The Absolutely Continuous Spectrum of One-Dimensional Schrödinger Operators with Decaying Potentials (original) (raw)
On the spectrum of the one-dimensional Schrödinger operator
Discrete and Continuous Dynamical Systems - Series B, 2008
The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy E belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.
First KdV Integrals¶and Absolutely Continuous Spectrum¶for 1-D Schrödinger Operator
Communications in Mathematical Physics, 2001
We consider 1-D Schrödinger operators on L 2 (R + ) with slowly decaying potentials. Under some conditions on the potential, related to the first integrals of the KdV equation, we prove that the a.c. spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable. Examples show that for special classes of sparse potentials these results can not be improved.
Communications in Mathematical Physics, 1993
We investigate one-dimensional discrete Schr odinger operators whose potentials are invariant under a substitution rule. The spectral properties of these operators can be obtained from the analysis of a dynamical system, called the trace map. We give a careful derivation of these maps in the general case and exhibit some speci c properties. Under an additional, easily veri able hypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying Schr odinger operator is singular and supported on a set of zero Lebesgue measure. A condition allowing to exclude point spectrum is also given. The application of our theorems is explained on a series of examples.
One-dimensional Schr�dinger operators with random decaying potentials
Communications in Mathematical Physics, 1988
We investigate the spectrum of the following random Schrόdinger operators: 72 where F(X t (ω)) is a Markovian potential studied by the Russian school [8]. We completely describe the transition of the spectrum from pure point type to absolutely continuous type as the decreasing order of a(t) grows. This is an extension to a continuous case of the result due to Delyon-Simon-Souillard [6], who deal with the lattice case.
On norm resolvent convergence of Schrödinger operators with δ′-like potentials
Journal of Physics A, 2010
For a function V : R → R that is integrable and compactly supported, we prove the norm resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schrödinger operators on the line of the form then the functions ε -2 V (x/ε) converge in the sense of distributions as ε → 0 to δ ′ (x), and the limit S 0 of S ε might be considered as a 'physically motivated' interpretation of the one-dimensional Schrödinger operator with potential δ ′ . In 1985, Šeba claimed that the limit operator S 0 is the direct sum of the free Schrödinger operators on positive and negative semi-axes subject to the Dirichlet condition at x = 0, which suggested that in dimension 1 there is no non-trivial Hamiltonian with potential δ ′ . In this paper, we show that in fact S 0 essentially depends on V : although the above results are true generically, in the exceptional (or 'resonant') case, the limit S 0 is non-trivial and is determined by the properties of an auxiliary Sturm-Liouville spectral problem associated with V . We then set V (ξ) = αΨ(ξ) with a fixed Ψ and show that there exists a countable set of resonances {α k } ∞ k=-∞ for which a partial transmission of the wave package occurs for S 0 .