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A \(C^\ast\)-algebra associated with dynamics on a graph of strings. (English) Zbl 1334.46058

A self-adjoint \(C^*\)-operator algebra \(\mathfrak{C}\) used in other papers of the first author is introduced. In the present paper, the dynamical system \[ \begin{cases} u_tt -\Delta u=0 &\text{in }\Omega \times (0,T),\\ u|_{t=0} =u_t |_{t=0} =0 &\text{in }\Omega,\\ u=f &\text{on }\Gamma \times [0,T], \end{cases} \] where \(\Omega\) is a finite compact metric graph, \(\Gamma\) is the set of its boundary vertices, \(\Delta\) is the Laplace operator in \(\Omega\) is considered. This shows the possible applications of the algebra \(\mathfrak{C}\) to inverse problems.

MSC:

46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
34B45 Boundary value problems on graphs and networks for ordinary differential equations
35R30 Inverse problems for PDEs
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
47N20 Applications of operator theory to differential and integral equations
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References:

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