Approximations of Weyl fractional-order integrals with insurance applications

Construction of the raising operator for Rosen-Morse eigenstates in terms of the Weyl fractional integral

The raising operator relating adjacent bound states for the general, non-symmetric Rosen-Morse potential is constructed explicitly. It is demonstrated that, in constrast to the symmetric (modified Pöschl-Teller) potential, the operator is non-local and must be expressed applying techniques from fractional calculus. A recurrence relation between adjacent states is derived applying the Weyl fractional integral, which, in contrast to standard recurrence relations, allows the efficient numerical computation of the coefficients of all Jacobi polynomials necessary for the evaluation of the bound sta…

A short proof of Weyl's law for fractional differential operators

We study spectral asymptotics for a large class of differential operators on an open subset of $\R^d$ with finite volume. This class includes the Dirichlet Laplacian, the fractional Laplacian, and also fractional differential operators with non-homogeneous symbols. Based on a sharp estimate for the sum of the eigenvalues we establish the first term of the semiclassical asymptotics. This generalizes Weyl's law for the Laplace operator.

Existence and symmetric result for Liouville-Weyl fractional nonlinear Schrödinger equation

We study the existence of positive solution for the one dimensional Schrödinger equation with mixed Lioville-Weyl fractional derivatives \begin{eqnarray*}\label{Eq00} _{t}D_{\infty}^α({_{-\infty}}D_{t}^αu(t)) + V(t) u(t) = & f(u(t)),\;\;t\in \mathbb{R}\\ u\in H^α(\mathbb{R}).\nonumber \end{eqnarray*} Furthermore, we analyse radial symmetry property of these solutions. The proof is carried out by using variational methods jointly with comparison and rearrangement argument.

Weyl Quantization of Fractional Derivatives

The quantum analogs of the derivatives with respect to coordinates q_k and momenta p_k are commutators with operators P_k and $Q_k. We consider quantum analogs of fractional Riemann-Liouville and Liouville derivatives. To obtain the quantum analogs of fractional Riemann-Liouville derivatives, which are defined on a finite interval of the real axis, we use a representation of these derivatives for analytic functions. To define a quantum analog of the fractional Liouville derivative, which is defined on the real axis, we can use the representation of the Weyl quantization by the Fourier transfor…

Multiplicity of solutions for fractional Hamiltonian systems with Liouville-Weyl fractional derivative

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: \begin{eqnarray}\label{eq00} _{t}D_{\infty}^α(_{-\infty}D_{t}^αu(t)) + L(t)u(t) = & \nabla W(t,u(t))\\ u\in H^α(\mathbb{R}, \mathbb{R}^{N}).\nonumber \end{eqnarray} where $α\in (1/2, 1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$, $L\in C(\mathbb{R}, \mathbb{R}^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R}\times \mathbb{R}^{n}, \mathbb{R})$, and $\nabla W$ is the gradient of $W$ at $u$. The novelty of this paper is tha…

谱分析中的加权魏尔分数算子：衰老、无限记忆与遗忘效应

Spectral Analysis of Weighted Weyl Fractional Operators: Aging, Infinite Memory, and the Amnesia Effect

本文提出了一种加权魏尔分数算子的谱分析方法，用于建模具有衰老和无限记忆的非局部系统，并揭示了快速衰老导致记忆丧失的机制。