ON THE ABSOLUTELY CONTINUOUS SPECTRUM OF MULTI-DIMENSIONAL SCHRÖDINGER OPERATORS WITH SLOWLY DECAYING POTENTIALS
1. In this short paper we extend the method of Laptev-Naboko-Safronov [15]. New estimates for the discrete spectrum obtained in this paper allow one to prove stronger results compared to [15]. The main technical tool of the paper [15] is the so called trace inequality, which relates properties of ne...
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| Language: | English |
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| Online Access: | http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.104.3098 http://www.math.kth.se/spect/preprints03_04/safronov.pdf |
| Summary: | 1. In this short paper we extend the method of Laptev-Naboko-Safronov [15]. New estimates for the discrete spectrum obtained in this paper allow one to prove stronger results compared to [15]. The main technical tool of the paper [15] is the so called trace inequality, which relates properties of negative eigenvalues to the properties of the a.c. spectrum. Based on this relation, the technique of our paper which treats the distribution function of the discrete spectrum, allows one to improve the known bounds for the eigenvalues of the Schrödinger operator. Let us state our main assertion. Theorem 0.1. Let d ≥ 3 and let V ∈ L ∞ (R d) be a real valued potential such that V (x) → 0 as |x | → ∞. Assume that V ∈ L d+1 (R d) and for some positive number δ> 0 the Fourier transform of V satisfies the estimate |ξ|<δ | ˆ V (ξ) | 2 dξ < ∞. Then the absolutely continuous spectrum of the operator − ∆ + V is essentially supported by the positive real line R+, i.e. the spectral projection corresponding to any set in R+ of positive Lebesgue measure is different from zero. This theorem almost coincides with the result of Deift and Killip [6] for the case d = 1 ( to be precise they do not impose the conditions V ∈ L ∞ and V → 0). Note that Denissov [9] proved recently some results about the a.c spectrum of Dirac operators. These results look quite optimal for the considered operator. However the same method can not be applied to the operators having a bottom of the spectrum ( Probably this explains where the condition on the Fourier transform for small energies comes from!). Different relations between the discrete and continuous spectrum appeared several times before in the case of one dimensional operator. We would like to mention a recent result of Damanik and Killip [4] which says that if one dimensional Schrödinger operators with potentials +V and −V have only finite number of eigenvalues then the positive spectrum is absolutely continuous. In connection to our result, a natural question for in |
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