Faculty Publications

Title

Consequences of the connection formulae for Sturm-Liouville spectral functions

Document Type

Conference

Journal/Book/Conference Title

Royal Society of Edinburgh - Proceedings A

Volume

132

Issue

2

First Page

387

Last Page

393

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 ρ′α(μ) for two values of α ∈ [0, π) at a fixed μ = Re{λ} ≥ Λ0, it is possible to uniquely determine ρ′β(μ), β ∈ [0, π). An explicit formula is derived to accomplish this. Further, in a more general case of the Sturm-Liouville problem for μ with ρ′α(μ), ρ′β(μ) both finite and positive, then the following inequality holds ρ′α(μ)ρ′β(μ) ≤ 1/π2sin2 (β - α).

Original Publication Date

1-1-2002

DOI of published version

10.1017/s0308210500001694

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