A partial inverse problem for quantum graphs with a loop
type: Item
instance of: scholarly article
title: A partial inverse problem for quantum graphs with a loop
zbMATH Open document ID: 1484.34068
author: Sheng-Yu Guan
author: Chuan-Fu Yang
author: Dong-Jie Wu
publication date: 2021-08-05T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34B45
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34L20
zbMATH DE Number: 7379127
zbMATH Keywords: quantum graph
zbMATH Keywords: Sturm-Liouville operator
zbMATH Keywords: partial inverse problem
zbMATH Keywords: jump condition
zbMATH Keywords: Riesz basis
reviewed by: Natalia Bondarenko
MaRDI profile type: Publication
full work available at URL: jiip-2020-0085
OpenAlex ID: W3092263758
cites work: A 2-edge partial inverse problem for the Sturm-Liouville operators with singular potentials on a star-shaped graph
cites work: A partial inverse problem for the Sturm-Liouville operator on a star-shaped graph
cites work: Partial inverse problems for the Sturm-Liouville operator on a star-shaped graph with mixed boundary conditions
cites work: Partial inverse problems for quadratic differential pencils on a graph with a loop
cites work: Q3545421
cites work: Q4530224
cites work: Discontinuous inverse eigenvalue problems
cites work: Riesz bases of solutions of Sturm-Liouville equations
cites work: Graph models for waves in thin structures
cites work: Quantum graphs: I. Some basic structures
cites work: Q3695595
cites work: Q4183612
cites work: Inverse problem for the Sturm--Liouville equation on a simple graph
cites work: Q3760860
cites work: Inverse nodal and inverse spectral problems for discontinuous boundary value problems
cites work: An interior inverse problem for discontinuous boundary-value problems
cites work: Inverse spectral problems for the Sturm-Liouville operator on a \(d\)-star graph
cites work: Inverse problems for the differential operator on a graph with cycles
cites work: A partial inverse problem for the Sturm-Liouville operator on the lasso-graph
cites work: An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions
cites work: Uniqueness theorems from partial information of the potential on a graph
cites work: Integral transforms connected with discontinuous boundary value problems
cites work: Boundary value problems with discontinuity conditions in an interior point of the interval
cites work: Inverse problems for Sturm-Liouville operators on graphs with a cycle
cites work: Q5391565
cites work: Inverse spectral problems for differential operators on spatial networks
cites work: Determination of the impulsive Sturm-Liouville operator from a set of eigenvalues
DOI: 10.1515/JIIP-2020-0085
Recommended article: A partial inverse problem for the Sturm-Liouville operator on the lasso-graph
Recommended article: Inverse Sturm-Liouville problem on a figure-eight graph
Recommended article: An inverse problem for Sturm-Liouville operators on arbitrary compact spatial networks
Recommended article: A partial inverse Sturm‐Liouville problem on an arbitrary graph
Recommended article: Inverse spectral problems for the Sturm-Liouville operator on a \(d\)-star graph
Recommended article: New trace formulae for Sturm-Liouville operators on the Lasso-graph
Recommended article: Q5040315
Recommended article: Partial inverse problems for the Sturm-Liouville operator on a star-shaped graph with different edge lengths
Recommended article: Inverse eigenvalue problem for a simple star graph
Recommended article: An inverse problem for an integro‐differential operator on a star‐shaped graph
published in: Journal of Inverse and Ill-posed Problems
description: scientific article; zbMATH DE number 7379127