Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs
Abstract
The 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 Schnoltype theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an subexponential 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').
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 June 2005
 DOI:
 10.1088/03054470/38/22/013
 arXiv:
 arXiv:mathph/0411003
 Bibcode:
 2005JPhA...38.4887K
 Keywords:

 35Q 35P 05C;
 Mathematical Physics;
 Mathematics  Mathematical Physics;
 35Q;
 35P;
 05C
 EPrint:
 4 eps figures, LATEX file, 21 pages Revised form: a cutandpaste blooper fixed