One-dimensional periodic fractional Schrödinger equations with exponential critical growth

In the present paper, we study the existence of nontrivial solutions of the following one-dimensional fractional Schr\“{o}dinger equation $$ (-\Delta)^{1/2}u+V(x)u=f(x,u), \ \ x\in \R, $$ where $(-\Delta)^{1/2}$ stands for the $1/2$-Laplacian, $V(x)\in \mathcal{C}(\R, (0,+\infty))$, and $f(x,u):\R\times\R\to \R$ is a continuous function with an exponential critical growth. Comparing with the existing works in the field of exponential-critical-growth fractional Schr\”{o}dinger equations, we encounter some new challenges due to the weaker assumptions on the reaction term $f$. By using some sharp energy estimates, we present a detailed analysis of the energy level, which allows us to establish the existence of nontrivial solutions for a wider class of nonlinear terms. Furthermore, we use the non-Nehari manifold method to establish the existence of Nehari-type ground state solutions of the one-dimensional fractional Schr\”{o}dinger equations.

Existence of Ground States of Fractional Schrödinger Equations

We consider ground states of the nonlinear fractional Schrödinger equation with potentials ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( x , u ) , s ∈ ( 0 , 1 ) , (-\Delta)^{s}u+V(x)u=f(x,u),\quad s\in(0,1), on the whole space ℝ N {\mathbb{R}^{N}} , where V is a periodic non-negative nontrivial function on ℝ N {\mathbb{R}^{N}} and the nonlinear term f has some proper growth on u. Under uniform bounded assumptions about V, we can show the existence of a ground state. We extend the result of Li, Wang, and Zeng to the fractional case.

Blow-up Dynamics for L2-Critical Fractional Schrödinger Equations

In this paper, we consider the $L^2$-critical fractional Schrödinger equation $iu_t-|D|^{\beta }u+|u|^{2\beta }u=0$ with initial data $u_0\in H^{\beta /2}(\mathbb{R})$ and $\beta $ close to $2$. We show that if the initial data have negative energy and slightly supercritical mass, then the solution blows up in finite time. We also give a specific description for the blow-up dynamics. This is an extension of the works of F. Merle and P. Raphaël for $L^2$-critical Schrödinger equations. However, the nonlocal structure of this equation and the lack of some symmetries make the analysis more complicated, hence some new strategies are required.

Fractional Quantum Chemistry

The realization of fractional quantum chemistry is presented. Adopting the integro-differential operators of the calculus of arbitrary-order, we develop a general framework for the description of quantum nonlocal effects in the complex electronic environments. After a brief overview of the historical and fundamental aspects of the calculus of arbitrary-order, various classes of fractional Schrödinger equations are discussed and pertinent controversies and open problems around their applications to model systems are detailed. We provide a unified approach toward fractional generalization of the quantum chemical models such as Hartree-Fock and Kohn-Sham density functional theory and develop fractional variants of the fundamental molecular integrals and correlation energy. Furthermore, we offer various strategies for modeling static and dynamic quantum nonlocal effects through constant- and variable-order fractional operators, respectively. Possible directions for future developments of fractional quantum chemistry are also outlined.