the achievements and memory of Professor Günter Hellwig.

Linear Operators Generated by a Countable Number of Quasi-differential Expressions

The theory of linear ordinary quasi-differential operators has been considered in Lebesgue locally integrable spaces on a single interval of the real line. Such spaces are not Banach spaces but can be considered as complete, locally convex, linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete, locally convex, linear topological space but now with the topology derived as a strict inductive limit. This article extends the previous single interval results to the case when a finite or countable number of intervals of the real line is considered. Conjugate and preconjugate linear quasi-differential operators are defined and relationships between these operators are developed.

A Problem Concerning the Gamma Function

It is known that (Γ is the gamma function)