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.
We consider Schrödinger equations with variable coefficients, and it is supposed to be a long-range type perturbation of the flat Laplacian on $R^n$. We characterize the wave front set of solutions to Schrödinger equations in terms of the initial state. Then it is shown that the singularity propagates following the classical flow, and it is formulated in a semiclassical settings. Methods analogous to the long-range scattering theory, in particular a modified free propagator, are employed.
We consider semiclassically scaled Schrodinger equations with an external potential and a highly oscillatory periodic potential. We construct asymptotic solutions in the form of semiclassical wave packets. These solutions are concentrated (both, in space and in frequency) around the effective semiclassical phase-space flow obtained by Peierl's substitution, and involve a slowly varying envelope whose dynamics is governed by a homogenized Schrodinger equation with time-dependent effective mass. The corresponding adiabatic decoupling of the slow and fast degrees of freedom is shown to be valid u…
We consider solutions of the time-dependent Schrödinger equation for a potential localised at the points of a Poisson process. We prove convergence of the phase-space distribution in the annealed Boltzmann-Grad limit to a semiclassical Wigner (or defect) measure and show that it is a solution of the linear Boltzmann equation. Our results hold for a large class of square-integrable initial data associated to Wigner measures, including Langragian states, WKB states and coherent states. This extends important previous work by Eng and Erdős.
We prove that radial, monotonic, superexponentially decaying potentials in R^n, n greater than or equal to 1 odd, are determined by the resonances of the associated semiclassical Schrodinger operator among all superexponentially decaying potentials in R^n.
We consider the semiclassical Schrödinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schrödinger equation, and we prove that, in the semiclassical limit and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.
We consider the semiclassical Schrödinger-Poisson system with a special initial data of WKB type such that the solution of the limiting hydrodynamical equation becomes time-global in dimensions at least three. We give an example of such initial data in the focusing case via the analysis of the compressible Euler-Poisson equations. This example is a large data with radial symmetry, and is beyond the reach of the previous results because the phase part decays too slowly. Extending previous results in this direction, we justify the WKB approximation of the solution with this data for an arbitrari…
We prove that the spectrum of an n-dimensional semiclassical radial Schrödinger operator determines the potential within a large class of potentials for which we assume no symmetry or analyticity. Our proof is based on the first two semiclassical trace invariants and on the isoperimetric inequality.
We consider the one dimensional focusing (cubic) Nonlinear Schrödinger equation (NLS) in the semiclassical limit with exponentially decaying complex-valued initial data, whose phase is multiplied by a real parameter. We prove smooth dependence of the asymptotic solution on the parameter. Numerical results supporting our estimates of important quantities are presented.
We derive a determinant formula for the WKB exponential of singularly perturbed Zakharov-Shabat system that corresponds to the semiclassical (zero dispersion) limit of the focusing Nonlinear Schr\" odinger equation. The derivation is based on the Riemann-Hilbert Problem (RHP) representation of the WKB exponential. We also prove its independence of the branchpoints of the corresponding hyperelliptic surface assuming that the modulation equations are satisfied.
We consider a magnetic Schrödinger operator $H^h$, depending on a semiclassical parameter $h>0$, on a compact Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the intensity of the magnetic field $b$ is strictly positive. We give a survey of the results on asymptotic behavior of the eigenvalues of the operator $H^h$ in the semiclassical limit.
We revisit the problem of semiclassical spectral asymptotics for a pure magnetic Schrödinger operator on a two-dimensional Riemannian manifold. We suppose that the minimal value $b_0$ of the intensity of the magnetic field is strictly positive, and the corresponding minimum is unique and non-degenerate. The purpose is to get the control on the spectrum in an interval $(hb_0, h(b_0 +γ_0)]$ for some $γ_0>0$ independent of the semiclassical parameter $h$. The previous papers by Helffer-Mohamed and by Helffer-Kordyukov were only treating the ground-state energy or a finite (independent of $h$) num…
In this paper, we establish (1) the classical limit of the Hartree equation leading to the Vlasov equation, (2) the classical limit of the $N$-body linear Schrödinger equation uniformly in N leading to the N-body Liouville equation of classical mechanics and (3) the simultaneous mean-field and classical limit of the N-body linear Schrödinger equation leading to the Vlasov equation. In all these limits, we assume that the gradient of the interaction potential is Lipschitz continuous. All our results are formulated as estimates involving a quantum analogue of the Monge-Kantorovich distance of ex…
A perturbation of a class of scalar Riemann-Hilbert problems (RHPs) with the jump contour as a finite union of oriented simple arcs in the complex plane and the jump function with a $z\log z$ type singularity on the jump contour is considered. The jump function and the jump contour are assumed to depend on a vector of external parameters $\vecβ$. We prove that if the RHP has a solution at some value $\vecβ_0$ then the solution of the RHP is uniquely defined in a some neighborhood of $\vecβ_0$ and is smooth in $\vecβ$. This result is applied to the case of semiclassical focusing NLS.
