We consider the long time asymptotics of (not necessarily small) odd solutions to the nonlinear Schrödinger equation with semi-linear and nonlocal Hartree nonlinearities, in one dimension of space. We assume data in the energy space H 1 (R) only, and we prove decay to zero in compact regions of space as time tends to infinity. We give three different results where decay holds: semilinear NLS, NLS with a suitable potential, and defocusing Hartree. The proof is based on the use of suitable virial identities, in the spirit of nonlinear Klein-Gordon models [29], and covers scattering sub, critical and supercritical (long range) nonlinearities. No spectral assumptions on the NLS with potential are needed.

The two dimensional crossover from independent particle towards collective motion is studied using 2 spinless fermions interacting via a U/r Coulomb repulsion in a L × L square lattice with periodic boundary conditions and nearest neighbor hopping t. Three regimes characterize the ground state when U/t increases. Firstly, when the fluctuation ∆r of the spacing r between the two particles is larger than the lattice spacing a, there is a scaling length L0 = √ 8π 2 (t/U ) such that the relative fluctuation ∆r/ r is a universal function of the dimensionless ratio L/L0, up to finite size corrections of order L −2 . L < L0 and L > L0 are respectively the limits of the free particle Fermi motion and of the correlated motion of a Wigner molecule. Secondly, when U/t exceeds a threshold U * (L)/t, ∆r becomes smaller than a, giving rise to a correlated lattice regime where the previous scaling breaks down and analytical expansions in powers of t/U become valid. A weak random potential reduces the scaling length and favors the correlated motion.PACS. 71 10-w Theories and models for many-electron systems -71 27+a Strongly correlated electron systems -73.20.Qt Electron solids

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The causes of dysphagia include neurological conditions and cancer. Swallowing impairment of liquids represents a risk of aspiration, pneumonia, dehydration, and nutritional deficiencies. Commercial thickeners often based on modified cornstarch address this issue. The herb Matico (Buddleja globosa Hope) is used as a wound‐healing adjuvant treatment for oral mucositis caused by cancer therapies. This study analyzes the flow behavior of Matico infusion with two thickeners, Thick & EasyTM, and Enterex® Food Thickener at three concentrations. A rheological assessment is performed (20 ± 1 °C subsequent intervals: 1–100 s−1, constant shear step at 100 s−1, and 100–1 s−1, each one with a 120 s span) at five time points (0–60 min). Rheological behavior is adjusted to the Herschel–Bulkley model. Significant differences (p < 0.05) are obtained for yield stress (σ0), consistency coefficient (K), and flow behavior index (n) between time‐points, supporting the increase in apparent viscosity (ηa) during rest. Shear thinning, Bingham plastic, and shear thickening behaviors are obtained for the three concentrations 4.5, 6.75, and 9.0 g 100 mL−1. Besides, a non‐reported rheopectic effect is found at the highest concentration, showing a consistent increase in ηa. These findings reveal the critical effect of time on thickener dispersions that might reduce safety and efficiency during the eating process.

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