# 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}
}```
• Published 1 February 2001
• Mathematics
• Journal of Functional Analysis
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.
58 Citations

### WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials

• Mathematics
• 2001
Abstract: Consider a Schrödinger operator on L2 of the line, or of a half line with appropriate boundary conditions. If the potential tends to zero and is a finite sum of terms, each of which has a

### Finite Gap Potentials and WKB Asymptotics¶for One-Dimensional Schrödinger Operators

• Mathematics
• 2001
Abstract: Consider the Schrödinger operator H=−d2/dx2+V(x) with power-decaying potential V(x)=O(x−α). We prove that a previously obtained dimensional bound on exceptional sets of the WKB method is

### On the absolutely continuous and negative discrete spectra of Schrödinger operators on the line with locally integrable globally square summable potentials

For one-dimensional Schrodinger operators with potentials q subject to ∑n=−∞∞(∫nn+1|q(x)|dx)2<∞, we prove that the absolutely continuous spectrum is [0,∞), extending the 1999 result due to

### Spectral properties of Schr\"odinger operators with locally \$H^{-1}\$ potentials

• Mathematics
• 2022
. We study half-line Schr¨odinger operators with locally H − 1 potentials. In the ﬁrst part, we focus on a general spectral theoretic framework for such operators, including a Last–Simon-type

### The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

• Mathematics
• 2008
We consider the quadratically semilinear wave equation on (ℝ d , 𝔤), d ≥ 3. The metric 𝔤 is non-trapping and approaches the Euclidean metric like ⟨x⟩−ρ. Using Mourre estimates and the Kato theory

### Cwikel and Quasi-Szegö Type Estimates for Random Operators

• Mathematics
• 2008
We consider Schrödinger operators with nonergodic random potentials. Specifically, we are interested in eigenvalue estimates and estimates of the entropy for the absolutely continuous part of the

### Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential

• Mathematics
• 2014
Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential.

### A short note on the appearance of the simplest antilinear ODE in several physical contexts

In this short note, we review several one-dimensional problems such as those involving linear Schrödinger equation, variable-coeﬃcient Helmholtz equation, Zakharov-Shabat system and Kubelka-Munk

### Schrödinger operators with many bound states

• Mathematics
• 2004
Consider the Schrodinger operators H± = −d2/dx2 ± V (x). We present a method for estimating the potential in terms of the negative eigenvalues of these operators. Among the applications are inverse

### A Note on the Appearance of the Simplest Antilinear ODE in Several Physical Contexts

We review several one-dimensional problems such as those involving linear Schrödinger equation, variable-coefficient Helmholtz equation, Zakharov–Shabat system and Kubelka–Munk equations. We show

## 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

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

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

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

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]