We study the Cauchy problem for Schrodinger equations with repulsive quadratic potential and power-like nonlinearity. The local problem is well-posed in the same space as that used when a confining harmonic potential is involved. For a defocusing nonlinearity, it is globally well-posed, and a scattering theory is available, with no long range effect for any superlinear nonlinearity. When the nonlinearity is focusing, we prove that choosing the harmonic potential sufficiently strong prevents blow-up in finite time. Thanks to quadratic potentials, we provide a method to anticipate, delay, or prevent wave collapse; this mechanism is explicit for critical nonlinearity.
We consider the scattering theory for the Schrodinger equation with $-Δ-|x|^α$ as a reference Hamiltonian, for $0< α\leq 2$, in any space dimension. We prove that when this Hamiltonian is perturbed by a potential, the usual short range/long range condition is weakened: the limiting decay for the potential depends on the value of $α$, and is related to the growth of classical trajectories in the unperturbed case. The existence of wave operators and their asymptotic completeness are established thanks to Mourre estimates relying on new conjugate operators. We construct the asymptotic velocity an…
We give an example of an absolutely continuous measure $μ$ on $\mathbb R^d$, for any $d \ge 1$, such that no minimizer of the $3$-marginal harmonic repulsive cost with all marginals equal to $μ$ is supported on a graph over the first variable.
In this note, we study existence of the outgoing/incoming resolvents of repulsive Schrödinger operators which may not be essentially self-adjoint on the Schwartz space. Moreover, we recover the classical result: The repulsive Schroödinger operators with large repulsive constant is not essentially self-adjoint on the Schwartz space.
In this paper, we consider the Cauchy problem for the nonlinear Schrödinger equations with repulsive inverse-power potentials \[ i \partial_t u + Δu - c |x|^{-σ} u = \pm |u|^αu, \quad c>0. \] We study the local and global well-posedness, finite time blow-up and scattering in the energy space $H^1$ for the equation. These results extend a recent work of Miao-Zhang-Zheng [Nonlinear Schrödinger equation with coulomb potential, arXiv:1809.06685] to a general class of inverse-power potentials and higher dimensions.
In each dimension n >= 2, we construct a class of nonnegative potentials that are homogeneous of order -sigma, chosen from the range 0 < sigma < 2, and for which the perturbed Schrodinger equation does not satisfy global in time Strichartz estimates.
Conventionally, bright solitary wave solutions can be obtained in self-focusing nonlinear Schrodinger equations with attractive self-interaction. However, when self-interaction becomes repulsive, it seems impossible to have bright solitary wave solution. Here we show that there exists symbiotic bright solitary wave solution of coupled nonlinear Schrodinger equations with repulsive self-interaction but strongly attractive interspecies interaction. For such coupled nonlinear Schrodinger equations in two and three dimensional domains, we prove the existence of least energy solutions and study the…
In this paper, a nonlinear Schrödinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both standing wave and equilibrium solutions do exist for certain parameter regimes. In addition, it is proved that both types of wave solutions are orbitally stable under the flow of the equation by minimizing the charge/energy functional.
We consider the focusing mass supercritical semilinear Schrödinger equation with a repulsive Dirac delta potential on the real line (deltaNLS). Our aim in the present paper is to find a necessary and sufficient condition on the data below the standing wave of NLS without a potential to determine the global behavior of the solution. Moreover, we determine the global dynamics of the radial solution whose mass-energy is larger than that of the standing wave and smaller than that of the standing wave of deltaNLS.
We consider the nonlinear Schrödinger equations on the star graph with the Kirchhoff boundary and the repulsive Dirac delta boundary at the origin. In the present paper, we show the scattering-blowup dichotomy result below the mass-energy of the ground state on the real line. The proof of the scattering part is based on a concentration compactness and rigidity argument. Our main contribution is to give a linear profile decomposition on the star graph by using a symmetrical decomposition.
We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to suitable vector-fields. Conversely, given an initial Cauchy datum, the solution is global in time and asymptotically free, provided that confinement affects one spatial direction only. This stems from anisotropic Morawetz estimates, involving a marginal of the position density.
