×

WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials. (English) Zbl 0985.34078

The main result of this paper gives asymptotics for solutions \(u_\pm(x,E)\) to \(-u''+(V-E)u=0\), where \(V=V_1+V_2\) on \(\mathbb{R}\), with (i) \(V_2\) bounded; (ii) \(V_1\) and \(V'_2\in \ell^p(L^1)(\mathbb{R})\) for some \(p\in[1,2)\) (which means that their \(L^1\) norms on \([n,n+1]\) form an \(\ell^p\) sequence). The authors find that, for almost all \(E>\limsup_{x\to\infty}V_2(x)\), there are solutions with “WKB asymptotics”. A part of the proof relies on a paper by the authors, which is to appear in the same journal.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Carleson, L., On convergence and growth of partial sums of Fourier series, Acta Math., 116, 135-157 (1966) · Zbl 0144.06402
[2] Christ, M.; Kiselev, A., Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results, J. Amer. Math. Soc., 11, 771-797 (1998) · Zbl 0899.34051
[3] M. Christ, and, A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal, to appear.; M. Christ, and, A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal, to appear. · Zbl 0974.47025
[4] M. Christ, and, A. Kiselev, On one-dimensional Schrödinger operators whose potentials have slowly decaying derivatives, Commun. Math. Phys, to appear.; M. Christ, and, A. Kiselev, On one-dimensional Schrödinger operators whose potentials have slowly decaying derivatives, Commun. Math. Phys, to appear.
[5] M. Christ, and, A. Kiselev, Absolutely continuous spectrum of Stark operators, in preparation.; M. Christ, and, A. Kiselev, Absolutely continuous spectrum of Stark operators, in preparation. · Zbl 1046.34095
[6] M. Christ, A. Kiselev, and Y. Last, Approximate eigenvectors and spectral theory, in Differential Equations and Mathematical Physics, Proceedings of an International Conference held at the University of Alabama at Birmingham, pp. 61-72.; M. Christ, A. Kiselev, and Y. Last, Approximate eigenvectors and spectral theory, in Differential Equations and Mathematical Physics, Proceedings of an International Conference held at the University of Alabama at Birmingham, pp. 61-72. · Zbl 1056.35129
[7] Deift, P.; Killip, R., On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Commun. Math. Phys., 203, 341-347 (1999) · Zbl 0934.34075
[8] R. Killip, Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum, preprint.; R. Killip, Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum, preprint. · Zbl 1021.34071
[9] Kiselev, A.; Last, Y., Solutions, spectrum and dynamics of Schrödinger operators on infinite domains, Duke Math. J., 102, 125-150 (2000) · Zbl 0951.35033
[10] Kiselev, A.; Last, Y.; Simon, B., Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators, Commun. Math. Phys., 194, 1-45 (1998) · Zbl 0912.34074
[11] Last, Y.; Simon, B., Modified Prüfer and EFGP transforms and deterministic models with dense point spectrum, J. Funct. Anal., 154, 513-530 (1998) · Zbl 0912.34075
[12] Remling, C., The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials, Commun. Math. Phys., 193, 151-170 (1998) · Zbl 0908.34067
[13] Simon, B., Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc., 124, 3361-3369 (1996) · Zbl 0869.34069
[14] Simon, B.; Spencer, T., Trace class perturbations and the absence of absolutely continuous spectra, Commun. Math. Phys., 125, 113-125 (1989) · Zbl 0684.47010
[15] Simon, B.; Stolz, G., Operators with singular continuous spectrum. V. Sparse potentials, Proc. Amer. Math. Soc., 124, 2073-2080 (1996) · Zbl 0979.34063
[16] Simon, B.; Zhu, Y. F., The Lyapunov exponents for Schrödinger operators with slowly oscillating potentials, J. Funct. Anal., 140, 541-556 (1996) · Zbl 0858.34074
[17] Stollmann, P.; Stolz, G., Singular spectrum for multidimensional Schrödinger operators with potential barriers, J. Operator Theory, 32, 91-109 (1994) · Zbl 0823.35043
[18] Stolz, G., Bounded solutions and absolute continuity of Sturm-Liouville operators, J. Math. Anal. Appl., 169, 210-228 (1992) · Zbl 0785.34052
[19] Stolz, G., Spectral theory for slowly oscillating potentials. II. Schrödinger operators, Math. Nachr., 183, 275-294 (1997) · Zbl 0951.34062
[20] Weidmann, J., Zur Spektral theorie von Sturm-Liouville Operatoren, Math. Z., 98, 268-302 (1968) · Zbl 0168.12301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.