ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract We study the spectrum of the HamiltonianH onl 2(ℤ) given by (Hψ)(n)=ψ(n+1)+ψ(n−1)+V(n)ψ(n) with the hierarchical (ultrametric) potentialV(2 m (2l+1))=λ(1−R m )/(1−R), corresponding to 1-, 2-, and 3-dimensional Coulomb potentials for 0〈R〈1,R=1 andR〉1, respectively, in a suitably chosen valuation metric. We prove that the spectrum is a Cantor set and gaps open at the eigenvaluese n (1)〈e n (2)〈...〈e n (2 n −1) of the Dirichlet problemHψ=Eψ, ψ(0)=ψ(2 n )=0,n≧1. In the gap opening ate n (k) the integrated density of states takes on the valuek/2 n . The spectrum is purely singular continuous forR≧1 when the potential is unbounded, and the Lyapunov exponent γ vanishes in the spectrum. The spectrum is purely continuous forR〈1 in σ(H)∩[−2, 2] and γ=0 here, but one cannot exclude the presence of eigenvalues near the border of the spectrum. We also propose an explicit formula for the Green's function.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01256499
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