Communications in Mathematical Physics

, Volume 211, Issue 2, pp 273–287

On Local Borg–Marchenko Uniqueness Results


  • Fritz  Gesztesy
    • Department of Mathematics, University of Missouri, Columbia, MO 65211, USA.¶E-mail:
  • Barry Simon
    • Division of Physics, Mathematics, and Astronomy, 253-37, California Institute of Technology,¶Pasadena, CA 91125, USA. E-mail:

DOI: 10.1007/s002200050812

Cite this article as:
Gesztesy, F. & Simon, B. Comm Math Phys (2000) 211: 273. doi:10.1007/s002200050812


We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl–Titchmarsh m-functions, mj(z), of two Schrödinger operators \(\), j≡ 1,2 in L2((0,R)), 0<R≤∞, are exponentially close, that is, \(\), 0<a<R, then q1q2 a.e. on [0,a]. The result applies to any boundary conditions at x≡ 0 and xR 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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© Springer-Verlag Berlin Heidelberg 2000