We prove optimal pointwise bounds on quasimodes of semiclassical Schrodinger operators with arbitrary smooth real potentials in dimension two. This end-point estimate was left open in the general study of semiclassical Lp bounds conducted by Koch-Tataru-Zworski. However, we show that their results imply the two dimensional end-point estimate by scaling and localization.
We consider the operator ${\mathcal A}_h=-Δ+iV$ in the semi-classical $h\rightarrow 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of ${\mathcal A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.
We consider a fractional Schrödinger-Poisson system in the whole space $\mathbb R^{N}$ in presence of a positive potential and depending on a small positive parameter $\varepsilon.$ We show that, for suitably small $\varepsilon$ (i.e. in the "semiclassical limit") the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of the set of minima of the potential.
In this paper, we study, in the semiclassical sense, the global approximate controllability in small time of the quantum density and quantum momentum of the 1-D semiclassical cubic Schrödinger equation with two controls between two states with positive quantum densities. We first control the asymptotic expansions of the zeroth and first order of the physical observables via Agrachev-Sarychev's method. Then we conclude the proof through techniques of semiclassical approximation of the nonlinear Schrödinger equation.
We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.
A concept of semiclassically concentrated solutions is formulated for the multidimensional nonlinear Schrödinger equation (NLSE) with an external field. These solutions are considered as multidimensional solitary waves. The center of mass of such a solution is shown to move along with the bicharacteristics of the basic symbol of the corresponding linear Schrödinger equation. The leading term of the asymptotic WKB-solution is constructed for the multidimensional NLSE. Special cases are considered for the standard one-dimensional NLSE and for NLSE in cylindrical coordinates.
In this article, we shall construct the resolvent of Laplacian at high energies near the spectrum on non-product conic manifolds with a single cone tip. Microlocally, the resolvent kernel is the sum of b-pseudodifferential operators, scattering pseudodifferential operators and intersecting Legendrian distributions. As an application, we shall establish Strichartz estimates for Schrödinger equations on non-compact manifolds with multiple non-product conic singularities.
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 study the spectral projection associated to a barrier-top resonance for the semiclassical Schrodinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrodinger group in terms of these resonances.
We consider a magnetic Schrödinger operator $H^h$, depending on the semiclassical parameter $h>0$, on a two-dimensional Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the magnetic field $b$ is strictly positive, and there exists a unique minimum point of $b$, which is non-degenerate. The main result of the paper is a complete asymptotic expansion for the low-lying eigenvalues of the operator $H^h$ in the semiclassical limit. We also apply these results to prove the existence of an arbitrary large number of spectral gaps in the semicla…
Via a Lyapunov-Schmidt reduction, we obtain multiple semiclassical solutions to a class of fractional nonlinear Schrödinger equations. Precisely, we consider \begin{equation*} \varepsilon^{2s}(-Δ)^{s}u+u+V(x)u=|u|^{p-1}u,\quad u\in H^s(\mathbf R^n), \end{equation*} where $0<s<1$, $n>4-4s$, $1<p<\frac{n+2s}{n-2s}$ (if $n>2s$) and $1<p<\infty$ (if $n\le 2s$), $V(x)$ is a non-negative potential function. If $V$ is a sufficiently smooth bounded function with a non-degenerate compact critical manifold $M$, then, when $\varepsilon$ is sufficiently small, there exist at least $l(M)$ semiclassical sol…
In this paper, we consider the following fractional logarithmic Schrödinger equation \begin{equation*} \varepsilon^{2s}(-Δ)^s u + V(x)u=u\log |u|^2\ \ \text{in}\ \R^N, \end{equation*} where $\varepsilon>0$, $N\ge 1$, $V(x)\in C(\R^N,[-1,+\infty))$. By introducing an interesting penalized function, we show that the problem has a positive solution $u_{\varepsilon}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$. There is no restriction on decay rates of $V$, especially it can be compactly supported.
We establish uniform bounds for the solutions $e^{itΔ}u$ of the Schrödinger equation on arithmetic flat tori, generalising earlier results by J. Bourgain. We also study the regularity properties of weak-* limits of sequences of densities of the form $|e^{itΔ}u_{n}|^{2}$ corresponding to highly oscillating sequences of initial data $(u_{n})$. We obtain improved regularity properties of those limits using previous results by N. Anantharaman and F. Macià on the structure of semiclassical measures for solutions to the Schrödinger equation on the torus.
