On the absence of absolutely continuous spectra for Schroedinger operators on radial tree graphs

Journal Article · Wed Dec 15 00:00:00 EST 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3526963· OSTI ID:21501213

Exner, Pavel ^[1]; Lipovsky, Jiri ^[2]

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1. Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Brehova 7, 11519 Prague (Czech Republic)
2. Nuclear Physics Institute ASCR, 25068 Rez (Czech Republic)

The subject of the paper is Schroedinger operators on tree graphs which are radial, having the branching number b{sub n} at all the vertices at the distance t{sub n} from the root. We consider a family of coupling conditions at the vertices characterized by (b{sub n}-1){sup 2}+4 real parameters. We prove that if the graph is sparse so that there is a subsequence of {l_brace}t{sub n+1}-t{sub n{r_brace}} growing to infinity, in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings, but on the the other hand, there are cases when the spectrum of such a Schroedinger operator can be purely absolutely continuous.

• OSTI ID: 21501213
• Journal Information: Journal of Mathematical Physics, Vol. 51, Issue 12; Other Information: DOI: 10.1063/1.3526963; (c) 2010 American Institute of Physics; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICAL METHODS AND COMPUTING
BRANCHING RATIO
DISTANCE
GRAPH THEORY
POTENTIALS
SCHROEDINGER EQUATION
DIFFERENTIAL EQUATIONS
DIMENSIONLESS NUMBERS
EQUATIONS
MATHEMATICS
PARTIAL DIFFERENTIAL EQUATIONS
WAVE EQUATIONS