In this paper we establish existence and nonexistence results concerning fully nontrivial minimal energy solutions of the nonlinear Schrödinger system \begin{align*} \begin{gathered} -Δu + \, u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q \quad\text{in}\R^n, -Δv + ω^2 v = |v|^{2q-2}v + b|u|^q|v|^{q-2}v\quad\text{in}\R^n. \end{gathered} \end{align*} We consider the repulsive case $b<0$ and assume that the exponent $q$ satisfies $1<q<\frac{n}{n-2}$ in case $n\geq 3$ and $1<q<\infty$ in case $n=1$ or $n=2$. For space dimensions $n\geq 2$ and arbitrary $b<0$ we prove the existence of fully nontrivial nonnegat…
Using certain Ito's equation, we introduce the probability on the space of paths and show its relevance to the scattering properties of multidimensional Schrodinger operator. To relate the geometry of the support of potential to the spectral type we develop a special variant of Potential theory and prove some estimates on the modified Harmonic measure.
We consider the cubic nonlinear Schrödinger equation with harmonic trapping on $\mathbb{R}^D$ ($1\leq D\leq 5$). In the case when all but one directions are trapped (a.k.a "cigar-shaped" trap), following the approach of Hani-Pausader-Tzvetkov-Visciglia, we prove modified scattering and construct modified wave operators for small initial and final data respectively. The asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the resonant system of the NLS equation with full trapping on $\mathbb{R}^{D-1}$. In the physical dimension $D=3$, this sy…
The cubic nonlinear Schrodinger equation with a lattice potential is used to model a periodic dilute gas Bose-Einstein condensate. Both two- and three-dimensional condensates are considered, for atomic species with either repulsive or attractive interactions. A family of exact solutions and corresponding potential is presented in terms of elliptic functions. The dynamical stability of these exact solutions is examined using both analytical and numerical methods. For condensates with repulsive atomic interactions, all stable, trivial-phase solutions are off-set from the zero level. For condensa…
For Schrödinger equations with a class of slowly decaying repulsive potentials, we show that the solution satisfies global-in-time Strichartz estimates for any admissible pairs. Our admissible class of potentials includes the positive homogeneous potential $Z|x|^{-μ}$ with $Z>0$ and $0<μ<2$ in three and higher dimensions, especially the repulsive Coulomb potential. The proof employs several techniques from scattering theory such as the long time parametrix construction of Isozaki-Kitada type, propagation estimates and local decay estimates.
In this paper, we determine the wave front sets of solutions to Schrödinger equations of a harmonic oscillator with sub-quadratic perturbation by using the representation of the Schrödinger evolution operator of a harmonic oscillator via the wave packet transform.
We investigate the dispersive properties of evolution equations on waveguides with a non flat shape. More precisely we consider an operator $H=-Δ_{x}-Δ_{y}+V(x,y)$ with Dirichled boundary condition on an unbounded domain $Ω$, and we introduce the notion of a \emph{repulsive waveguide} along the direction of the first group of variables $x$. If $Ω$ is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation $Hu-λu=f$. As consequences we prove smoothing estimates for the Schrödinger and wave equations associated to $H$, and Strichartz estimates for the Schrödinger equation. Ad…
We study the asymptotic behaviour of solutions to semi-classical nonlinear Schrodinger equations with a potential, for concentrating and oscillating initial data, when the nonlinearity is repulsive and the potential is a polynomial of degree at most two. We describe the separate roles of the nonlinearity and of the potential, with tools which seem to be specific to the class of potentials that we consider. We also discuss the case of more general subquadratic potentials.
In this paper we obtain some Strichartz estimates for the Schrödinger equation associated to the harmonic oscillator and the Laplacian. Our main tool will be some embeddings between Lebesgue spaces and suitable Triebel-Lizorkin spaces.
We prove some local smoothing estimates for the Schrödinger initial value problem with data in $L^2(\mathbb{R}^d)$, $d \geq 2$ and a general class of potentials. In the repulsive setting we have to assume just a power like decay $(1+|x|)^{-γ}$ for some $γ>0$. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory.
We consider the Cauchy problem of a dissipative nonlinear Schrödinger equation with a time dependent harmonic potential. We find a critical situation that the $L^2$-norm of dissipative solutions decays or not and which is decided by a nonlinear power and time decay order of harmonic potential.
