Scattering theory for Schrödinger operators with Bessel-type potentials (Q2890361)

The authors show that the classical direct and inverse scattering theory on the semi-axis \(\mathbb{R}_{+}\) for the one-dimensional Schrödinger operators \(H(q)\), defined by NEWLINE\[NEWLINE H(q)y:=-y''+q(x)y, NEWLINE\]NEWLINE with the real-valued Faddeev-Marchenko potentials: NEWLINE\[NEWLINE q\in L_{1}^{1}(\mathbb{R}_{+}):=\left\{f\in L^{1}(\mathbb{R}_{+}):\int_{0}^{\infty}x|f(x)|\,dx<\infty\right\}, \qquad\text{and}\qquad \mathrm{Im}\,q=0, NEWLINE\]NEWLINE can be successfully extended to the operators \(H(q_{\kappa})\) with the Bessel-type potentials \(q_{\kappa}\): NEWLINE\[NEWLINE q_{\kappa}(x):=\dfrac{\kappa(\kappa+1)}{x^{2}},\qquad \kappa\in [-\frac{1}{2},\frac{1}{2}). NEWLINE\]NEWLINE For the operators \(H(q_{\kappa})\), the authors construct transformation operators, Jost solutions, the scattering function \(S(H(q_{\kappa});\omega)\), derive the Marchenko equation and demonstrate that its solution reconstructs the potential they have started with.NEWLINENEWLINEIn the case of the 1-d Schrödinger operators \(H(q)\) with real-valued Faddeev-Marchenko potentials \(q\), the scattering function \(S(H(q);\omega)\) is continuous on the whole real axis \(\mathbb{R}\) and is close to 1 at infinity in the sense that the difference \(1-S(H(q);\omega)\) belongs to the Hilbert space \(L^{2}(\mathbb{R})\).NEWLINENEWLINEFor the operators \(H(q_{\kappa})\) with the Bessel-type potentials \(q_{\kappa}\), the authors discover a new phenomenon. In contrast with the classical case, they find that NEWLINE\[NEWLINE S(H(q_{\kappa});\omega)=e^{-\pi i\kappa}\quad\text{if}\quad \omega>0\qquad\text{and}\qquad S(H(q_{\kappa});\omega)=e^{\pi i\kappa}\quad\text{if}\quad \omega<0, NEWLINE\]NEWLINE i.e., the scattering function \(S(H(q_{\kappa});\omega)\) is no longer continuous, but rather has two jump discontinuities at the origin and the other at infinity. The jump at the origin is in some sense caused by the behaviour of the potential at infinity, while the behaviour of \(S(H(q_{\kappa});\omega)\) at infinity is determined by the singularity of the potential at the origin.