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(2m(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 (2n−1) of the Dirichlet problemHψ=Eψ, ψ(0)=ψ(2n)=0,n≧1. In the gap opening ate n (k) the integrated density of states takes on the valuek/2n. 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.
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Communicated by J. Fröhlich
Work supported by the Fonds National Suisse de la Recherche Scientifique, Grant No. 2.042-0.86 (H.K. and R.L.) and 2.483-0.87 (A.S.)
On leave from the Dipartimento di Fisica, Università degli Studi di Firenze, Largo E. Fermi 2, I-50125 Firenze, Italy
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Kunz, H., Livi, R. & Sütő, A. Cantor spectrum and singular continuity for a hierarchical Hamiltonian. Commun.Math. Phys. 122, 643–679 (1989). https://doi.org/10.1007/BF01256499
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DOI: https://doi.org/10.1007/BF01256499