Hostname: page-component-89b8bd64d-nlwjb Total loading time: 0 Render date: 2026-05-07T04:31:48.386Z Has data issue: false hasContentIssue false
  • English
  • Français

SINGULAR SCHRÖDINGER OPERATORS IN ONE DIMENSION

Part of: Ordinary differential operators Boundary value problems

Published online by Cambridge University Press:  12 April 2012

E. B. Davies
E. B. Davies*
Affiliation:
Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, U.K. (email: E.Brian.Davies@kcl.ac.uk)
Get access

Abstract

We consider a class of singular Schrödinger operators H that act in L2(0,∞), each of which is constructed from a positive function ϕ on (0,∞). Our analysis is direct and elementary. In particular it does not mention the potential directly or make any assumptions about the magnitudes of the first derivatives or the existence of second derivatives of ϕ. For a large class of H that have discrete spectrum, we prove that the eigenvalue asymptotics of H does not depend on rapid oscillations of ϕ or of the potential. Similar comments apply to our treatment of the existence and completeness of the wave operators.

MSC classification

Secondary: 34B27: Green functions 34L05: General spectral theory 34L20: Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions 34L25: Scattering theory, inverse scattering

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 141 - 159
Copyright
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Atkinson, F. V., The asymptotic solution of second-order differential equations. Ann. Mat. Pura Appl. (4) 37 (1954), 347–378.CrossRefGoogle Scholar
[2]Atkinson, F. V., Everitt, W. N. and Zettl, A., Regularization of a Sturm–Liouville problem with an interior singularity using quasi-derivatives. Differ. Integral Equ. Appl. 1(2) (1988), 213–221.Google Scholar
[3]Ben-Artzi, M. and Devinatz, A., Spectral and scattering theory for the adiabatic oscillator and related potentials. J. Math. Phys. 20(4) (1979), 594–607.CrossRefGoogle Scholar
[4]Benguria, R. and Loss, M., A simple proof of a theorem of Laptev and Leidl. Math. Res. Lett. 7 (2000), 195–203.CrossRefGoogle Scholar
[5]Bennewitz, C. and Everitt, W. N., On second-order left-definite boundary value problems. In Ordinary Differential Equations and Operators (Dundee, 1982) (Lecture Notes in Mathematics 1032), Springer (Berlin, 1983), 31–67.CrossRefGoogle Scholar
[6]Combescure, M., Spectral and scattering theory for a class of strongly oscillating potentials. Comm. Math. Phys. 73 (1980), 43–62.CrossRefGoogle Scholar
[7]Davies, E. B., Heat Kernels and Spectral Theory, Cambridge University Press (Cambridge, 1989).CrossRefGoogle Scholar
[8]Davies, E. B., Heat kernel bounds, conservation of probability and the Feller property. J. Anal. Math. 58 (1992), 99–119.CrossRefGoogle Scholar
[9]Davies, E. B., Spectral Theory and Differential Operators, Cambridge University Press (Cambridge, 1995).CrossRefGoogle Scholar
[10]Davies, E. B., Linear Operators and their Spectra, Cambridge University Press (Cambridge, 2007).CrossRefGoogle Scholar
[11]Davies, E. B. and Harrell II, E. M., Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation. J. Differential Equations 66 (1987), 165–188.CrossRefGoogle Scholar
[12]Eckhardt, J., Gesztesy, F., Nichols, R. and Teschl, G., Weyl–Titchmarsh theory for Sturm–Liouville operators with distributional potential coefficients, 2011(in preparation).Google Scholar
[13]Eckhart, J. and Teschl, G., Sturm–Liouville operators with measure-valued coefficients. Preprint, 2011, arXiv:1105.3755.Google Scholar
[14]Hryniv, R. O. and Mykytyuk, Y. V., Eigenvalue asymptotics for Sturm–Liouville operators with singular potentials. J. Funct. Anal. 238 (2006), 27–57.CrossRefGoogle Scholar
[15]Kappeler, T. and Möhr, C., Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator with singular potentials. J. Funct. Anal. 186 (2001), 62–91.CrossRefGoogle Scholar
[16]Matveev, V. B. and Skriganov, M. M., Wave operators for the Schrödinger equation with rapidly oscillating potential. Soviet Math. Dokl. 13 (1972), 185–188.Google Scholar
[17]Mingarelli, A. B., Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions (Lecture Notes in Mathematics 989), Springer (Berlin, 1983).CrossRefGoogle Scholar
[18]Nesterov, P. N., Construction of the asymptotic behavior of solutions of the one-dimensional Schrödinger equation with a rapidly oscillating potential. Mat. Zametki 80(2) (2006), 240–250, (in Russian, translation in Math. Notes 80(2006), 233–243).Google Scholar
[19]Pearson, D. B., Scattering theory for a class of oscillating potentials. Helv. Phys. Acta 52 (1979), 541–554.Google Scholar
[20]Reed, M. and Simon, B., Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators, Academic Press (New York, 1978).Google Scholar
[21]Reed, M. and Simon, B., Methods of Modern Mathematical Physics, Vol. 3, Scattering Theory, Academic Press (New York, 1979).Google Scholar
[22]Reid, W. T., Riccati Differential Equations, Academic Press (New York, 1972).Google Scholar
[23]Savchuk, A. M. and Shkalikov, A. A., Sturm–Liouville operators with singular potentials. Math. Notes 66 (1999), 741–753.CrossRefGoogle Scholar
[24]Savchuk, A. M. and Shkalikov, A. A., On the eigenvalues of the Sturm–Liouville operator with potentials from Sobolev spaces. Mat. Zametki 80 (2006), 864–884 (in Russian, translation in Math. Notes 80(2006), 814–832).Google Scholar
[25]Skriganov, M. M., The spectrum of a Schrödinger operator with rapidly oscillating potential. Tr. Mat. Inst. Steklov 125 (1973), 187–195 (in Russian, translation in Proc. Steklov Inst. Math. 125 (1973), 177–185).Google Scholar
[26]Teschl, G., Mathematical Methods in Quantum Mechanics, with Applications to Schrödinger Operators (Graduate Studies in Mathematics 99), American Mathematical Society (Providence, RI, 2009).Google Scholar
[27]White, D. A. W., Schrödinger operators with rapidly oscillating central potentials. Trans. Amer. Math. Soc. 275 (1983), 641–677.Google Scholar
[28]Wielens, N., The essential self-adjointness of generalized Schrödinger operators. J. Funct. Anal. 61 (1985), 98–115.CrossRefGoogle Scholar