Spectral concentration in the Sturm-Liouville differential equation
Kallenbach, Jeffrey C.
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The spectral function associated with the Sturm-Liouville differential equation is investigated. We explore first a refinement of an existing result by Gilbert and Harris by deriving conditions for the potential (coefficient) function which allow us to compute the spectral function. We next further refine the same result using a particular choice for a component of the potential. The next section imposes slightly more stringent restrictions on the potential which guarantee that the spectral function w ill be eventually of one sign. Finally, we generalize the first result by expanding the class of potentials to which the result applies. Throughout, we use a method employed by Gilbert and Harris that relates the solution of the differential equation to a series solution of an associated Riccati differential equation. Our new theorems are preceeded by existing results which provide background for the new material. When needed we also provide examples of the computation of explicit parameters for which our results hold.