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In this paper, we study the existence of infinitely many solutions for a fractional Kirchhoff–Schrödinger–Poisson system. Based on variational methods, especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by Kajikiya for the critical case, we obtain infinitely many solutions for the system under certain assumptions. The novelties of this article lie in the appearance of the possibly degenerate Kirchhoff function and weak assumptions on the nonlinear term which are quite mild. [ABSTRACT FROM AUTHOR]

In this paper, the existence of a nontrivial least energy solution is considered for the nonlinear fractional Schrödinger-Poisson systems (−Δ)su + V(x)u + ϕu = |u|p−1u and (−Δ)tϕ = u2 in R 3 , where (−Δ)α is the fractional Laplacian for α = s, t ∈ (0, 1) with s < t and 2s + 2t > 3. Under some appropriate assumptions on the non-constant potential V(x), we prove the existence of a nontrivial least energy solution when 2 < p < 2 s * − 1 = (3 + 2 s) / (3 − 2 s) by using some new analytical skills and the Nehari-Pohožaev type manifold. [ABSTRACT FROM AUTHOR]

In this paper, we study the following critical fractional Schrödinger equation: ( − Δ ) s u + V ( x ) u = | u | 2 s * − 2 u + λ f ( x , u ) , x ∈ R N , where λ > 0, 0 < s < 1, N > 2s, 2 s * = 2 N N − 2 s , (−Δ)s denotes the fractional Laplacian of order s, and f is a continuous superlinear but subcritical function. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by the Nehari method. [ABSTRACT FROM AUTHOR]

In the present paper, it is proved that there are many quasi-periodic solutions of a class of space fractional nonlinear Schrödinger equations with the Riesz fractional derivative by means of KAM (Kolmogorov-Arnold-Moser) theorem. [ABSTRACT FROM AUTHOR]

This paper considers the fractional Schrödinger-Poisson system in R³. We prove that the problem has m-bump solutions under some given conditions which are given in the Introduction. Moreover, the system has more and more multi-bump solutions as ϵ → 0. [ABSTRACT FROM AUTHOR]

In this paper the generalized fractional Schrödinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function. The Green function for free particle is also presented in this paper. Finally, we discuss the relationship between the cases of the generalized fractional Schrödinger equation and the ones in standard quantum. [ABSTRACT FROM AUTHOR]

This paper is a direct offspring of the work of Garbaczewski and Stephanovich ["L'evy flights and nonlocal quantum dynamics," J. Math. Phys. 54, 072103 (2013)] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems. Presently, we give a decisive computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computer-assisted outcomes (potentially defective, in view of various cutoffs), i.e., detailed eigenvalues and shapes of eigenfunctions, has been analyzed. [ABSTRACT FROM AUTHOR]

In this paper, the fractional Schrödinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, 'On the nonlocality of the fractional Schrödinger equation,' J. Math. Phys. 51, 062102 (2010)] and [S. S. Bayin, 'On the consistency of the solutions of the space fractional Schrödinger equation,' J. Math. Phys. 53, 042105 (2012)] published in this journal, controversial opinions regarding solutions to the fractional Schrödinger equation for a particle moving in the infinite potential well that were derived by Laskin ['Fractals and quantum mechanics,' Chaos 10, 780-790 (2000)] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power kernels. Even if the equations look not very complicated, no solution to these equations in explicit form is known. Still, the obtained equations are used to show that the eigenvalues and eigenfunctions of the fractional Schrödinger equation for a particle moving in the infinite potential well given by Laskin ['Fractals and quantum mechanics,' Chaos 10, 780-790 (2000)] and many other papers by different authors cannot be valid as has been first stated by Jeng et al. ['On the nonlocality of the fractional Schrödinger equation,' J. Math. Phys. 51, 062102 (2010)]. [ABSTRACT FROM AUTHOR]

This paper considers the fractional Schrödinger equation with unbounded potential. We prove the existence of ground state by Lagrange multiplier method. The standing wave with prescribed frequency is also obtained by Nehari's manifold approach. In the last part, we obtain that the standing wave is bound state. [ABSTRACT FROM AUTHOR]