Perturbations of one-dimensional Schrodinger operators preserving the absolutely continuous spectrum

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Continuous analogs of polynomials orthogonal on the unit circle. Krein systems (2009, methodology): “These ideas were also used quite recently [41].”
On New Relations Between Spectral Properties of Jacobi Matrices and Their Coefficients (2003, methodology): “Remark 2. Closely related results can be found in [ 16 ], Theorem 4, where the author used a “locally spectral technique” in order to obtain a continuous version of Theorem 1.3.”
Sum rules for Jacobi matrices and their applications to spectral theory (2001, methodology): “) Research in this direction began with Kiselev [29] and culminated in the work of Christ-Kiselev [11], Remling [47], Deift-Killip [13], and Killip [28].”
Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics (2005, methodology): “In turn, we were all motivated by some arguments of Killip [23].”