Infinite bound states and $1/n$ energy spectrum induced by a Coulomb-like potential of type III in a flat band system

In this work, we investigate the bound states in a one-dimensional spin-1 flat band system with a Coulomb-like potential of type III, which has a unique non-vanishing matrix element in basis $|1\rangle$. It is found that, for such a kind of potential, there exists infinite bound states. Near the threshold of continuous spectrum, the bound state energy is consistent with the ordinary hydrogen-like atom energy level with Rydberg correction. In addition, the flat band has significant effects on the bound states. For example, there are infinite bound states which are generated from the flat band. Furthermore, when the potential is weak, the bound state energy is proportional to the potential strength $\alpha$. When the bound state energies are very near the flat band, they are inversely proportional to the natural number $n$ (e.g., $E_n\propto 1/n, n=1,2,3,...$). Further we find that the energy spectrum can be well described by quasi-classical approximation (WKB method). Finally, we give a critical potential strength $\alpha_c$ at which the bound state energy reaches the threshold of continuous spectrum. After crossing the threshold, the bound states in the continuum (BIC) would exist in such a flat band system.

Schrödinger operator with decreasing potential in a cylinder

The Schrödinger operator − Δ + V ( x , y ) -\Delta + V(x,y) is considered in a cylinder R m × U \mathbb {R}^m \times U , where U U is a bounded domain in R d \mathbb {R}^d . The spectrum of such an operator is studied under the assumption that the potential decreases in longitudinal variables, | V ( x , y ) | ≤ C ⟨ x ⟩ − ρ |V(x,y)| \le C \langle x\rangle ^{-\rho } . If ρ > 1 \rho > 1 , then the wave operators exist and are complete; the Birman invariance principle and the limiting absorption principle hold true; the absolute continuous spectrum fills the semiaxis; the singular continuous spectrum is empty; the eigenvalues can accumulate to the thresholds only.

It Is Time to Study Overlapping Molecular and Circuit Pathophysiologies in Alzheimer’s and Lewy Body Disease Spectra

Alzheimer’s disease (AD) is the leading cause of dementia due to neurodegeneration and is characterized by extracellular senile plaques composed of amyloid β1–42 (Aβ) as well as intracellular neurofibrillary tangles consisting of phosphorylated tau (p-tau). Dementia with Lewy bodies constitutes a continuous spectrum with Parkinson’s disease, collectively termed Lewy body disease (LBD). LBD is characterized by intracellular Lewy bodies containing α-synuclein (α-syn). The core clinical features of AD and LBD spectra are distinct, but the two spectra share common cognitive and behavioral symptoms. The accumulation of pathological proteins, which acquire pathogenicity through conformational changes, has long been investigated on a protein-by-protein basis. However, recent evidence suggests that interactions among these molecules may be critical to pathogenesis. For example, Aβ/tau promotes α-syn pathology, and α-syn modulates p-tau pathology. Furthermore, clinical evidence suggests that these interactions may explain the overlapping pathology between AD and LBD in molecular imaging and post-mortem studies. Additionally, a recent hypothesis points to a common mechanism of prion-like progression of these pathological proteins, via neural circuits, in both AD and LBD. This suggests a need for understanding connectomics and their alterations in AD and LBD from both pathological and functional perspectives. In AD, reduced connectivity in the default mode network is considered a hallmark of the disease. In LBD, previous studies have emphasized abnormalities in the basal ganglia and sensorimotor networks; however, these account for movement disorders only. Knowledge about network abnormalities common to AD and LBD is scarce because few previous neuroimaging studies investigated AD and LBD as a comprehensive cohort. In this paper, we review research on the distribution and interactions of pathological proteins in the brain in AD and LBD, after briefly summarizing their clinical and neuropsychological manifestations. We also describe the brain functional and connectivity changes following abnormal protein accumulation in AD and LBD. Finally, we argue for the necessity of neuroimaging studies that examine AD and LBD cases as a continuous spectrum especially from the proteinopathy and neurocircuitopathy viewpoints. The findings from such a unified AD and Parkinson’s disease (PD) cohort study should provide a new comprehensive perspective and key data for guiding disease modification therapies targeting the pathological proteins in AD and LBD.

Monotonicity in the averaging process

We investigate an averaging process that describes how interacting agents approach consensus through binary interactions. In each elementary step, two agents are selected at random and they reach compromise by adopting their opinion average. We show that the fraction of agents with a monotonically decreasing opinion decays as $e^{-\alpha t}$, and that the exponent $\alpha=\tfrac{1}{2}-\tfrac{1+\ln \ln 2}{4\ln 2}$ is selected as the extremum from a continuous spectrum of possible values. The opinion distribution of monotonic agents is asymmetric, and it becomes self-similar at large times. Furthermore, the tails of the opinion distribution are algebraic, and they are characterized by two distinct and nontrivial exponents. We also explore statistical properties of agents with an opinion strictly above average.

Properties of 4ΛH Hypernucleus in Three-Cluster Microscopic Models

Within the framework of microscopic three-cluster algebraic models with possible consideration of clustering types (D + n) + Λ, (D + Λ) + n, and (n + Λ) + D, the properties of discrete spectrum states of hypernucleus 4ΛH and continuous spectrum states in the 3H + Λ channel are studied. It is shown that the cluster structure is almost completely determined by the clustering (D + n) + Λ with a rather appreciable effect from the polarization of the binary subsystem (D + n) due to its interaction with the Λ particle.