We prove local and global well-posedness for semi-relativistic, nonlinear Schrödinger equations $i \partial_t u = \sqrt{-Δ+ m^2} u + F(u)$ with initial data in $H^s(\mathbb{R}^3)$, $s \geq 1/2$. Here $F(u)$ is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type self-interactions. For focusing $F(u)$, which arise in the quantum theory of boson stars, we derive a sufficient condition for global-in-time existence in terms of a solitary wave ground state. Our proof of well-posedness does not rely on Strichartz type estimates, and it enables us to add external potentials of a general class.
In a recent paper published in Nonlinear Analysis: Theory, Methods & Applications, C. Argaez and M. Melgaard studied excited states for pseudo-relativistic multi-configuration methods. Their paper follows a previous work of mine in the non-relativistic case (Arch. Rat. Mech. Anal., 171, 2004). The main results of the paper of C. Argaez and M. Melgaard are correct, but the proofs are both wrong and incomplete.
We prove uniqueness of ground states $Q$ in $H^{1/2}$ for pseudo-relativistic Hartree equations in three dimensions, provided that $Q$ has sufficiently small $L^2$-mass. This result shows that a uniqueness conjecture by Lieb and Yau in [CMP 112 (1987),147--174] holds true at least under a smallness condition. Our proof combines variational arguments with a nonrelativistic limit, which leads to a certain Hartree-type equation (also known as the Choquard-Pekard or Schroedinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredien…
We consider the three-dimensional semi-relativistic Hartree model for fast quantum mechanical particles moving in a self-consistent field. Under appropriate assumptions on the initial density matrix as a (fully) mixed quantum state we prove, using Wigner transformation techniques, that its classical limit yields the well known relativistic Vlasov-Poisson system. The result holds for the case of attractive and repulsive mean-field interaction, with an additional size constraint in the attractive case.
We prove blow-up in finite time for radially symmetric solutions to the pseudo-relativistic Hartree-Fock equation with negative energy. The non-linear Hartree-Fock equation is commonly used in the physics literature to describe the dynamics of white dwarfs. We extend thereby recent results by Fröhlich and Lenzmann, who established in \cite{FL1,FL2} blow-up for solutions to the pseudo-relativistic Hartree equation. As key ingredient for handling the exchange term we use the conservation of the expectation of the square of the angular momentum operator.
We justify WKB analysis for Hartree equation in space dimension at least three, in a regime which is supercritical as far as semiclassical analysis is concerned. The main technical remark is that the nonlinear Hartree term can be considered as a semilinear perturbation. This is in contrast with the case of the nonlinear Schrodinger equation with a local nonlinearity, where quasilinear analysis is needed to treat the nonlinearity.
In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation $\sqrt{-\varepsilon^2 Δ+ m^2}u + V u = (I_α* |u|^{p}) |u|^{p-2}u$ in $\mathbb{R}^N$ where $m>0$, $2 \leq p < \frac{2N}{N-1}$, $V \colon \mathbb{R}^N \to \mathbb{R}$ is an external scalar potential, $I_α(x) = \frac{c_{N,α}}{|x|^{N-α}}$ is a convolution kernel, $c_{N,α}$ is a positive constant and $(N-1)p-N<α<N$. For $N=3$, $α=p=2$, our equation becomes the pseudo-relativistic Hartree equation with Coulomb kernel.
We study the existence of ground state solutions for a class of non-linear pseudo-relativistic Schrödinger equations with critical two-body interactions. Such equations are characterized by a nonlocal pseudo-differential operator closely related to the square-root of the Laplacian. We investigate such a problem using variational methods after transforming the problem to an elliptic equation with a nonlinear Neumann boundary conditions.
We consider a class of evolution equations in Lindblad form, which model the dynamics of dissipative quantum mechanical systems with mean-field interaction. Particularly, this class includes the so-called Quantum Fokker-Planck-Poisson model. The existence and uniqueness of global-in-time, mass preserving solutions is proved, thus establishing the existence of a nonlinear conservative quantum dynamical semigroup. The mathematical difficulties stem from combining an unbounded Lindblad generator with the Hartree nonlinearity.
We prove existence of positive ground state solutions to the pseudo-relativistic Schrödinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-Δ+m^2} u +Vu = \left( W * |u|^θ \right)|u|^{θ-2} u \quad\text{in $\mathbb{R}^N$}\\ u \in H^{1/2}(\mathbb{R}^N) \end{array} \right. \end{equation*} where $N \geq 3$, $m >0$, $V$ is a bounded external scalar potential and $W$ is a convolution potential, radially symmetric, satisfying suitable assumptions. We also furnish some asymptotic decay estimates of the found solutions.
With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[(-Δ+m^2)^σu+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $0<σ<1$, $V$ is a bounded continuous potential and $F$ the primitive of $f$. We also show results about the regularity of any solution of this problem.