We argue that the critical behaviour near the point of ``gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation $ iεψ_t +\frac{ε^2}2ψ_{xx}+ |ψ|^2 ψ=0$ with analytic initial data of the form $ψ(x,0;ε) =A(x) e^{\frac{i}ε S(x)}$ is approximately described by a particular solution to the Painlevé-I equation.
In this article, we study the asymptotic fields of the Yukawa particle-field model of quantum physics, in the semiclassical regime $\hslash\to 0$, with an interaction subject to an ultraviolet cutoff. We show that the transition amplitudes between final (respectively initial) states converge towards explicit quantities involving the outgoing (respectively incoming) wave operators of the nonlinear Schrödinger-Klein-Gordon (S-KG) equation. Thus, we rigorously link the scattering theory of the Yukawa model to that of the Schrödinger-Klein-Gordon equation. Moreover, we prove that the asymptotic va…
We continue our study of a magnetic Schrödinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic field has degenerate magnetic wells. The main result of the paper is an asymptotics of the groundstate energy of the operator in the semiclassical limit. The upper bounds are improved in the case when we have a localization by a miniwell effect of lowest order. These results are applied to prove the existence of an arbitrary large number of spectral gaps in the…
We consider the following Schrödinger equation $$ - \hslash ^2 Δu + V(x)u = Γ(x) f(u) \quad \mathrm{in} \ \mathbb{R}^N, $$ where $u \in H^1 (\mathbb{R}^N)$, $u > 0$, $\hslash > 0$ and $f$ is superlinear and subcritical nonlinear term. We show that if $V$ attains local minimum and $Γ$ attains global maximum at the same point or $V$ attains global minimum and $Γ$ attains local maximum at the same point, then there exists a positive solution for sufficiently small $\hslash>0$.
In this note, we study solutions to semiclassical Schrodinger equations on a real analytic manifold with a real analytic potential and prove the semiclassical version of Cauchy estimates on derivatives. As an application, we use Donnelly and Fefferman's method to prove the upper and lower bounds for (n-1)-dimensional Hausdorff measure of the nodal sets of the solutions to semiclassical Schrodinger equations.
In this paper, we study the following semilinear Schrödinger equation $$ -ε^2\triangle u+ u+ V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}), $$ where $N\geq 2$ and $ε>0$ is a small parameter. The function $V$ is bounded in $\mathbb{R}^N$, $\inf_{\mathbb{R}^N}(1+V(x))>0$ and it has a possibly degenerate isolated critical point. Under some conditions on $f,$ we prove that as $ε\rightarrow 0,$ this equation has a solution which concentrates at the critical point of $V$.}
We study the long time dynamics of the Schrödinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrödinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrödinger operators: we prove that adding a potential to the Laplacian on the sphere results on the existence of geodesics $γ$ such that $δ_γ$ cannot be obtained as a semiclassical measure for some sequence of eigenfunctions. We also show that the same phenomenon occurs for the free Lapl…
We are interested in a WKB analysis of the Logarithmic Non-Linear Schrödinger Equation with "Riemann-like" variables in an analytic framework in semiclassical regime. We show that the Cauchy problem is locally well posed uniformly in the semiclassical constant and that the semiclassical limit can be performed. In particular, our framework is not only compatible with the Gross-Pitaevskii equation with logarithmic nonlinearity, but also allows initial data (and solutions) which can converge to $0$ at infinity.
In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying that the $L^2$-norm of a solution on any open subset of the torus controls the full $L^2$-norm.
This paper is devoted to the magnetic nonlinear Schrödinger equation \[ \Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u=f(| u|^{2})u \text{ in } \mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous functions and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ function having exponential critical growth. Under a global assumption on the potential $V$, we use variational methods and Ljusternick-Schnirelmann theory to prove existence, multiplicity, concentration, and decay of nont…
We consider perturbations of the semiclassical Schr{ö}dinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the perturbation, the solutions associated to initial data in a small spectral window become equidistributed in the semiclassical limit. As an application of our method, we also derive some properties of the quantum Loschmidt echo below and beyond the Ehrenfest time for initial data in a small spectral window.