This paper is concerned with the Cauchy problem for an inhomogeneous nonlinear Schrodinger equation with exponential growth nonlinearity and harmonic potential in two space dimensions. We prove global well-posedness, existence of the associated ground state and instability of the standing wave.
We provide necessary and sufficient geometric conditions for the exact controllability of the one-dimensional fractional free and fractional harmonic Schrödinger equations. The necessary and sufficient condition for the exact controllability of fractional free Schrödinger equations is derived from the Logvinenko-Sereda theorem and its quantitative version established by Kovrijkine, whereas the one for the exact controllability of fractional harmonic Schrödinger equations is deduced from an infinite dimensional version of the Hautus test for Hermite functions and the Plancherel-Rotach formula.
Nonlinear Schrodinger Equations (NLS) of the Hartree type occur in the modeling of quantum semiconductor devices. Their "semiclassical" limit of vanishing (scaled) Planck constant is both a mathematical challenge and practically relevant when coupling quantum models to classical models. With the aim of describing the semi-classical limit of the 3D Schrodinger--Poisson system with an additional harmonic potential, we study some semi-classical limits of the Hartree equation with harmonic potential in space dimension n>1. The harmonic potential is confining, and causes focusing periodically in ti…
We study the cubic-quartic nonlinear Schrödinger equation (NLS) in two and three spatial dimension. This equation arises in the mean-field description of Bose-Einstein condensates with Lee-Huang-Yang correction. We first prove global existence of solutions in natural energy spaces which allow for the description of self-bound quantum droplets with vorticity. Existence of such droplets, described as central vortex states in 2D and 3D, is then proved using an approach via constrained energy minimizers. A natural connection to the NLS with repulsive inverse-square potential in 2D arises, leading …
We find positive non-radial solutions for a system of Schrödinger equations in a weak fully attractive or repulsive regime in presence of an external radial trapping potential that exhibits a maximum or a minimum at infinity.
For the Schrödinger flow from $R^2 \times R^+$ to the 2-sphere $S^2$, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blow up time (if any), and that they blow up if and only if the length scale of the nearest harmonic map goes to zero.
We consider the final-state problem for the nonlinear Schrödinger equations (NLS) with a suitable time-decaying harmonic oscillator. In this equation, the power of nonlinearity $|u|^ρu $ is included in the long-range class if $0 < ρ\leq 2/(n(1- λ)) $ with $0 \leq λ<1/2$, which is determined by the harmonic potential and a coefficient of Laplacian. In this paper, we find the final state for this system and obtain the decay estimate for asymptotics.
We prove the scattering for a defocusing nonlinear Schrödinger equation with a sum of two repulsive potentials with strictly convex level surfaces, thus providing a scattering result in a trapped setting similar to the exterior of two strictly convex obstacles.
In this paper,we show that spherical bounded energy solution of the defocusing 3D energy critical Schrödinger equation with harmonic potential, $(i\partial_t + \frac Δ2+\frac {|x|^2}2)u=|u|^4u$, exits globally and scatters to free solution in the space $Σ=H^1\bigcap\mathcal F H^1$. We preclude the concentration of energy in finite time by combining the energy decay estimates.
We consider the initial-value problem for the $1d$ cubic nonlinear Schrödinger equation with a repulsive delta potential. We prove that small initial data in a weighted Sobolev space lead to global solutions that decay in $L^\infty$ and exhibit modified scattering.
We consider the nonlinear Schrödinger equation with a repulsive Dirac delta potential in one dimensional Euclidean space. We classify the global dynamics of even solutions with the same action as the high-frequency ground state standing wave solutions.
In this article we study the Schrödinger equation associated with Harmonic oscillator in the form of Strichartz type inequality. We give simple proofs for Strichartz type inequalities using purely the $L^2 \to L^p$ operator norm estimates of the spectral projections associated harmonic oscillator proved in \cite{KT}. Our Strichartz type estimates are sharp in sense of regularity of initial data.
