Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators

@article{Last1999EigenfunctionsTM,
  title={Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schr{\"o}dinger operators},
  author={Yoram Last and Barry Simon},
  journal={Inventiones mathematicae},
  year={1999},
  volume={135},
  pages={329-367}
}
  • Y. Last, B. Simon
  • Published 15 January 1999
  • Mathematics
  • Inventiones mathematicae
Neat stuff about eigenfunctions, transfer matrices, and a.c. spectrum of one-dimensional Schrodinger operators 
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References

SHOWING 1-10 OF 93 REFERENCES
Operators with singular continuous spectrum, V. Sparse potentials
By presenting simple theorems for the absence of positive eigenvalues for certain one-dimensional Schrodinger operators, we are able to construct explicit potentials which yield purely singular…
Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials
We prove that for any one-dimensional Schrödinger operator with potentialV(x) satisfying decay condition|V(x)|≦Cx−3/4−ε, the absolutely continuous spectrum fills the whole positive semi-axis. The…
Trace class perturbations and the absence of absolutely continuous spectra
We show that various Hamiltonians and Jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices.
Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators
We prove that one-dimensional Schrödinger operators with even almost periodic potential have no point spectrum for a denseGδ in the hull. This implies purely singular continuous spectrum for the…
One-dimensional Schrödinger operators with random or deterministic potentials: New spectral types
Dimensional Hausdorff properties of singular continuous spectra.
We present an extension of the Gilbert-Pearson theory of subordinacy, which relates dimensional Hausdorff spectral properties of one-dimensional Schrodinger operators to the behavior of solutions of…
Subordinacy and spectral theory for infinite matrices
The notion of subordinacy, previously used as a tool in the spectral theory of ordinary differential operators, is extended and applied to solutions of three-term recurrence relations. The resulting…
Support theorems for random Schrödinger operators
In the multi-dimensional case it is shown that the increase of the topological support of the probability measure describing the randomness of potentials implies the increase of the spectrum. In the…
JACOBI MATRICES WITH RANDOM POTENTIALS TAKING FINITELY MANY VALUES
It is proved that Jacobi matrices with random potentials taking finitely many values have no absolutely continuous spectrum unless the potentials are periodic.
Singular continuous spectrum in a class of random Schrödinger operators
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