Convergence of Spectra of Mesoscopic Systems Collapsing onto a Graph

By Peter Kuchment, Hongbiao Zeng. 2001

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Nonclassical Sturm-Liouville problems and Schrodinger operators on radial trees (2000, background): “More recently there have been mathematical studies, some concerned with the interpretation of differential operators on graphs as limits of partial differential operators on thin domains [10, 12, 18, 33, 35], while others focus on novel problems of spectral or scattering theory [4, 5, 6, 14, 23].”
Graph models for waves in thin structures (2002, background): “The result (and the proof provided in [123]) of theorem 1 still holds while the exterior and interior sizes of the vertex neighbourhoods decay as d α , where α ∈ (0.”
Quantum graphs: I. Some basic structures (2004, background): “The reader can find more examples in [34, 43, 59].”