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 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∞(Rd) be a real valued potential such that V (x) → 0 as |x| → ∞. Assume that V ∈ Ld+1(Rd) and for some positive number δ > 0 the Fourier transform of V satisfies the estimate ∫