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.

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1. T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin - Heidelberg - New York, 1980.
2. Y. Katznelson, An Introduction to Harmonic Analysis, 3rd ed., Cambridge University Press, 2004.
3. V.A. Marchenko, Sturm-Liouville Operators and their Applications, Kiev: Naukova Dumka, 1977 [in Russian]; Engl. transl., Basel., Birkhäuser, 1986.
4. Ya. Mykytyuk, N. Sushchyk, Inverse scattering problems for half-line Schrödinger operators and Banach algebras, Opuscula Math. 38 (2018), 719-731.
5. W. Rudin, Functional Analysis, McGraw Hill, New York, 1973.
6. E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed., Oxford University Press, 1948.
7. N. Wiener, Tauberian theorems, Ann. of Math. 33 (1932), 1-100.
8. N. Wiener, Generalized Harmonic Analysis and Tauberian Theorems, MIT Press, Cambridge, MA/London, 1966.
• Communicated by P.A. Cojuhari.
• Revised: 2019-03-19.
• Accepted: 2019-03-20.
• Published online: 2019-05-23.