Bruce A. Watson

Inverse Spectral Problems for Sturm–Liouville Equations with Eigenparameter Dependent Boundary Conditions

Inverse Sturm–Liouville problems with eigenparameter-dependent boundary conditions are considered. Theorems analogous to those of both Hochstadt and Gelfand and Levitan are proved. In particular, let l y = (1/ r )(−( py ′)′+ qy ), l˜ y = (1/ r˜ )(−( p˜ y ′)′+ q˜ y ), formula here where det Δ = δ > 0, c ≠ 0, det [sum ] > 0, t ≠ 0 and ( cs + dr − au − tb ) 2 cr − ta )( ds − ub ). Denote by ( l ; α; Δ) the eigenvalue problem ly = λ y with boundary conditions y (0)cosα+ y ′(0)sinα = 0 and ( a λ+ b ) y (1) = ( c λ+ d )( py ′)(1). Define (l˜; α; Δ) as above but with l replaced by l˜ . Let w n denote the eigenfunction of ( l ; α; Δ) having eigenvalue λ n and initial conditions w n (0) = sin α and pw ′ n (0) = −cos α and let γ n = − aw n (1)+ cpw ′ n (1). Define w˜ n and γ˜ n similarly. As sample results, it is proved that if ( l ; α; Δ) and (l˜; α; Δ) have the same spectrum, and ( l ; α; Σ) and (l˜; α; Σ) have the same spectrum or ∫ 1 0 [mid ] w n [mid ] 2 rdt +([mid ]γ n [mid ] 2 /δ) = ∫ 1 0 [mid ] w˜ n [mid ] 2 r˜ dt +([mid ]γ˜ n [mid ] 2 /δ) for all n , then q / r = q˜/r˜.

Discrete-time stochastic processes on Riesz spaces

Abstract It has been recognised that order is closely linked with probability theory, with lattice theoretic approaches being used to study Markov processes but, to our knowledge, the complete theory of (sub, super) martingales and their stopping times has not been formulated on Riesz spaces. We generalize the concepts of stochastic processes, (sub, super) martingales and stopping times to Riesz spaces. In this paper we consider discrete time processes with bounded stopping times.

Inverse nodal problems for Sturm–Liouville equations on graphs

We consider inverse nodal problems on graphs. Eigenfunction and eigenvalue asymptotic approximations are used to provide an asymptotic expression for the spacing of nodal points on each edge of the graph. Based on this, the uniqueness of the potential for given nodal data is proved and we give a construction of q as a limit, in , of a sequence of functions whose nth term is dependent only on the nth eigenvalue and its associated nodal data.

Inverse spectral problems for weighted Dirac systems

In this work, inverse spectral problems for weighted one-dimensional Dirac systems are studied. Three uniqueness theorems are proved corresponding to the following given spectral data: two spectra; one spectrum and the associated norming constants; and one spectrum under the assumption that the coefficient matrices have a suitable symmetry. Weighted Dirac systems, as opposed to unweighted Dirac systems, occur in situations where the underlying space is a Riemannian manifold, rather than flat Euclidean space.

Convergence of Riesz space martingales

Abstract We prove a martingale convergence for sub and super martingales on Riesz spaces. As a consequence we can form Krickeberg and Riesz like decompositions. The minimality of the Krickeberg decomposition yields a natural ordered lattice structure on the space of convergent martingales making this space into a Dedekind complete Riesz space. Finally we show that the Riesz space of convergent martingales is Riesz isomorphic to the order closure of the union of the ranges of the conditional expectations in the filtration. Consequently we can characterize the space of order convergent martingales both in Riesz spaces and in the setting of probability spaces.

TRANSFORMATIONS BETWEEN STURM–LIOUVILLE PROBLEMS WITH EIGENVALUE DEPENDENT AND INDEPENDENT BOUNDARY CONDITIONS

Explicit relationships are given connecting ‘almost’ isospectral Sturm–Liouville problems with eigen-value dependent, and independent, boundary conditions, respectively. Application is made to various direct and inverse spectral questions.

Inverse spectral problems for left-definite Sturm–Liouville equations with indefinite weight

Abstract Inverse Sturm–Liouville problems with indefinite weight are considered. Theorems analogous to those of both Hochstadt and Gelfand–Levitan are proved. As a sample result, under suitable symmetry conditions, we show that half of one spectrum suffices to determine the potential function uniquely.

Markov processes on Riesz spaces

Measure-free discrete time stochastic processes in Riesz spaces were formulated and studied by Kuo, Labuschagne and Watson. Aspects relating martingales, stopping times, convergence of these processes as well as various decomposition were considered. Here we formulate and study Markov processes in a measure-free Riesz space setting.

Spectral Isomorphisms between Generalized Sturm-Liouville Problems

We characterize all isospectral norming constant preserving maps between certain classes of Sturm-Liouville problems with eigenparameter dependent and constant boundary conditions. In consequence we obtain existence and uniqueness inverse spectral results for Sturm-Liouville problems with eigenparameter dependent boundary conditions.

The M-matrix inverse problem for the Sturm-Liouville equation on graphs ∗

AbstractWe consider an inverse spectral problemfor Sturm-Liouville boundaryvalue problemson a graph with formally self-adjoint boundary conditions at the nodes, where the giveninformation is the M-matrix. Based on the results found in S. Currie, B.A. Watson,M-matrix asymptotics for Sturm-Liouville problems on graphs, J. Com. Appl. Math.,doi: 10.1016/j.cam.2007.11.019, using the Green’s function, we prove that the poles ofthe M-matrix are at the eigenvalues of the associated boundary value problem and aresimple, located on the real axis and that the residue at a pole is a negative semi-definitematrix with rank equal to the multiplicity of the eigenvalue. We define the so callednorming constants and relate them to the spectral measure and the M-matrix. Thisenables us to recover, from the M-matrix, the boundary conditions and the potential,up to a unitary equivalence for co-normal boundary conditions. ∗ Keywords: inverse problem, differential operators on graphs, m-function, Sturm-Liouville (2000)MSC:34A55, 34B45, 34B20, 34L05, 34B27 .