We study the scattering problem for a long range potential, which is time dependent. We prove the existence and completeness of the scattering wave operators, and find some properties of the weakly localized, non-scattering part of the solution. The method we use follows recent methods introduced and applied to short range systems.

For wave propagation problems in homogeneous, isotropic media, exponentially convergent domain truncation techniques based on perfectly matched damping layers or optimal rational approximations to the exact radiation operator are known. However, these methods fail for systems with wave families satisfying general dispersion relations, forcing practitioners to resort to ad hoc procedures based on grid stretching and artificial damping. Here we propose a new method for constructing convergent approximations on truncated domains, the phase space filter, which unlike other methods is completely general and mathematically justified. Based on the fact that outgoing waves can be characterized as waves located near the boundary of the computational domain with group velocities pointing outward, the key idea of the phase space filtering algorithm consists of applying a filter to the solution that removes outgoing waves only. The method introduced in this work is a simplified version of the original phase space filter, which was proposed for the Schro"\dinger equation. The method is applied to anisotropic wave models for which existing techniques are unstable, namely free-space problems governed in the far field by the Euler equations linearized about a uniform mean flow and Maxwell's equations in an anisotropic medium. We also present experiments in waveguide geometry and for isotropic problems which indicate that a more sophisticated multiscale extension of the algorithm is needed to obtain good long-time accuracy for waveguide problems, or to match the performance of the optimized local radiation conditions available in the isotropic case. Theoretical results concerning the convergence and computational costs of the phase space filter are discussed and stability is proven.

The dynamics of Schrödinger equation with time dependent potentials of general time dependence is considered. It is shown that for localized in space potentials, there is propagation of regularity which is uniformly bounded in higher Sobolev norms. Unlike the cases where the solution scatter, and then propagation is proved via a standard bootstrap argument, the solutions considered here have a part that does not scatter, as expected in general. For this we introduce propagation estimates that work directly in (e.g.) H^2(ℛ^3).

For the Schr & ouml;dinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and a weakly localized part. The proof is based on constructing in an adapted way the Free Channel Wave Operator, and further tools from the recent works [21,22,35]. This work generalizes the results of the first part of [21,22] to arbitrary dimension, and non-radial data. (c) 2025 Published by Elsevier Inc.

This paper investigates the L-P boundedness of wave operators for the Laplace operator with finite rank perturbations H = -Delta+ Sigma(N)(i=1) phi(i) on R-d. For dimensions d >= 3, we prove that the wave operators W +/-(H, H-0 are bounded on L-P for the full range 1 <= p <= infinity. This extends the work of Nier and the third author [27] by resolving the previously unexplored question of boundedness at the endpoint cases p = 1 and p = infinity. In lower dimensions d = 1 - 2 we establish the L-P -boundedness of the wave operators the first time. Furthermore, we reveal an intriguing dichotomy in the endpoint case p = 1: If integral(Rd)phi(i)(x)dx = 0 holds for every 1 <= i <= N then the wave bounded on L-P(R-d) 1 <= p <= infinity. If there exists at least one i (1 <= i <= N) such that integral(Rd)phi(i)(x)dx not equal 0, then the wave operators remain bounded for 1 < p < infinity and satisfy weak type (1,1) estimates, but fail to be bounded on L-1(R-d). (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We study the long-time dynamics of small solutions to the one-dimensional nonlinear Schrödinger equation i∂_t v+∂_x^2v-βv^2v +(x)v^4v+γi∂_xv=0, where is spatially localized. The cubic nonlinearity is long range and produces the logarithmic phase correction, whereas the localized quintic term is short range at leading order. We construct a global forward modified wave operator for small complex asymptotic profiles and prove quantitative final-state estimates. For small data in the weighted energy space, we also establish global existence, sharp t^-1/2 decay, and forward modified scattering with a unique asymptotic profile. The principal new phenomenon occurs beyond this leading law. The exact Duhamel tail generated by the localized quintic term admits a quantitative inner scaling limit at the distinguished frequency ζ=-γ/2 on the scale ζ+γ/2∼ t^-1/2. Its universal shape is explicit and depends on the value of the scattering profile on the distinguished ray and on the zeroth moment of . When both quantities are nonzero, the limit is nontrivial, belongs optimally to C^2,1_loc, and is not C^3 at the center. Away from the corresponding self-similar ray ξ=-γ, we construct rigorously defined higher-order outer expansions through every integer order allowed by the decay of , and to each fixed finite order when is rapidly decreasing.

