The Lyapunov exponents for Schrodinger operators and Jacobi matrices with slowly oscillating potentials
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In the first part, we study the one-dimensional half-line Schrodinger operator [...] (1) with 0 < [...] < 1. For each [...], let [...] denote the unique self-adjoint realizatio...
| Summary: | NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
In the first part, we study the one-dimensional half-line Schrodinger operator [...] (1) with 0 < [...] < 1. For each [...], let [...] denote the unique self-adjoint realization of [...] on [...] with boundary condition at 0 given by [...] = 0.
By studying the integrated density of states, we prove the existence of the Lyapunov exponent and the Thouless formula for (1). This yields an explicit formula for these Lyapunov exponents. By applying rank one perturbation theory, we also obtain some spectral consequences. Our main results are the following.
Theorem. Let [...] = [...] and [...] = [...]. Then for all [...], where [...] is the resonance set for (1) which has both Lebesgue measure zero and Hausdorff dimension zero, we have [...] = [...] where [...] is the Lyapunov exponent for [...], and [...] is the integrated density of states for [...].
Theorem. For all [...], where [...] is the resonance set for (1) which has both Lebesgue measure zero and Hausdorff dimension zero, the operator [...] in (1) has Lyapunov behavior with the Lyapunov exponent given by [...] = [...] (2)
Theorem. For [...] (with respect to Lebesgue measure), [...] has dense pure point spectrum on (-1,1), and the eigenfunction of [...] to all eigenvalues [...] (-1,1) decay like [...] at [...] for almost every [...], where [...] is the Lyapunov exponent for (1) which is given by (2).
Theorem. For [...], the singular continuous part, [...], of the spectral measure [...] for [...] is supported on a Hausdorff dimension zero set.
In the second part, we extend the above arguments to the Jacobi matrix on [...] which is a discrete analog of the Schrodinger operator (1). Let [...] = [...] (3) with [...] < 2 and 0 < [...] < 1.
Similarly, by studying the integrated density of states for (3), we can prove the existence of the Lyapunov exponents and the Thouless formula for (3). Then, we can compute an explicit formula for these Lyapunov exponents. By applying rank one perturbation theory again, we can also obtain some interesting spectral consequences for [...]. We have the following theorems.
Theorem. There exists a Lebesgue measure zero and Hausdorff dimension zero set [...], which we call the resonance set for (3). For all [...] has Lyapunov behavior with the Lyapunov exponent given by [...] = [...] (4)
Theorem. For almost all [...] < 2 (with respect to Lebesgue measure), [...] has dense pure point spectrum on [...], and the eigenvectors to all eigenvalues E decay like [...] at infinity, where [...] is the Lyapunov exponent for (3) which is given by (4).
Theorem. For [...], [...], the singular continuous part of the spectral measure [...] for [...], is supported on a Hausdorff dimension zero set. |
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