Inverse spectral problems for radial Schrödinger operators and closed systems
The paper is devoted to the following boundary value problem \[ L(l,q)(y):=-y''(x)+\frac{l(l+1)}{x^{2}}y(x)+q(x)y(x)=\lambda y(x),\quad x\in(0,1],\tag{1} \] subject to the boundary conditions \[ \lim_{x\rightarrow0}\frac{y(x)}{x^{l+1}}<\infty,\quad y'(1)+\beta y(1)=0.\tag{2} \] Here \(\lambda\) is the spectral parameter, \(l\in\mathbb{N}\cup\{0\}\), the function \(q\), called the potential, is real-valued and belongs to \(L^{p}(0,1)\) for \(p\in(1,+\infty)\). The parameter \(\beta\) belongs to \(\mathbb{R}\cup\{\infty\}\). If \(\beta=\infty\), then the boundary condition \(y'(1)+\beta y(1)=0\) means the Dirichlet condition \(y(1)=0\). The radial Schrödinger operators (or perturbed Bessel operators) \(L(l, q)\) arise when the separation of variables is used for the study of the Schrödinger operators \(-\Delta+q(r)\) on a ball in \(\mathbb{R}^{3}\). This paper is concerned with an inverse problem of spectral analysis for the operator induced by (1)--(2). Such problems consist in the recovery of differential operators from their spectral characteristics. The authors study an inverse eigenvalue problem for the radial Schrödinger operators on the unit interval. This problem consists in the recovery of the potential on a subinterval \((0,a), a\leq1\), from eigenvalues corresponding to the boundary value problems with different boundary conditions. They obtain a sufficient condition for the unique specification of the radial Schrödinger operator by a set of eigenvalues and a part of the potential function on \((a, 1)\) in terms of the cosine system closedness. The Borg-type and the Hochstadt-Lieberman type results are obtained as corollaries of the main result. Furthermore, under an additional hypothetical condition, they show that their condition is not only sufficient but also necessary for the uniqueness of the inverse problem solution. The main tool of the proof technique is the singular transformation operator representation for the solution of the radial Schrödinger equation.
scientific article; zbMATH DE number 7615172
scientific article