An inverse problem for quantum trees with observations at interior vertices

Sergei Avdonin; Julian Edward; Sergei Avdonin; Julian Edward

doi:10.3934/nhm.2021008

Networks and Heterogeneous Media

2021, Volume 16, Issue 2: 317-339. doi: 10.3934/nhm.2021008

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An inverse problem for quantum trees with observations at interior vertices

Sergei Avdonin ^{1
,},
Julian Edward ^{2
,
,}

1.
Department of Mathematics and Statistics, University of Alaska at Fairbanks, Fairbanks, AK 99775, USA
2.
Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA

Received: 01 April 2020 Revised: 01 January 2021 Published: 12 March 2021
Primary: 35R30; Secondary: 35R02, 35L05, 35L20

In this paper we consider a non-standard dynamical inverse problem for the wave equation on a metric tree graph. We assume that positive masses may be attached to the internal vertices of the graph. Another specific feature of our investigation is that we use only one boundary actuator and one boundary sensor, all other observations being internal. Using the Dirichlet-to-Neumann map (acting from one boundary vertex to one boundary and all internal vertices) we recover the topology and geometry of the graph, the coefficients of the equations and the masses at the vertices.
- Inverse problem,
- quantum graph,
- wave equation,
- boundary and interior measurements,
- attached masses
Citation: Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices[J]. Networks and Heterogeneous Media, 2021, 16(2): 317-339. doi: 10.3934/nhm.2021008

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Abstract

In this paper we consider a non-standard dynamical inverse problem for the wave equation on a metric tree graph. We assume that positive masses may be attached to the internal vertices of the graph. Another specific feature of our investigation is that we use only one boundary actuator and one boundary sensor, all other observations being internal. Using the Dirichlet-to-Neumann map (acting from one boundary vertex to one boundary and all internal vertices) we recover the topology and geometry of the graph, the coefficients of the equations and the masses at the vertices.

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