Uniqueness of Reflectionless Jacobi Matrices and the Denisov-rakhmanov Theorem
نویسنده
چکیده
If a Jacobi matrix J is reflectionless on (−2, 2) and has a single an0 equal to 1, then J is the free Jacobi matrix an ≡ 1, bn ≡ 0. I’ll discuss this result and its generalization to arbitrary sets and present several applications, including the following: if a Jacobi matrix has some portion of its an’s close to 1, then one assumption in the Denisov-Rakhmanov Theorem can be dropped. 1. Statement of results A Jacobi matrix is a difference operator of the following type: (Ju)n = anun+1 + an−1un−1 + bnun Here, an > 0 and bn ∈ R, and we also always assume that a, b are bounded sequences. Alternatively, one can represent J by a tridiagonal matrix with respect to the standard basis of `(Z): J = . . . . . . . . . a−2 b−1 a−1 a−1 b0 a0 a0 b1 a1 . . . . . . . . . Half line operators J+, on ` (Z+), say, are defined similarly, by considering a suitable truncation of this matrix. The case an ≡ 1 is of particular interest; these operators are called (discrete) Schrödinger operators. The Denisov-Rakhmanov (DR) Theorem [5, 10] says the following: Date: June 14, 2010.
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