On Local Borg–Marchenko Uniqueness Results
نویسندگان
چکیده
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 Hj = − d2 dx2 + qj , j = 1, 2 in L2((0, R)), 0 < R ≤ ∞, are exponentially close, that is, |m1(z) − m2(z)| = |z|→∞ O(e −2 Im(z)a), 0 < a < R, then q1 = q2 a.e. on [0, a]. The result applies to any boundary conditions at x = 0 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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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 ((0, R)), 0 < R ≤ ∞, are exponentially close, that is, |m 1 (z) − m 2 (z)| = |z|→∞ O(e −2 Im(z 1/2)a), 0 < a < R, then q 1 = q 2 a.e. on [0, a]. The result applies to any boundary conditions at x...
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