Hongyou Wu

Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign

Fora givenSturm-Liouville equation whoseleading coefficient function changessign, we es- tablish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues(unbounded from both below and above) for a separated self-adjoint bound- ary condition can be numbered in terms of the Prangle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also re- late this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.

Regular Approximations of Singular Sturm-Liouville Problems with Limit-Circle Endpoints

For any self-adjoint realization S of a singular Sturm-Liouville equation on an interval (a,b) with limit-circle endpoints, we construct a family of self-adjoint realizations Sr,r ∈ (0,∞), of this equation on subintervals (ar,br) of (a,b) such that every eigenvalue of S is the limit of a continuous eigenvalue branch of this family. Of particular interest are the cases when at least one endpoint is oscillatory or the leading coefficient function changes sign. In these cases, we show that the index determining each continuous eigenvalue branch has an infinite number of jump discontinuities and give an explicit characterization of these discontinuities.

Geometric aspects of Sturm-Liouville problems II. Space of boundary conditions for left-definiteness

For a given regular Sturm-Liouville equation with an indefinite weight function, we explicitly describe the space of left-definite selfadjoint boundary conditions. The description only uses one value of a fundamental solution of the matrix form of the equation. As a consequence we show that this space has the shape of a solid consisting of two cones sharing a common base.

On some eigenvalue problems in fuel–cell dynamics

An eigenvalue problem arising from the study of gas dynamics in a loaded tubular solid oxide fuel cell is considered. An asymptotic theory from which the equations are derived is reviewed, and the results of analysis on the small and large parameter asymptotics are presented. These results suggest an interesting and hitherto unknown property of a class of Sturm–Liouville problems in which the first eigenvalue approaches zero but subsequent ones approach infinity as a parameter approaches zero. This was first discovered numerically and later confirmed asymptotically and rigorously.