A partial inverse problem for quantum graphs with a loop
The Sturm-Liouville problem on the lasso-type graph with the boundary edge of length \(l_1 = 1\) and the loop of length \(l_2 = 2\) is considered: \[ -y_j'' + q_j(x_j) y_j = \lambda y_j, \quad x_j \in (0, l_j), \quad j = 1, 2, \] with the standard matching conditions in the internal vertex \[ y_1(1) = y_2(0) = y_2(2), \quad y_1'(1) + y_2'(2) = y_2'(0), \] the Dirichlet boundary condition \(y_1(0) = 0\), and the discontinuity conditions \[ y_1(d + 0) = \beta y_1(d - 0), \quad y_1'(d + 0) = \beta^{-1} y_1'(d - 0) + \alpha y_1(d - 0), \quad d \in (0, 1), \] where \(q_j \in L_2(0, l_j)\) are real-valued functions, \(\lambda\) is the spectral parameter, \(d \in \mathbb Q\), \(\alpha \in \mathbb R\), \(\beta > 0\). The authors obtain eigenvalue asymptotics for this boundary value problem and study the partial inverse problem that consists in the recovery of the potential \(q_1\) on the boundary edge from a part of the spectrum, while the potential \(q_2\) on the loop is known a priori. Under some additional assumptions, it is proved that the solution of this partial inverse problem is unique and a reconstruction algorithm is developed. The authors use the method of [the reviewer, Anal. Math. Phys. 8, 155--168 (2018; Zbl 1391.34037)], which is based on the Riesz-basis property of a special vector-functional system. This method allows to reduce a partial inverse problem on a graph to the classical inverse problem on an inverval.
scientific article; zbMATH DE number 7379127
scientific article