Abstract. We apply weighted Mourre commutator theory to prove the limiting absorption principle for the discrete Schrödinger operator perturbed by the sum of a Wigner-von Neumann and long-range type potential. In particular, this implies a new result concerning the absolutely continuous spectrum for these operators even for the one-dimensional operator. We show that methods of classical Mourre theory based on differential inequalities and on the generator of dilation cannot apply to the mentionned Schrödinger operators.

We compare the isotropic equivalent 15 − 2000 keV γ-ray energy, E γ , emitted by a sample of 91 swift Gamma-Ray Bursts (GRBs) with known redshifts, with the isotropic equivalent fireball energy, E fb , as estimated within the fireball model framework from X-ray afterglow observations of these bursts. The uncertainty in E γ , which spans the range of ∼ 10 51 erg to ∼ 10 53.5 erg, is ≈ 25% on average, due mainly to the extrapolation from the BAT detector band to the 15 − 2000 keV band. The uncertainty in E fb is approximately a factor of 2, due mainly to the X-ray measurements' scatter. We find E γ and E fb to be tightly correlated. The average(std) of η 11hr γ ≡ log 10 (E γ /(3ε e E 11hr fb )) are −0.34(0.60), and the upper limit on the intrinsic spread of η γ is approximately 0.5 (ε e is the fraction of shocked plasma energy carried by electrons and E xhr fb is inferred from the X-ray flux at x hours). If the uncertainties in the determinations of E γ and E fb are twice larger than we estimated, then the data imply no intrinsic variance in η γ . We also find that E fb inferred from X-ray observations at 3 and 11 hours are similar, with an average(std) of log 10 (E 3hr fb /E 11hr fb ) of 0.04(0.28). The small variance of η γ implies that burst-to-burst variations in ε e and in the efficiency of fireball energy conversion to γ-rays are small, and suggests that both are of order unity. The small variance of η γ and the similarity of E 3hr fb and E 11hr fb further imply that ε e does not vary significantly with shock Lorentz factor, and that for most bursts the modification of fireball energy during the afterglow phase, by processes such as radiative losses or extended duration energy injection, are not significant. Finally, our results imply that if fireballs are indeed jets, then the jet opening angle satisfies θ ≥ 0.1 for most cases. Extending our analysis to late times we find a significant reduction in E fb , < E 3hr fb /E 2d fb >= 1.4, consistent with jet breaks on a 1 d time scale in a significant fraction of the bursts. These results are consistent with the main results of Freedman & Waxman (2001), which were based on a much smaller sample of GRBs.

We consider discrete Schrödinger operators on Z d for which the perturbation consists of the sum of a long range type potential and a Wigner-von Neumann type potential. Still working in a framework of weighted Mourre theory, we improve the limiting absorption principle (LAP) that was obtained in [Ma1]. To our knowledge, this is a new result even in the one-dimensional case. The improvement consists in a weakening of the assumptions on the long range potential and better LAP weights. The improvement relies only on the fact that the generator of dilations (which serves as conjugate operator) is bounded from above by the position operator. To exploit this, Loewner's theorem on operator monotone functions is invoked.

Following the method of Froese and Herbst, we show for a class of potentials V that an eigenfunction ψ with eigenvalue E of the multi-dimensional discrete Schrödinger operator H = ∆ + V on Z d decays sub-exponentially whenever the Mourre estimate holds at E. In the one-dimensional case we further show that this eigenfunction decays exponentially with a rate at least of cosh −1 ((E − 2)/(θE − 2)), where θE is the nearest threshold of H located between E and 2. A consequence of the latter result is the absence of eigenvalues between 2 and the nearest thresholds above and below this value. The method of Combes-Thomas is also reviewed for the discrete Schrödinger operators.

Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov–Vainberg Schrödinger operators, Δ + V and D + V on ℓ2(Zd), with emphasis on d = 1, 2, 3. Considered are electric potentials V satisfying a long range condition of the following type: V−τjκV decays appropriately at infinity for some κ∈N and all 1 ≤ j ≤ d, where τjκV is the potential shifted by κ units on the jth coordinate. More comprehensive results are obtained for small values of κ, e.g., κ = 1, 2, 3, 4. We work in a simplified framework in which the main takeaway appears to be the existence of bands where a limiting absorption principle holds, and hence, pure absolutely continuous spectrum exists. Other decay conditions at infinity for V arise from an isomorphism between Δ and D in dimension 2. Oscillating potentials are examples in application.