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

DOI10.1090/S0894-0347-98-00276-8zbMath0899.34051arXivmath/9706221MaRDI QIDQ4211061

Alexander Kiselev, Michael Christ

Publication date: 10 September 1998

Published in: Journal of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/9706221

zbMATH Keywords

Schrödinger operatorsdecaying potentialnorm estimatesa.e. convergenceWKB asymptoticsabsolutely continuous spectra

Mathematics Subject Classification ID

Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) (34L40) Singular and oscillatory integrals (Calderón-Zygmund, etc.) (42B20) Maximal functions, Littlewood-Paley theory (42B25) Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) Perturbation theories for operators and differential equations in quantum theory (81Q15) Singular perturbations, turning point theory, WKB methods for ordinary differential equations (34E20)

Related Items (37)

The spectral measure of a Jacobi matrix in terms of the Fourier transform of the perturbation ⋮ Twelve tales in mathematical physics: An expanded Heineman prize lecture ⋮ Lorentz space extension of Strichartz estimates ⋮ Spectral and regularity properties of a pseudo-differential calculus related to Landau quantization ⋮ Stability of spectral types for Jacobi matrices under decaying random perturbations ⋮ Stability of spectral types of quasi-periodic Schrödinger operators with respect to perturbations by decaying potentials ⋮ Preservation of a.c.\ spectrum for random decaying perturbations of square-summable high-order variation ⋮ The absence of singular continuous spectrum for perturbed Jacobi operators ⋮ Dirac operators with operator data of Wigner-von Neumann type ⋮ On new relations between spectral properties of Jacobi matrices and their coefficients ⋮ Nonlinear Fourier transform—towards the construction of nonlinear Fourier modes ⋮ Absolutely continuous spectrum of Stark operators ⋮ Lower bounds on the eigenvalue sums of the Schrödinger operator and the spectral conservation law ⋮ Spectral theory?50 years of progress and a conclusion ⋮ Noncompact complete Riemannian manifolds with singular continuous spectrum embedded into the essential spectrum of the Laplacian, I. The hyperbolic case ⋮ Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian ⋮ Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains ⋮ Schrödinger operators in the twentieth century ⋮ Tosio Kato's work on non-relativistic quantum mechanics. I ⋮ Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators ⋮ Maximal functions associated to filtrations ⋮ WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials ⋮ Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials ⋮ The spectrum of differential operators of order \(2n\) with almost constant coefficients ⋮ Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials ⋮ Imbedded singular continuous spectrum for Schrödinger operators ⋮ Multi-dimensional Schrödinger operators with some negative spectrum ⋮ Stability of asymptotics of Christoffel-Darboux kernels ⋮ Sum rules and spectral measures of Schrödinger operators with \(L^2\) potentials ⋮ Effective perturbation methods for one-dimensional Schrödinger operators ⋮ Tosio Kato’s work on non-relativistic quantum mechanics, Part 2 ⋮ On the order of growth of generalized eigenfunctions of the Sturm-Liouville operator. The Shnol' theorem ⋮ On the absolutely continuous spectrum of multi-dimensional Schrödinger operators with slowly decaying potentials ⋮ Schrödinger operators with decaying potentials: some counterexamples ⋮ Revisiting the Christ–Kiselev’s multi-linear operator technique and its applications to Schrödinger operators ⋮ Jacobi matrices with power-like weights -- grouping in blocks approach ⋮ Transfer matrix for long-range potentials

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• One-dimensional Schrödinger operators with random decaying potentials
• Spectral theory of ordinary differential operators
• Transient and recurrent spectrum
• Bounded solutions and absolute continuity of Sturm-Liouville operators
• Asymptotic integration of adiabatic oscillators
• The existence of wave operators in scattering theory
• Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators
• Effective perturbation methods for one-dimensional Schrödinger operators
• Some Schrödinger operators with power-decaying potentials and pure point spectrum
• Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture
• The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials
• Wave operators and positive eigenvalues for a Schrödinger equation with oscillating potential
• Absolutely continuous spectra of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials
• Schrodinger Operators with Rapidly Oscillating Central Potentials
• Absolute Continuity of Hamiltonians with von Neumann Wigner Potentials
• Absolutely Continuous Spectra of Second Order Differential Operators with Short and Long Range Potentials
• Erratum: Spectral and scattering theory for the adiabatic oscillator and related potentials [J. Math. Phys. 2 0, 594(1979)]
• An interpolation theorem related to the a.e. convergence of integral operators
• Some Schrödinger operators with dense point spectrum
• Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators
• Maximal functions associated to filtrations

This page was built for publication: Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results