A new approach to inverse spectral theory, I. Fundamental formalism

1999/06/30 by Barry Simon · 2 citations
Mathematics · #math.SP #msc:34A55

paper · pdf

published as Ann. of Math. (2) 150 (1999), no. 3, 1029-1057 · 29 pages, published version

arxiv created 1999/11/01 · arxiv updated 2009/11/30

Abstract

We present a new approach (distinct from Gel'fand-Levitan) to the theorem of Borg-Marchenko that the m-function (equivalently, spectral measure) for a finite interval or half-line Schrödinger operator determines the potential. Our approach is an analog of the continued fraction approach for the moment problem. We prove there is a representation for the m-function m(-κ²) = -κ- ∫₀^b A(α) e^-2ακ dα+ O(e^-(2b-ε)κ). A on [0,a] is a function of q on [0,a] and vice-versa. A key role is played by a differential equation that A obeys after allowing x-dependence: (∂ A)/(∂ x) = (∂ A)/(∂ α) + ∫₀^αA(β, x) A(α-β, x) dβ. Among our new results are necessary and sufficient conditions on the m-functions for potentials q₁ and q₂ for q₁ to equal q₂ on [0,a].

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• The spectrum of Dirichlet-to-Neumann maps for radial conductivities
• Gelfand-Levitan condition for Dirac operators

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