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We consider the linear, second-order, differential equationwith the boundary condition yð0Þ cos a þ y 0 ð0Þ sin a ¼ 0 for some aA½0; pÞ: ðÃÃÞWe suppose that qðxÞ is real-valued, continuously differentiable and that qðxÞ-0 as x-N with qeL 1 ½0; NÞ: Our main object of study is the spectral function r a ðlÞ associated with (Ã) and (ÃÃ). We derive a series expansion for this function, valid for lXL 0 where L 0 is computable and establish a L 1 ; also computable, such that (Ã) and (ÃÃ) with a ¼ 0; have no points of spectral concentration for lXL 1 : We illustrate our results with examples. In particular we consider the case of the Wigner-von Neumann potential.

While students are solving a proportion problem, their work in a measure space will enable teachers to take the measure of their thinking.

No abstract

For a special case of the Sturm-Liouville equation, −(py′)′ + qy = λwy on [0, ∞) with the initial condition y(0) cos α + p(0)y′(0) sin α = 0, α ∈ [0, π), it is shown that, given the spectral derivative for two values of α ∈ [0, π) at a fixed μ = Re{λ} ≥ Λ0, it is possible to uniquely determine . An explicit formula is derived to accomplish this. Further, in a more general case of the Sturm-Liouville problem for μ with both finite and positive, then the following inequality holds