# WKB Asymptotic Behavior of Almost All Generalized Eigenfunctions for One-Dimensional Schrödinger Operators with Slowly Decaying Potentials

@article{Christ2001WKBAB, title={WKB Asymptotic Behavior of Almost All Generalized Eigenfunctions for One-Dimensional Schr{\"o}dinger Operators with Slowly Decaying Potentials}, author={Michael Christ and Alexander V. Kiselev}, journal={Journal of Functional Analysis}, year={2001}, volume={179}, pages={426-447} }

We prove the WKB asymptotic behavior of solutions of the differential equation −d2u/dx2+V(x) u=Eu for a.e. E>A where V=V1+V2, V1∈Lp(R), and V2 is bounded from above with A=lim supx→∞ V(x), while V′2(x)∈Lp(R), 1⩽p<2. These results imply that Schrodinger operators with such potentials have absolutely continuous spectrum on (A, ∞). We also establish WKB asymptotic behavior of solutions for some energy-dependent potentials.

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## References

SHOWING 1-10 OF 27 REFERENCES

### Scattering and Wave Operators for One-Dimensional Schrödinger Operators with Slowly Decaying Nonsmooth Potentials

- Mathematics
- 2001

Abstract.We prove existence of modified wave operators for one-dimensional Schrödinger equations with potential in
$L^p (\mathbb{R}),p < 2.$ If in addition the potential is conditionally integrable,…

### Modified Prüfer and EFGP Transforms and the Spectral Analysis of One-Dimensional Schrödinger Operators

- Mathematics
- 1998

Abstract:Using control of the growth of the transfer matrices, wediscuss the spectral analysis of continuum and discrete half-line Schrödinger operators with slowly decaying potentials. Among our…

### Absolutely continuous spectrum for one-dimensional Schrodinger operators with slowly decaying potentials: Some optimal results

- Mathematics
- 1997

The absolutely continuous spectrum of one-dimensional Schr\"odinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely…

### Spectral Theory for Slowly Oscillating Potentials II. Schrödinger Operators

- Mathematics
- 1997

The absolutely continuous and singular spectrum of one‐dimensional Schrödinger operators with slowly oscillating potentials and perturbed periodic potentials is studied, continuing similar…

### Modified Prüfer and EFGP Transforms and Deterministic Models with Dense Point Spectrum

- Mathematics
- 1998

We provide a new proof of the theorem of Simon and Zhu that in the region |E|<λ for a.e. energies, −(d^2/dx^2)+λ cos(x^α), 0<α<1 has Lyapunov behavior with a quasi-classical formula for the Lyapunov…

### The Absolutely Continuous Spectrum of One-Dimensional Schrödinger Operators with Decaying Potentials

- Mathematics
- 1998

Abstract:We investigate one-dimensional Schrödinger operators with asymptotically small potentials. It will follow from our results that if with , then is an essential support of the absolutely…

### Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators

- Mathematics
- 1996

We provide a short proof of that case of the Gilbert-Pearson theorem that is most often used: That all eigenfunctions bounded implies purely a.c. spectrum. Two appendices illuminate Weidmann's result…

### Solutions, Spectrum, and Dynamica for Schr\"odinger Operators on Infinite Domains

- Mathematics
- 1999

Let H be a Schr\"odinger operator defined on an unbounded domain D in R^d with Dirichlet boundary conditions (D may equal R^d in particular). Let u(x,E) be a solution of the Schr\"odinger equation…

### On the Absolutely Continuous Spectrum¶of One-Dimensional Schrödinger Operators¶with Square Summable Potentials

- Mathematics
- 1999

Abstract:For continuous and discrete one-dimensional Schrödinger operators with square summable potentials, the absolutely continuous part of the spectrum is essentially supported by [0,∞) and [−2,2]…