We introduce a microlocal phase-space-filtered physics-informed neural network (PINN–TDPSF or Microlocal PINNFilter) framework for wave propagation on unbounded domains. The method combines a slabwise neural residual approximation of the interior evolution with a time-dependent phase-space filter applied in a buffer surrounding the physical computational domain. The central idea is to replace local artificial-boundary penalties by a phase-space radiation mechanism: a component is removed only when it is localized near the artificial boundary and its group velocity points outward. The proposed method is not intended to replace FFT, spectral, or split-step solvers for known-coefficient forward problems where such methods are available and highly accurate. Instead, it embeds the time-dependent phase-space filter into a residual-based neural framework. This coupling is useful when open-domain wave propagation must be combined with nonlinear residuals, sparse or off-grid observations, unknown coefficients, variable interior media, or other non-FFT-diagonalizable physics. Numerical experiments for linear Schrödinger propagation, potential scattering, anisotropic Schrödinger dynamics, nonlinear Schrödinger wave packets, soliton stress tests, linearized Euler waves, and sparse-data recovery of a localized acoustic defect show that the method reduces artificial reflection and wraparound, uses group velocity correctly in anisotropic media, preserves physically incoming branch components, and provides diagnostics when the assumptions behind outgoing-packet filtering are violated.

Efficient simulation of many-body quantum systems is central to advances in physics, chemistry, and quantum computing, with a key question being whether the simulation cost scales polynomially with the system size. In this work, we analyze many-body quantum systems with Coulomb interactions, which are fundamental to electronic and molecular systems. We prove that Trotterization for such unbounded Hamiltonians achieves a 1/4-order convergence rate, with explicit polynomial dependence on the number of particles. The result holds for all initial wavefunctions in the domain of the Hamiltonian, and the 1/4-order convergence rate is optimal, as previous work has numerically demonstrated that it can be saturated by a specific initial ground state. The main challenges arise from the many-body structure and the singular nature of the Coulomb potential. Our proof strategy differs from prior state-of-the-art Trotter analyses, addressing both difficulties in a unified framework. Our analysis treats the Coulomb potential as an unbounded operator without modification or regularization, and does not rely on spatial discretization, making it compatible with both first- and second-quantized circuit constructions.

This paper establishes the Lp boundedness of wave operators localized at high-frequency for linear Schrodinger equations in R3 with timedependent potentials. The approach to the proof is based on new cancellation lemmas. As a typical application based on this method, combined with Strichartz estimates is the existence and scattering for nonlinear dispersive equations. For example, we prove global existence and uniform boundedness in L infinity, for a class of Hartree nonlinear Schrodinger equations in L2(R3), allowing the presence of solitons. We also prove the existence of free channel wave operators in Lp(R3) for p > 6.

We study the large-time behavior of global energy class (H^1) solutions of the one-dimensional nonlinear Schrödinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions decompose into a free wave and a weakly localized part which is asymptotically orthogonal to any fixed free wave. We further show that the L^2 norm of this weakly localized part is concentrated in the region |x| ≤ t^1/2+, and that the energy (Ḣ^1) norm is concentrated in |x| ≤ t^1/3+. Our results hold for solutions with arbitrarily large initial data.

We give a proof of local decay estimates for Schrödinger-type equations, which is based on the knowledge of Asymptotic Completeness. This approach extends to time dependent potential perturbations, as it does not rely on Resolvent Estimates or related methods. Global in time Strichartz estimates follow for quasi-periodic time-dependent potentials from our results.

Inspiring by a recent work [57], we analyse a 3-wave kinetic equation, derived from the elastic beam wave equation on the lattice. The ergodicity condition states that two distinct wavevectors are supposed to be connected by a finite number of collisions. In this work, we prove that once the ergodicity condition is violated, the domain is broken into disconnected domains, called no-collision and collisional invariant regions. If one starts with a general initial condition, whose energy is finite, then in the long-time limit, the solutions of the 3-wave kinetic equation remain unchanged on the no-collision region and relax to local equilibria on the disjoint collisional invariant regions. The equilibration temperature will differ from region to region. This behavior of 3-wave systems was first described by Spohn in [55], without a detailed rigorous proof. Our proof follows Spohn's physically intuitive arguments.

We consider the Bogoliubov approximation for the many-body quantum dynamics of weakly interacting Bose gases and establish a uniform-in-time validity of the Bogoliubov theory. The proof relies on a detailed analysis of the dispersive behavior of the symplectic Bogoliubov dynamics, which allows for a rigorous derivation of the Bogoliubov theory as an effective description of quantum fluctuations around the Bose-Einstein condensate on all time scales.

In the present paper, we consider global solutions of a class of nonlinear wave equations of the form u= N(x,t,u)u, where the nonlinearity N(x, t, u)u is assumed to satisfy appropriate boundedness assumptions. Under these appropriate assumptions, we prove that the free channel wave operator exists. Moreover, if the interaction term N(x, t, u)u is localized, then we prove that the global solution of the full nonlinear equation can be decomposed into a ‘free’ part and a ‘localized’ part.

