Inverse scattering problems for Sturm-Liouville operators with spectral parameter dependent on boundary conditions (Q1646337)

The paper considers the scattering problem for a Schödinger operator on the half-line with Robin condition depending on an energy-dependent constant \[ -y''(x)+q(x)y(x) = zy(x) \] with \(x\in [0,\infty)\) and boundary condition \[ \frac{y'(0)}{y(0)} = f(z)\,. \] The authors assume that the potential satisfies \[ \int_0^\infty (1+x)|q(x)|\,\mathrm{d}x < \infty\,. \] The function \(f\) is a meromorphic Herglotz function defined by \(f(z) = \frac{p_1(z)}{p_2(z)}\), where \(p_j\) are polynomials and their roots satisfy certain interlacing property. The paper proves that the potential \(q\) can be uniquely recovered from the scattering data. First, the Jost solution \(e(x,\lambda) = \mathrm{e}^{i\lambda x} + \int_x^\infty K(x,t) \,\mathrm{e}^{i\lambda t}\,\mathrm{d}t\), with \(z = \lambda^2\), which is for \(\mathrm{Im\,}\lambda\geq 0\) a unique solution of the above Schrödinger equation, is introduced. The scattering function \(S(\lambda)\) is defined using polynomials \(p_j\), Jost solution, its conjugate and their derivatives. By the scattering data the authors mean the scattering function and the sets \(\{\lambda_k,m_k\}\), where \(i\lambda_k\) are poles of the scattering function in the upper half-plane and \(m_k\) are norming numbers defined using \(e\), \(\lambda_k\) and \(p_j\). The equation for the kernel \(K\) is derived. From the scattering data this equation can be constructed. Solving it one obtains \(K\) and then the potential is obtained as \(q(x) = -2 \frac{\mathrm{d}K(x,x)}{\mathrm{d}x}\).