Reconstruction of the potential from I-function

2001/02/22 by A. G. Ramm, А. Г. Рамм, Ramm, A. G.
Computer Science · Mathematics · Physics and Astronomy · #Matrix Theory and Algorithms #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:34R30

paper · pdf · doi:10.48550/arxiv.math-ph/0102028

arxiv created 2001/03/31 · arxiv updated 2009/11/30

Abstract

If f(x,k) is the Jost solution and f(x) = f(0,k), then the I-function is I(k) := (f^′(0,k))/(f(k)). It is proved that I(k) is in one-to-one correspondence with the scattering triple \mathcal S :=\S(k), k_j, s_j, 1 ≤ j ≤ J\ and with the spectral function ρ(λ) of the Sturm-Liouville operator l= -(d²)/(dx²) + q(x) on (0, ∞) with the Dirichlet condition at x=0 and q(x) ∈ L_1,1 := \q: q= q, ∫^∞₀ (1+x) |q(x) dx < ∞\. Analytical methods are given for finding \mathcal S from I(k) and I(k) from \mathcal S, and ρ(λ) from I(k) and I(k) from ρ(λ). Since the methods for finding q(x) from \mathcal S or from ρ(λ) are known, this yields the methods for finding q(x) from I(k).

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