With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[\sqrt{-Δ+m^2}\, u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $V$ is a bounded potential, not necessarily continuous, and $F$ the primitive of $f$. We also show that any of this problem is a classical solution. Furthermore, we prove that the ground state solution has exponential decay.
In this paper, we prove a quantitative version of the semiclassical limit from the Hartree to the Vlasov equation with singular interaction, including the Coulomb potential. To reach this objective, we also prove the propagation of velocity moments and weighted Schatten norms which implies the boundedness of the space density of particles uniformly in the Planck constant.
We are concerned with the following constrained minimization problem: $$e(a_{1},a_{2},β) := \inf\left\{E_{a_{1},a_{2},β}(u_{1},u_{2}): \|u_{1}\|_{L^{2}(\mathbb{R}^{3})} = \|u_{2}\|_{L^{2}(\mathbb{R}^{3})} = 1\right\},$$ where $E_{a_{1},a_{2},β}$ is the energy functional associated to two coupled pseudo-relativistic Hartree equations involving three parameters $a_{1}, a_{2}, β$ and two trapping potentials $V_1(x)$ and $V_2(x)$. In this paper, we obtain the existence of minimizers of $e(a_{1},a_{2},β)$ for possible $a_{1}, a_{2}$ and $β$ under suitable conditions on the potentials, which general…
In this paper, we investigate the constrained minimization problem \begin{equation}\label{eq:0.1} e(a):=\inf_{\{u\in \mathcal{H},\|u\|_2^2=1\}}E_a(u), \end{equation} where the energy functional \begin{equation} \label{eq:0.2} E_a(u)=\int_{\mathbb{R}^3}(u\sqrt{-Δ+m^2}\,u+Vu^2)\,dx -\frac{a}{2}\int_{\mathbb{R}^3}(|x|^{-1}*u^2)u^2\,dx \end{equation} with $m\in \mathbb{R}$, $a>0$, is defined on a Sobolev space $\mathcal{H}$. We show that there exists a threshold $a^*>0$ so that $e(a)$ is achieved if $0<a<a^*$, and has no minimizers if $a\geq a^*$. We also investigate the asymptotic behavior of non…
We prove the existence of a solution to the semirelativistic Hartree equation $$\sqrt{-Δ+m^2}u+ V(x) u = A(x)\left( W * |u|^p \right) |u|^{p-2}u $$ under suitable growth assumption on the potential functions $V$ and $A$. In particular, both can be unbounded from above.
We prove several existence and non existence results of solitary waves for a class of nonlinear pseudo-relativistic Hartree equations with general nonlinearities. We use variational methods and some new variational identities involving the half Laplacian.
We consider a class of pseudo-relativistic Hartree equations in presence of general nonlinearities not satisfying the Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory, we show the existence of two non trivial signed solutions, one positive and one negative.
The semi-classical regime of standing wave solutions of a Schrödinger equation in presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is show that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.
We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.
We study nondegeneracy of ground states of the Hartree equation $$ -Δu+u=(I_{2}\ast u^2)u\quad\mbox{ in }\mathbb R^n $$ where $n=3,4,5$ and $I_2$ is the Newton potential. As an application of the nondegeneracy result, we use a Lyapunov-Schmidt reduction argument to construct multiple semiclassical solutions to the following Hartree equation with an external potential $$-\varepsilon^2Δu+u+V(x)u=\varepsilon^{-2}(I_{2}\ast u^2)u\quad \mbox{ in }\mathbb R^n.$$
We prove that the initial value problem for the equation \[ - i\partial_t u + \sqrt{m^2-Δ} \, u= (\frac{e^{-μ_0 |x|}}{|x|} \ast |u|^2)u \ \text{in} \ \mathbb R^{1+3}, \quad m\ge 0, \ μ_0 >0\] is globally well-posed and the solution scatters to free waves asymptotically as $t \to \pm \infty$ if we start with initial data which is small in $H^s(\mathbb R^{3})$ for $s>\frac12$, and if $m>0$. Moreover, if the initial data is radially symmetric we can improve the above result to $m\ge 0$ and $s>0$, which is almost optimal, in the sense that $L^2(\mathbb R^{3})$ is the critical space for the equatio…
In this paper we study the small data scattering of Hartree type semirelativistic equation in space dimension $3$. The Hartree type nonlinearity is $[V * |u|^2]u$ and the potential $V$ which generalizes the Yukawa has some growth condition. We show that the solution scatters to linear solution if an initial data given in $ H^{s,1}$ is sufficiently small and $s>\frac14$. Here, $H^{s, 1}$ is Sobolev type space taking in angular regularity with norm defined by $\|φ\|_{ H^{s, 1}} = \|φ\|_{ H^{s}} + \|\nabla_{\mathbb S} φ\|_{H^{s}}$. To establish the results we employ the recently developed Stricha…
