Page 65 - Textos de Matemática Vol. 34

ON THE PROBLEM OF THE NEGATIVE SPECTRUM
SUSANA D. MOURA
Dedicated to Professor J. A. Sampaio Martins on the occasion of his sixtieth birthday
Abstract. We study the behaviour of the negative spectrum of some fractal Schr¨odinger operators.
1. Introduction
In quantum mechanics it is of interest the behaviour of the so called negative
spectrum of Schr¨odinger operators of type Hβ=−∆+βV as β→∞,
where V is a potential in Rn (of course, n = 3 is the case of physical relevance). What happens if one replaces V by fractals or related finite Radon measures? In [28] Triebel has studied this type of problems (in a slightly modified form) when V = b(γ)μ(dγ), b ∈ Lr(Γ), and Γ is a d-set with associated Radon measure μ. In [24] we dealt also with this type of issue for Γ a perturbed d-set, so called (d, Ψ)-set, where Ψ is an admissible function in the sense of [16, 17]. In this note we improve our earlier results. On the one hand, we consider functions Ψ in a larger class than the class of admissible functions – namely, the class of slowly varying functions – and, on the other hand, we obtain explicit estimates not only for iterated logarithms as we did in [24].
We finish this introduction by collecting some general notation used throughout this paper.
As usual, Rn denotes the n-dimensional real Euclidean space, N the collection of all natural numbers and N0 = N ∪ {0}. We use the equivalence “∼” in
ak ∼bk or φ(x)∼ψ(x)
2000 Mathematics Subject Classification. 28A80, 35P20, 46E35, 47B06.
Key words and phrases. Besov spaces with generalised smoothness, Birman-Schwinger princi- ple, entropy numbers, fractals, negative spectrum, slowly varying functions.
This work was supported by CMUC (Centre of Mathematics of the University of Coimbra).
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