dc.contributor.author Carlson, Robert ( ) dc.date.accessioned 2018-08-30T14:43:04Z dc.date.available 2018-08-30T14:43:04Z dc.date.issued 1997-12-18 dc.identifier.citation Carlson, R. (1997). Hill's equation for a homogeneous tree. Electronic Journal of Differential Equations, 1997(23), pp. 1-30. en_US dc.identifier.issn 1072-6691 dc.identifier.uri https://digital.library.txstate.edu/handle/10877/7659 dc.description.abstract The 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.format Text dc.format.extent 30 pages dc.format.medium 1 file (.pdf) dc.language.iso en en_US dc.publisher Southwest Texas State University, Department of Mathematics en_US dc.source Electronic Journal of Differential Equations, 1997, San Marcos, Texas: Southwest Texas State University and University of North Texas. dc.subject Spectral Graph Theory en_US dc.subject Hill's equation en_US dc.subject Periodic potential en_US dc.title Hill's Equation for a Homogeneous Tree en_US dc.type publishedVersion txstate.documenttype Article dc.rights.license This work is licensed under a Creative Commons Attribution 4.0 International License.
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