具有分数幂的薛定谔型算子的最小速度界

Propagation property and its application to inverse scattering for fractional powers of the negative Laplacian

Enss (1983) proved a propagation estimate for the usual free Schroedinger operator that turned out later to be very useful for inverse scattering in the work of Enss--Weder (1995). Since then, this method has been called the Enss--Weder time-dependent method. We study the same type of propagation estimate for the fractional powers of the negative Laplacian and, as with the Enss--Weder method, we apply our estimate to inverse scattering. We find that the high-velocity limit of the scattering operator uniquely determines the short-range interactions.

On the continuum limit for discrete NLS with long-range lattice interactions

We consider a general class of discrete nonlinear Schroedinger equations (DNLS) on the lattice $h \mathbb{Z}$ with mesh size $h>0$. In the continuum limit when $h \to 0$, we prove that the limiting dynamics are given by a nonlinear Schroedinger equation (NLS) on $\mathbb{R}$ with the fractional Laplacian $(-Δ)^α$ as dispersive symbol. In particular, we obtain that fractional powers $1/2 < α< 1$ arise from long-range lattice interactions when passing to the continuum limit, whereas NLS with the non-fractional Laplacian $-Δ$ describes the dispersion in the continuum limit for short-range lattice…

Moment Analysis for Localization in Random Schroedinger Operators

We study localization effects of disorder on the spectral and dynamical properties of Schroedinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift a…

Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori

In a previous paper we established Cwikel-type estimates on noncommutative tori and used them to get analogues in this setting of the Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators. In this paper, we focus on "curved" NC tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The Cwikel-type estimates of our previous paper are extended to pseudodifferential operators and powers of Laplace-Beltrami operators. There are several applications of these e…

Localized States and Dynamics in the Nonlinear Schroedinger / Gross-Pitaevskii Equation

This article is a review of results on the nonlinear Schroedinger / Gross-Pitaevskii equation (NLS / GP). Nonlinear bound states and aspects of their stability theory are discussed from variational and bifurcation perspectives. Nonlinear bound states, in the particular cases where the potential is a single-well, double-well potential and periodic potential, are discussed in detail. We then discuss particle-like dynamics of solitary wave solutions interacting with a potential over a large, but finite, time interval. Finally we turn to the very long time behavior of solutions. We focus on the im…

Bound states in the continuum of fractional Schrödinger equation in the Earth's gravitational field and their effects in the presence of a minimal length: applications to distinguish ultralight particles

In this paper, the influence of the fractional dimensions of the Lévy path under the Earth's gravitational field is studied, and the phase transitions of energy and wave functions are obtained: the energy changes from discrete to continuous and wave functions change from non-degenerate to degenerate when dimension of Lévy path becomes from integer to non-integer. By analyzing the phase transitions, we solve two popular problems. First, we find an exotic way to produce the bound states in the continuum (BICs), our approach only needs a simple potential, and does not depend on interactions betwe…

$\mathcal{N}=1$超共形极小模型的分解及其分数量子霍尔波函数

Decomposition of $\mathcal{N}=1$ superconformal minimal models and their fractional quantum Hall wavefunctions

本文通过共集构造分解$\mathcal{N}=1$超共形极小模型，并利用自由场方法计算分数量子霍尔态的基态波函数，推导出多项式求和项的显式表达式。

分数宇宙学中的共形与非最小耦合

Conformal and Non-Minimal Couplings in Fractional Cosmology

本文在分数阶微积分框架下研究引力与标量场的非最小耦合宇宙学，通过动力系统与渐近展开方法分析修正宇宙学方程，为无需暗能量解释晚期宇宙加速提供定性描述。

Limit shape of minimal difference partitions and fractional statistics

The class of minimal difference partitions MDP($q$) (with gap $q$) is defined by the condition that successive parts in an integer partition differ from one another by at least $q\ge 0$. In a recent series of papers by A. Comtet and collaborators, the MDP($q$) ensemble with uniform measure was interpreted as a combinatorial model for quantum systems with fractional statistics, that is, interpolating between the classic Bose-Einstein ($q=0$) and Fermi-Dirac ($q=1$) cases. This was done by formally allowing values $q \in (0,1)$ using an analytic continuation of the limit shape of the correspondi…

极小模型、模线性微分方程与分数权模形式

Minimal models, modular linear differential equations and modular forms of fractional weights

本文利用 (2,p) 型极小模型特征标与 η^{c_eff} 的乘积解释 Ibukiyama 分数权模形式，重新证明其 SL_2(Z) 表示公式，并构造相关向量值模形式及模线性微分方程。

Fractional $S$-duality, Classification of Fractional Topological Insulators and Surface Topological Order

In this paper, we propose a generalization of the $S$-duality of four-dimensional quantum electrodynamics ($\text{QED}_4$) to $\text{QED}_4$ with fractionally charged excitations, the fractional $S$-duality. Such $\text{QED}_4$ can be obtained by gauging the $\text{U(1)}$ symmetry of a topologically ordered state with fractional charges. When time-reversal symmetry is imposed, the axion angle ($θ$) can take a nontrivial but still time-reversal invariant value $π/t^2$ ($t\in\mathbb{Z}$). Here, $1/t$ specifies the minimal electric charge carried by bulk excitations. Such states with time-reversa…

Rational matrix pseudodifferential operators

The skewfield K(d) of rational pseudodifferential operators over a differential field K is the skewfield of fractions of the algebra of differential operators K[d]. In our previous paper we showed that any H from K(d) has a minimal fractional decomposition H=AB^(-1), where A,B are elements of K[d], B is non-zero, and any common right divisor of A and B is a non-zero element of K. Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-zero element of K[d]. In the present paper we study the ring M_n(K(d)) of nxn matrices over the skewfie…