Hamiltonian dynamics on finite metric graphs with prescribed energy-preserving vertex isomorphisms form a global Liouville-preserving one-parameter group on the quotient phase space after excluding finitely many energy levels.
Quantum scattering theory on graphs with tails
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abstract
We consider quantum walks on a finite graphs to which infinite tails are attached. We explore how the propagating and bound states depend on the structure of the finite graph. The S-matrix for such graphs is defined. Its unitarity is proved as well as some other of its properties such as its transformation under time reversal. A spectral decomposition of the identity for the Hamiltonian of the graph is derived using its eigenvectors. We derive formulas for the S-matrix of a graph under certain operation such as cutting a tail, attaching a tail or connecting two tails to form an edge.
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2026 1verdicts
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Liouville-Preserving Hamiltonian Scattering on Finite Metric Graphs
Hamiltonian dynamics on finite metric graphs with prescribed energy-preserving vertex isomorphisms form a global Liouville-preserving one-parameter group on the quotient phase space after excluding finitely many energy levels.