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On Local Borg–Marchenko Uniqueness Results

Gesztesy, Fritz and Simon, Barry (2000) On Local Borg–Marchenko Uniqueness Results. Communications in Mathematical Physics, 211 (2). pp. 273-287. ISSN 0010-3616. doi:10.1007/s002200050812.

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We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl–Titchmarsh m-functions, m_j(z), of two Schrödinger operators, H_j = -d^2/dx^2 + q_j, j = 1,2 in L^2((O,R)), O < R ≤ ∞, are exponentially close, that is, |m_1(z) - m_2(z)|_|z|→∞ = O(e^(-2 IM(z^1/2)a), O < ɑ <R, then q_1 = q_2 a.e. on [O,ɑ]. The result applies to any boundary conditions at x = O and x = R and should be considered a local version of the celebrated Borg–Marchenko uniqueness result (which is quickly recovered as a corollary to our proof). Moreover, we extend the local uniqueness result to matrix-valued Schrödinger operators.

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Simon, Barry0000-0003-2561-8539
Additional Information:© 2000 Springer-Verlag. Received: 22 October 1999; Accepted: 2 November 1999. This material is based upon work supported by the National Science Foundation under Grant No. DMS-9707661. F. G. thanks T. Tombrello for the hospitality of Caltech where this work was done.
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Issue or Number:2
Record Number:CaltechAUTHORS:20170801-072515055
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Official Citation:Gesztesy, F. & Simon, B. Comm Math Phys (2000) 211: 273.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79655
Deposited By: Ruth Sustaita
Deposited On:01 Aug 2017 15:55
Last Modified:15 Nov 2021 17:49

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