Don B. Hinton

Hamiltonian systems of limit point or limit circle type with both endpoints singular

Abstract A linear Hamiltonian system J y′ = ( λA + B ) y is considered on an open interval ( a , b ), where both a and b are singular. The system is assumed to be of limit point or limit circle type at the endpoints. A theory of boundary problems for such systems is developed. Explicit boundary conditions are given, resolvent operators constructed and unique solutions established. The results given extend to Hamiltonian systems a theory of singular boundary value problems due to M. H. Stone and K. Kodaira.

The Effect of Variable Change on Oscillation and Disconjugacy Criteria with Applications to Spectral Theory and Asymptotic Theory

This study of the effect of variable change on differential operators was motivated by recent papers of Tipler [ 131 and Hinton and Lewis [37] which appeared in the same issue of the Journal of D@‘erential Equations. Tipler gave a geometric proof [ 13, p. 1671 of the following result of Hawking and Penrose (see [29, 301). Let F(t) be a real valued continuous, nonnegative, function on (-W, co) such that F(b) > 0 for some point b. Then there exists a pair of conjugate points on (-co, co) for the differential equation

Lyapunov Inequalities and their Applications

For nearly 50 years Lyapunov inequalities have been an important tool in the study of differential equations. In this survey, building on an excellent 1991 historical survey by Cheng, we sketch some new developments in the theory of Lyapunov inequalities and present some recent disconjugacy results relating to second and higher order differential equations as well as Hamiltonian systems.

Inverse scattering on the line for a Dirac system

The whole‐line version of the Gelfand–Levitan–Marchenko (GLM) equation for a Dirac system is studied. A new derivation of the GLM equation is given, under weaker hypotheses than Frolov’s earlier treatment [Sov. Math. Dokl. 13, 1468 (1972)], and the complete inversion is carried out in some explicit cases in which a spectral gap is present. Previous calculations of this type are restricted either to a scalar potential or degenerate gap. Applications are discussed in connection with optical couplers and soliton equations.

Titchmarsh-Weyl Theory for Hamiltonian Systems

For linear Hamiltonian systems, a Titchmarsh-Weyl matrix M(λ) function is defined. The systems formulation used by Atkinson is employed under a limit point hypothesis. A theory analogous to the second order scalar case of Chaudhuri and Everitt is developed. Characterizations are given for the resolvent set, point spectrum, continuous spectrum and point-continuous spectrum. An invariance with respect to boundary conditions is established for certain parts of the spectrum.

Sturm’s 1836 Oscillation Results Evolution of the Theory

We examine how Sturm’s oscillation theorems on comparison, separation, and indexing the number of zeros of eigenfunctions have evolved. It was Bocher who first put the proofs on a rigorous basis, and major tools of analysis where introduced by Picone, Prufer, Morse, Reid, and others. Some basic oscillation and disconjugacy results are given for the second-order case. We show how the definitions of oscillation and disconjugacy have more than one interpretation for higher-order equations and systems, but it is the definitions from the calculus of variations that provide the most fruitful concepts; they also have application to the spectral theory of differential equations. The comparison and separation theorems are given for systems, and it is shown how they apply to scalar equations to give a natural extension of Sturm’s second-order case. Finally we return to the second-order case to show how the indexing of zeros of eigenfunctions changes when there is a parameter in the boundary condition or if the weight function changes sign.

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.

Eigenfunction expansions and spectral matrices of singular differential operators

We consider the eigenfunction expansions associated with a symmetric differential operator M[·] of order 2 n with coefficients defined on an open interval (a, b) . Each singular endpoint of ( a , b ) is assumed to be of limit- n type. A direct convergence theory is established for the eigenfunction series expansion of a function y in a set Termwise differentiation of the series is established for the derivatives of order up to n . For O ≤ i ≤ n − 1, the i -fold differentiated series converges absolutely and uniformly to y ( i ) on compact intervals; the n −fold differentiated series converges to y n in the mean. The expansion theory is valid also when an essential spectrum is present. An explicit formula is given for the calculation of the spectral matrix.

Asymptotic phase, asymptotic modulus, and Titchmarsh-Weyl coefficient for a Dirac system

Abstract This paper studies a series representation for the Jost function associated with a Dirac system. Under suitable hypotheses the series yields the asymptotic form of the Titchmarsh-Weyl coefficient, valid not only in sectors of the upper halfplane but also the real axis. The asymptotic form of the spectral function, higher derivatives of the Jost function, and the limiting form of the asymptotic phase are also considered. Extensions of certain scattering formulas known in the Schrodinger case are given for the Dirac system.

Positive eigenvalues of second order boundary value problems and a theorem of M. G. Krein

Conditions are given which guarantee that the least real eigenvalue is positive for certain boundary value problems for the vector-matrix equation -y + p(x)y = γω(x)y. This leads to conditions which guarantee the stable boundedness, according to Krein, for solutions of y + λp(x)y = 0 with certain real values of A. As a consequence, a result first stated by Krein is proven.