Solutions of semi-classical Schrodinger equation with isotropic harmonic potential focus periodically in time. We study the perturbation of this equation by a nonlinear term. If the scaling of this perturbation is critical, each focus crossing is described by a nonlinear scattering operator, which is therefore iterated as many times as the solution passes through a focus. The study of this nonlinear problem is made possible by the introduction of two operators well adapted to Schrodinger equations with harmonic potential, and by suitable Strichartz inequalities.
Time-decaying harmonic oscillators yield dispersive estimates with weak decay, and change the threshold power of the nonlinearity between the short and the long range. In the non-critical case for the time-decaying harmonic oscillator, this threshold can be characterized by polynomial nonlinearities. However, in the critical case, it is difficult to characterize the threshold using only polynomial terms, and thus we use logarithmic nonlinear terms.
We derive conditional stability estimates for inverse scattering problems related to time harmonic magnetic Schrödinger equation. We prove logarithmic type estimates for retrieving the magnetic (up to a gradient) and electric potentials from near field or far field maps. Our approach combines techniques from similar results obtained in the literature for inhomogeneous inverse scattering problems based on the use of geometrical optics solutions.
We obtain existence and multiplicity results for the solutions of a class of coupled semilinear bi-harmonic Schrödinger equations. Actually, using the classical Mountain Pass Theorem and minimization techniques, we prove the existence of critical points of the associated functional constrained on the \emph{Nehari manifold}. Furthermore, we show that using the so-called \emph{fibering method} and the \emph{Lusternik-Schnirel'man theory} there exist infinitely many solutions, actually a countable family of critical points, for such a semiliner bi-hamonic Schrödinger system under study in this wo…
The initial value problem for some coupled nonlinear Schrodinger system with unbounded potential is investigated. In the defocusing case, global well-posedness is obtained. For the focusing sign, existence of global and non global solutions is discussed via potential well method. Moreover, existence of ground state and instability of standing waves are proved.
We study strong instability of standing waves $e^{iωt} ϕ_ω(x)$ for nonlinear Schrödinger equations with $L^2$-supercritical nonlinearity and a harmonic potential, where $ϕ_ω$ is a ground state of the corresponding stationary problem. We prove that $e^{iωt} ϕ_ω(x)$ is strongly unstable if $\partial_λ^2 E(ϕ_ω^λ) |_{λ=1}\le 0$, where $E$ is the energy and $v^λ(x)=λ^{N/2} v(λx)$ is the $L^2$-invariant scaling.
This note studies the asymptotic behavior of global solutions to the fourth-order Schrödinger equation $$i\dot u+Δ^2 u+F(x,u)=0 .$$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to Dodson-Murphy \cite{dm}.
In this paper, we consider the energy critical nonlinear Schrödinger equation with a repulsive inverse square potential. In particular, we deal with radial initial data, whose energy is equal to the energy of static solution to the corresponding nonlinear Schrödinger equation without a potential. We investigate time behavior of the radial solutions with such initial data.
We study the instability of standing-wave solutions $e^{iωt}ϕ_ω(x)$ to the inhomogeneous nonlinear Schrödinger equation $$iϕ_t=-\triangleϕ+|x|^2ϕ-|x|^b|ϕ|^{p-1}ϕ, \qquad \in\mathbb{R}^N, $$ where $ b > 0 $ and $ ϕ_ω $ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $ω$ of wave and the power of nonlinearity $p $ for any fixed $ b > 0. $
We consider Calderón's inverse boundary value problems for a class of nonlinear Helmholtz Schrödinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the Dirichlet-to-Neumann map of the corresponding equations. The local uniqueness of the linearized partial data Calderón's inverse problem is obtained following \cite{DKSU}. The Runge approximation properties and unique continuation principle allow us to extend to global situations. Simultaneous recovery of some unknown cavity$/$boundary and coefficients are given as some appl…
We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrödinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e. the ones describing self-action and non-linear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonli…
We consider a 3d cubic focusing nonlinear Schrödinger equation with a potential $$i\partial_t u+Δu-Vu+|u|^2u=0,$$ where $V$ is a real-valued short-range potential having a small negative part. We find criteria for global well-posedness analogous to the homogeneous case $V=0$ (Duyckaerts-Holmer-Roudenko). Moreover, by the concentration-compactness approach, we prove that if $V$ is repulsive, such global solutions scatter.