We consider the semiclassical limit of nonlinear Schrödinger equations with wavepacket initial data. We recover the Wigner measure of the problem, a macroscopic phase-space density which controls the propagation of the physical observables such as mass, energy and momentum. Wigner measures have been used to create effective models for wave propagation in random media, quantum molecular dynamics, mean field limits, and the propagation of electrons in graphene. In nonlinear settings, the Vlasov-type equations obtained for the Wigner measure are often ill-posed on the physically interesting space…
In this paper, a class of Schrödinger-Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied with the same argument of the nonlinear term with only a positive potential, because the weight potentials set $\{Q_i(x)|1\le i \le m\}$ contains nonpositive, sign-changing, and nonnegative elements. By introducing the ground energy function and subtle analysis, we first prove the existence of ground state solution $v_\varepsilon$ in the semiclassical limit via the Nehari manifold and concentration-compactness principle. Then…
The purpose of this paper is to present a comparison between the modified nonlinear Schrödinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schrödinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the anal…
We construct solutions to the nonlinear magnetic Schrödinger equation $$ \left\{ \begin{aligned} - \varepsilon^2 Δ_{A/\varepsilon^2} u + V u &= \lvert u\rvert^{p-2} u & &\text{in}\ Ω,\\ u &= 0 & &\text{on}\ \partialΩ, \end{aligned} \right. $$ in the semiclassical régime with strong magnetic fields. In contrast with the well-studied mild magnetic field régime, the limiting energy depends on the magnetic field allowing to recover the Lorentz force in the semi-classical limit. Our solutions concentrate around global or local minima of a limiting energy that depends on the electric potential and t…
In this paper, we consider the singularly perturbed fractional Schrödinger equation \begin{equation*} ε^{2α}(-Δ)^αu+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $ε>0$ is a small parameter, $α\in(0,1)$, $(-Δ)^α$ is the fractional Laplacian operator of order $α$, $V$ possesses global minimum points, and $f$ is asymptotically linear at infinity. We investigate the relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its global minimum. We also construct multiple concentrating solutions if $V$ has several strict global minim…
We study the following fractional Schrödinger equation \begin{equation*}\label{eq0.1} \varepsilon^{2s}(-Δ)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} where $s\in(0,1)$. Under some conditions on $f(u)$, we show that the problem has a family of solutions concentrating at any finite given local minima of $V$ provided that $V\in C(\R^N,[0,+\infty))$. All decay rates of $V$ are admissible. Especially, $V$ can be compactly supported. Different from the local case $s=1$ or the case of single-peak solutions, the nonlocal effect of the operator $(-Δ)^s$ makes the peaks of the candidate s…
We consider the long-time properties of the an obstruction in the Riemann-Hilbert approach to one dimensional focusing Nonlinear Schrödinger equation in the semiclassical limit for a one parameter family of initial conditions. For certain values of the parameter a large number of solitons in the system interfere with the $g$-function mechanism in the steepest descent to oscillatory Riemann-Hilbert problems. The obstruction prevents the Riemann-Hilbert analysis in a region in $(x,t)$ plane. We obtain the long time asymptotics of the boundary of the region (obstruction curve). As $t\to\infty$ th…
We begin a study of a multi-parameter family of Cauchy initial-value problems for the modified nonlinear Schrödinger equation, analyzing the solution in the semiclassical limit. We use the inverse scattering transform for this equation, along with the steepest descent method of Deift and Zhou. The initial conditions are selected both to allow all relevant scattering data to be calculated without approximation and also to place the governing equation in a transsonic state in which the quantum fluid dynamical system formally approximating it is of hyperbolic type for some $x$ and of elliptic typ…
This paper focuses on the linearly coupled critical fractional Schrödinger system \begin{equation*} \begin{cases} ε^{2s}(-\triangle)^s u +a(x)u=u^p+λv\quad &\text{in}\ \mathbb{R}^N,\\ ε^{2s}(-\triangle)^s v +b(x)v=v^{2_s^*-1}+λu\quad &\text{in}\ \mathbb{R}^N, \end{cases} \end{equation*} where $N>2s,$ $s\in(0,1),$ $p\in(1,2_s^*),$ $ε$ and $λ$ are positive parameters, $a,b\in C{(\mathbb{R}^N)}$ are positive potentials, and $(-\triangle)^s$ is the fractional Laplacian operator. Under certain assumptions on $a$ and $λ,$ we obtain the existence, decay estimates and concentration property of positiv…
Some focusing coupled Schrodinger equations are investigated. First, existence of ground state is obtained. Second, global and non global existence of solutions are discussed via potential-well method. Finally, strong instability of standing waves is established.
We investigate some focusing fourth-order coupled Schrodinger equations. Existence of ground state and global well-posedness are obtained. Moreover, the best constant of some Gagliardo-Nirenberg inequality is studied.
