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In this paper, we study the existence and instability of normalized standing waves for the fractional Schrödinger equation i∂tψ = (−Δ)sψ − f(ψ), where 0 < s < 1, f(ψ) = |ψ|pψ with 4 s N < p < 4 s N − 2 s or f(ψ) = (|x|−γ*|ψ|2)ψ with 2s < γ < min{N, 4s}. To do this, we consider normalized solutions of the associated stationary equation (−Δ)su + ωu − f(u) = 0. By constructing a suitable submanifold of a L2-sphere and considering an equivalent minimizing problem, we prove the existence of normalized solutions. In particular, based on this equivalent minimizing problem, we can easily obtain the sharp threshold of global existence and blow-up for the time-dependent equation. Moreover, we can show that all normalized ground state standing waves are strongly unstable by blow-up. Our results are a complementary to the results of Peng and Shi [J. Math. Phys. 59, 011508 (2018)] and Zhang and Zhu [J. Dyn. Differ. Equations 29, 1017–1030 (2017)], where the existence and stability of normalized standing waves have been studied in the L2-subcritical case. [ABSTRACT FROM AUTHOR]

Stochastic Schrödinger equations are a special type of stochastic evolution equations in complex Hilbert spaces, which arise in the study of open quantum systems. Quantum Bernoulli noises refer to annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. In this paper, we investigate a linear stochastic Schrödinger equation of exclusion type in terms of quantum Bernoulli noises. Among others, we prove the well-posedness of the equation, illustrate the results with examples, and discuss the consequences. Our main work extends that of Chen and Wang [J. Math. Phys. 58(5), 053510 (2017)]. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and asymptotic properties of solutions to the fractional Schrödinger equation (− Δ) σ u = λ u + | u | q − 2 u + μ I α * | u | p | u | p − 2 u under the normalized constraint ∫ R N u 2 = a 2 , where N ≥ 2, σ ∈ (0, 1), α ∈ (0, N), q ∈ (2 + 4 σ N , 2 N N − 2 σ ] , p ∈ [ 1 + 2 σ + α N , N + α N − 2 σ ) , a, μ > 0, Iα(x) = |x|α−N, and λ ∈ R appears as a Lagrange multiplier. By using a refined version of the min-max principle, we show that the above problem admits a mountain pass type solution ûμ for some λ ̂ < 0 under suitable assumptions on the related parameters. In particular, we can prove that ûμ is a ground state if p ≤ q 2 + α N . Furthermore, we give some asymptotic properties of the solutions. We mainly extend the results in the work of Bhattarai [J. Differ. Equations 263, 3197–3229 (2017)] and Feng et al. [J. Math. Phys. 60, 1–12(2019)] concerning the above problem from the L2-subcritical setting to L2-critical and L2-supercritical settings with respect to p, involving the Sobolev critical case q = 2 N N − 2 σ especially. [ABSTRACT FROM AUTHOR]

We derive two equivalent analytical expressions for an lth partial-wave scattering length al for central potentials with long-range tails of the form V (r) = − ℏ 2 2 m B r n − 4 (r n − 2 + R n − 2) 2 − ℏ 2 2 m C r 2 (r n − 2 + R n − 2) , (r ⩾ rs, R > 0). For C = 0, this family of potentials reduces to the Lenz potentials discussed in a similar context in our earlier works [R. Szmytkowski, Acta Phys. Pol. A 79, 613 (1991); J. Phys. A: Math. Gen. 28, 7333 (1995)]. The formulas for al that we provide in this paper depend on the parameters B, C, and R characterizing the tail of the potential, on the core radius rs, as well as on the short-range scattering length als, the latter being due to the core part of the potential. The procedure, which may be viewed as an analytical extrapolation from als to al, is relied on the fact that the general solution to the zero-energy radial Schrödinger equation with the potential given above may be expressed analytically in terms of the generalized associated Legendre functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the following quasilinear Schrödinger equations − Δ u + V (x) u + κ 2 Δ ( u 2 ) u = f (u) + μ | u | 2 * − 2 u , x ∈ R N , where N ≥ 3, κ > 0, μ ≥ 0, and V : R N → R satisfy suitable assumptions. First, by using a change of variable and some new skills, we obtain the ground states for this problem with subcritical growth via the Pohozaev manifold. Second, we establish the existence of ground state solutions with critical growth via L∞estimates, which use the method developed by Brezis and Nirenberg [Commun. Pure Appl. Math. 36, 437–477 (1983)] and Jeanjean [Proc. R. Soc. Edinburgh, Sect A. 129, 787–809 (1999)]. Moreover, we give the nonexistence of positive solutions for this problem, where the nonlinear term allow general asymptotically linear growth. Our results extend and supplement the results obtained by Severo et al. [J. Differ. Equations 263, 3550–3580 (2017)], Xu and Chen [J. Differ. Equations 265, 4417–4441 (2018)], and Lehrer and Maia [J. Funct. Anal. 266, 213–246 (2014)] and some other related literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the orbital stability of standing waves for the fractional Schrödinger equations with combined power-type and Choquard-type nonlinearities. By using variational methods, when one nonlinearity is focusing and L2-critical, the other is defocusing and L2-supercritical, we prove that there exist the orbitally stable standing waves. We extend the results of Bhattarai [J. Differ. Equations 263, 3197–3229 (2017)] and Feng-Zhang [Comput. Math. Appl. 75, 2499–2507 (2018)] to the L2-critical and L2-supercritical nonlinearities. [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]