We study the asymptotics of the Schrödinger equation with time-dependent potential in dimension one. Assuming that the potential decays sufficiently rapidly as |x| →∞, we prove that the solution can be written as the sum of a free wave e^-itΔ u_+ and a weakly bound component u_wb(t). Moreover, we show that the weakly bound part decomposes as u_wb(t) = u_loc(t) + o_Ḣ^1(1), where ∂_x u_loc(t) is localized near the origin uniformly in time. Since decay conditions on the potential do not preclude resonances unless d ≥ 5, our results can be seen as a natural extension of [Terence Tao. Dynamics of Partial Differential Equations, 5(2), 2008] and [Avy Soffer, Xiaoxu Wu. arXiv:2304.04245] to the lower-dimensional case.

This paper investigates the L^p-bounds of wave operators for higher-order Schrödinger operators H = (-Δ)^m + V on ℝ^n, with m ≥ 2 and real-valued decaying potentials V. Our main objective is to establish the sharp L^p-boundedness of the wave operators W_±(H; (-Δ)^m) in the presence of all types of zero-resonance singularities, for all odd dimensions 1 ≤ n ≤ 4m - 1. Specifically, for odd n with 1 ≤ n ≤ 4m - 1, there exist m_n types of zero resonances for H, along with a critical type k_c (both depending on n and m). If zero is a regular point of H or a 𝐤-th kind resonance with 1 ≤𝐤≤ k_c, the wave operators W_±(H; (-Δ)^m) are bounded on L^p(ℝ^n) for all 1 < p < ∞. If zero is a 𝐤-th kind resonance with k_c < 𝐤≤ m_n, we show that the range of p-boundedness for W_±(H; (-Δ)^m) narrows to 1 < p < p_𝐤, where p_𝐤 = n/n - 2m + 𝐤 + k_c - 1. Additionally, if zero is an eigenvalue of H (i.e., 𝐤 = m_n + 1), then W_±(H; (-Δ)^m) are bounded on L^p(ℝ^n) for all 1 < p < 2n/n - 1. Furthermore, it is shown that the wave operators W_±(H; (-Δ)^m) are unbounded on L^p(ℝ^n) for all p_𝐤 < p ≤∞ if k_c < 𝐤≤ m_n, and for all 2n/n - 1 < p ≤∞ if zero is an eigenvalue of H with a non-zero solution ϕ to Hϕ = 0 in ⋂_s < -1/2 L^2_s(ℝ^n) ∖ L^2(ℝ^n)(referred to as a p-wave resonance). The key idea of the proof is to reduce the L^p-unboundedness to establishing the optimality of time-decay estimates for e^itHP_ac(H) in weighted L^2 spaces.

This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential $$u_{t t}+\big(\Delta^2+V\big)u=0, \,\ u(0, x)=f(x),\ u_{t}(0, x)=g(x)$$ in dimension three, where $V$ is a real-valued and decaying potential on $\R^3$. Assume that zero is a regular point of $H:= \Delta^2+V $, we first prove the following optimal time decay estimates of the solution operators \begin{equation*} \big\|\cos (t\sqrt{H})P_{ac}(H)\big\|_{L^{1} \rightarrow L^{\infty}} \lesssim|t|^{-\frac{3}{2}}\ \ \hbox{and} \ \ \Big\|\frac{\sin(t\sqrt{H})}{\sqrt{H}} P_{a c}(H)\Big\|_{L^{1} \rightarrow L^{\infty}} \lesssim|t|^{-\frac{1}{2}}. \end{equation*} Moreover, if zero is a resonance of $H$, then time decay of the solution operators above also are considered. It is noticed that the first kind resonance does not effect the decay rates for the propagator operators $\cos(t\sqrt{H})$ and $\frac{\sin(t\sqrt{H})}{\sqrt{H}}$, but their decay will be dramatically changed for the second and third resonance types.

We consider the Schroedinger equation with a general interaction term, which is localized in space. The interaction may be x, t dependent and non-linear. Purely non-linear parts of the interaction are localized via the radial Sobolev embedding. Under the assumption of radial symmetry and boundedness in H1(R3) of the solution, uniformly in time. we prove it is asymptotic in L2 (and H1) in the strong sense, to a free wave and a weakly localized solution. The general properties of the localized solutions are derived. The proof is based on the introduction of phase-space analysis of the nonlinear dispersive dynamics and relies on a new class of (exterior) a priory propagation estimates. This approach allows a unified analysis of general linear time-dependent potentials and non-linear interactions.

I present a review of the recent advancements in scattering theory, which provides a unified approach to studying dispersive and hyperbolic equations with general interaction terms and data. These equations encompass time-dependent potentials, as well as NLS, NLKG, and NLW equations. Additionally, I discuss a series of open problems along with their significance and potential future applications in scattering and inverse scattering.

We give a short description of the proof of asymptotic-completeness for NLS-type equations with radial data in three dimensions. We also show some aspects of the method by giving a new proof of Asymptotic Completeness for the two-body Quantum Scattering case.