Eigenvalue Estimates for the Weighted Laplacian on Metric Trees

By K. Naimark, Michael Solomyak. 2000

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Quantum graphs: I. Some basic structures (2004, background): “references therein), the corresponding theory of quantum graphs is just developing (e.g., examples of such studies in [3, 5, 6, 7, 8, 9, 15], [16] - [20], [25], [29] - [35], [39], [43] - [49], [56] - [58], [ 64 , 65, 70, 74, 79, 80] and further references in [54]).”
Nonclassical Sturm-Liouville problems and Schrodinger operators on radial trees (2000, background): “Additional mathematical work includes applications to nerve impluse transmission [24], and the study of evolution equations on networks [2, 19].”
SCHRÖDINGER OPERATORS ON HOMOGENEOUS METRIC TREES: SPECTRUM IN GAPS (2001, background): “ F1 v,...,F b v are mutually orthogonal and their orthogonal sum fF v = F 1 v ⊕ ...⊕ Fb v contains Fv. Denote by F′ v = Ffv ⊖ Fv the orthogonal complement. The next theorem is a direct consequence of [17], Theorem 5.1 and Lemma 5.2, where a more general class of trees was considered. Later the resul”
Inverse spectral problem for quantum graphs (2006, background): “The spectral problem for quantum graphs has been investigated recently by Naimark, Sobolev and Solomyak [28, 29, 32–35].”
On the spectrum of the Laplacian on regular metric trees (2004, methodology): “This realization was used in the papers [9, 10, 13] and [14].”