Peter Kuchment 2005 J. Phys. A: Math. Gen. 38 4887 doi:10.1088/0305-4470/38/22/013
Peter Kuchment
Show affiliationsThe paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics and other areas. A Schnol-type theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an sub-exponential growth integral estimate is available. A theorem on spectral gap opening for 'decorated' quantum graphs is established (its analogue is known for the combinatorial case). It is also shown that if a periodic combinatorial or quantum graph has a point spectrum, it is generated by compactly supported eigenfunctions ('scars').
02.10.Ox Combinatorics; graph theory
81Q50 Quantum chaos (See also 37Dxx)
05Cxx Graph theory (For applications of graphs, see 68R10, 90C35, 94C15)
Issue 22 (3 June 2005)
Received 31 October 2004, in final form 30 December 2004
Published 18 May 2005
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