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dc.contributor.authorCarlson, Robert
dc.date.accessioned2018-08-30T14:43:04Z
dc.date.available2018-08-30T14:43:04Z
dc.date.issued1997-12-18
dc.date.submitted1997-08-24
dc.identifier.citationCarlson, R. (1997). Hill's equation for a homogeneous tree. "Electronic Journal of Differential Equations," Vol. 1997, No. 23, pp. 1-30.en_US
dc.identifier.issn1072-6691
dc.identifier.urihttps://digital.library.txstate.edu/handle/10877/7659
dc.description.abstractThe analysis of Hill’s operator −D2 + q(x) for q even and periodic is extended from the real line to homogeneous trees T. Generalizing the classical problem, a detailed analysis of Hill’s equation and its related operator theory on L2(T ) is provided. The multipliers for this new version of Hill’s equation are identified and analyzed. An explicit description of the resolvent is given. The spectrum is exactly described when the degree of the tree is greater than two, in which case there are both spectral bands and eigenvalues. Spectral projections are computed by means of an eigenfunction expansion. Long time asymptotic expansions for the associated semigroup kernel are also described. A summation formula expresses the resolvent for a regular graph as a function of the resolvent of its covering homogeneous tree and the covering map. In the case of a finite regular graph, a trace formula relates the spectrum of the Hill’s operator to the lengths of closed paths in the graph.en_US
dc.formatText
dc.format.extent30 pages
dc.format.medium1 file (.pdf)
dc.language.isoen_USen_US
dc.publisherTexas State University, Department of Mathematicsen_US
dc.sourceElectronic Journal of Differential Equations, 1997, San Marcos, Texas: Southwest Texas State University and University of North Texas.
dc.subjectSpectral Graph Theoryen_US
dc.subjectHill's equationen_US
dc.subjectPeriodic potentialen_US
dc.titleHill's Equation for a Homogeneous Treeen_US
txstate.documenttypeArticle
dc.rights.licenseCreative Commons Attribution 4.0 International License [https://creativecommons.org/licenses/by/4.0/]


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