Opuscula Math. 39, no. 4 (2019), 557-576
https://doi.org/10.7494/OpMath.2019.39.4.557
Opuscula Mathematica
Description of the scattering data for Sturm-Liouville operators on the half-line
Yaroslav Mykytyuk
Nataliia Sushchyk
Abstract. We describe the set of the scattering data for self-adjoint Sturm-Liouville operators on the half-line with potentials belonging to \(L_1(\mathbb{R}_+,\rho(x)\,\text{d} x)\), where \(\rho:\mathbb{R}_+\to\mathbb{R}_+\) is a monotonically nondecreasing function from some family \(\mathscr{R}\). In particular, \(\mathscr{R}\) includes the functions \(\rho(x)=(1+x)^{\alpha}\) with \(\alpha\geq 1\).
Keywords: inverse scattering, Schrödinger operator, Banach algebra.
Mathematics Subject Classification: 34L25, 34L40, 47L10, 81U40.
- T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin - Heidelberg - New York, 1980.
- Y. Katznelson, An Introduction to Harmonic Analysis, 3rd ed., Cambridge University Press, 2004.
- V.A. Marchenko, Sturm-Liouville Operators and their Applications, Kiev: Naukova Dumka, 1977 [in Russian]; Engl. transl., Basel., Birkhäuser, 1986.
- Ya. Mykytyuk, N. Sushchyk, Inverse scattering problems for half-line Schrödinger operators and Banach algebras, Opuscula Math. 38 (2018), 719-731.
- W. Rudin, Functional Analysis, McGraw Hill, New York, 1973.
- E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed., Oxford University Press, 1948.
- N. Wiener, Tauberian theorems, Ann. of Math. 33 (1932), 1-100.
- N. Wiener, Generalized Harmonic Analysis and Tauberian Theorems, MIT Press, Cambridge, MA/London, 1966.
- Yaroslav Mykytyuk
https://orcid.org/0000-0002-5037-7143- Lviv National University, 1 Universytets'ka Str., 79602 Lviv, Ukraine
- Nataliia Sushchyk
- https://orcid.org/0000-0001-9395-3094
- Lviv National University, 1 Universytets'ka Str., 79602 Lviv, Ukraine
- Communicated by P.A. Cojuhari.
- Received: 2019-01-26.
- Revised: 2019-03-19.
- Accepted: 2019-03-20.
- Published online: 2019-05-23.
