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Masaki Kawamoto

Publications and source records attributed to Masaki Kawamoto.

At least 19 recordsLinked to original sources
arXiv2026

Large-data modified wave operators for the defocusing nonlinear Schrödinger equation in one space dimension with subcritical long-range nonlinearity

We study long-time behavior of the solutions to the final state problem for the defocusing nonlinear Schrödinger equation (NLS) in one space dimension with the power nonlinearity $|u|^{2σ}u$ in the subcritical long-range regime $\frac{2}{\sqrt{7}}<σ<1$. Given a prescribed asymptotic profile in a weighted $L^2$-space, without size restriction, obtained by modifying the free solution with a nonlinear polynomial phase correction, we construct a unique global solution of the NLS that scatters to this profile, thereby proving the existence of modified wave operators. The proof relies on two new ingredients. Extending our previous work for the cubic case, we incorporate the leading part of the nonlinear term into the linear part as a linear potential by linearizing the NLS around the asymptotic profile and prove a global modified energy estimate for the linearized equation. We also exploit a specific structure of the nonlinearity arising from the linearization, which gives rise to a crucial cancellation when estimating the nonlinear terms in the modified energy space and enables us to control the polynomial growth of the nonlinear phase correction in the subcritical case.

math.AP↗
arXiv2026

Correction to the article "Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance"

This article resolves some errors in the paper ``Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance. Math. Ann. 392, 1051-1097 (2025)''. The errors are in the proof of contraction of a map associated with our equation in two and three dimensions.

math.AP↗
arXiv2026

Modified Scattering for the Time-Dependent Kohn--Sham Equation

We study the long-time behavior of the (critical) Kohn--Sham equation in two and three dimensions, i.e.,\[ \mathrm{i} \partial_t γ = \Big[-\frac{1}{2}Δ+ λ\, |\cdot|^{-1} \ast ρ_{γ} + μ\, ρ_{γ}^{1/d}, γ \Big] \quad \text{for} \quad d=2,3. \] By introducing a suitable ''square root'' of the density matrix and exploiting the pseudo-conformal transform, we establish global well-posedness for small initial data in an appropriate weighted Schatten norm. We also prove the optimal time decay of the particle density and establish modified scattering for small and localized solutions. In particular, our results provide a resolution to the open problems proposed by Pusateri and Sigal (2021) for the critical and subcritical regimes, rigorously proving their conjectures regarding modified scattering in the critical case and linear scattering in the subcritical cases. Our results place these scattering phenomena in the operator-valued setting of density matrices, thereby extending the classical scalar theory to a broader framework.

math.AP↗
arXiv2026

Modified wave operators for the defocusing cubic nonlinear Schrödinger equation in one space dimension with large scattering data

In the present paper, we construct modified wave operators for the defocusing cubic nonlinear Schrödinger equation (NLS) in one space dimension without size restriction on scattering data. In the proof, we introduce a new formulation of the problem based on the linearization of the NLS around a prescribed asymptotic profile. For the linearized equation which is a system of Schrödinger equations with non-symmetric, time-dependent long-range potentials, we show a modified energy identity, as well as an associated energy estimate, which allow us to apply a simple energy method to construct the modified wave operators. As a byproduct, we also obtain in the focusing case an improved explicit upper bound for the size of scattering data to ensure the existence of modified wave operators. Our argument relies neither on the complete integrability nor on the framework of analytic function spaces, and also works for short-range perturbations of the cubic nonlinearity.

math.AP↗
arXiv2025

On Schrödinger equation with square and inverse-square potentials

In this paper, we study the linear and nonlinear Schrödinger equations with a time-decaying harmonic oscillator and inverse-square potential. This model retains a form of scale invariance, and using this property, we demonstrate the asymptotic completeness of wave operators and Strichartz estimates for linear propagators.

math.AP↗
arXiv2025

Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance

In this paper, we consider the nonlinear Schrödinger equation (NLS) with a general homogeneous nonlinearity in dimensions up to three. We assume that the degree (i.e., power) of the nonlinearity is such that the equation is mass-subcritical and short-range. We establish global well-posedness (GWP) and scattering for small data in the standard weighted space for a class of homogeneous nonlinearities, including non-gauge-invariant ones. Additionally, we include the case where the degree is less than or equal to the Strauss exponent. When the nonlinearity is not gauge-invariant, the standard Duhamel formulation fails to work effectively in the weighted Sobolev space; for instance, the Duhamel term may not be well-defined as a Bochner integral. To address this issue, we introduce an alternative formulation that allows us to establish GWP and scattering, even in the presence of poor time continuity of the Duhamel term.

math.AP↗
arXiv2025

Modified scattering for nonlinear Schrödinger equations with long-range potentials

We study the final state problem for the nonlinear Schrödinger equation with a critical long-range nonlinearity and a long-range linear potential. Given a prescribed asymptotic profile which is different from the free evolution, we construct a unique global solution scattering to the profile. In particular, the existence of the modified wave operators is obtained for sufficiently localized small scattering data. The class of potential includes a repulsive long-range potential with a short-range perturbation, especially the positive Coulomb potential in two and three space dimensions. The asymptotic profile is constructed by combining Yafaev's type linear modifier [38] associated with the long-range part of the potential and the nonlinear modifier introduced by Ozawa [29]. Finally, we also show that one can replace Yafaev's type modifier by Dollard's type modifier under a slightly stronger decay assumption on the long-range potential. This is the first positive result on the modified scattering for the nonlinear Schrödinger equation in the case when both of the nonlinear term and the linear potential are of long-range type.

math.AP↗
arXiv2024

Modified scattering for the cubic nonlinear Schrödinger equation with long-range potentials in one space dimension

We consider the cubic nonlinear Schrödinger equation with long-range linear potentials in one space dimension, and prove the modified scattering in the energy space for the associated final state problem with a prescribed small asymptotic profile. Compared with the leading term of the free solution, the asymptotic profile has an additional phase correction depending both on the long-range part of the potential and on the nonlinear term. The proof is based on a simple energy method and does not rely on global-in-time Strichartz estimates for Schrödinger equations with linear potentials. In particular, the class of potentials to which our theorem applies is large enough to accommodate slowly decaying negative potentials so that the associated Schrödinger operators may have negative eigenvalues.

math.AP↗
arXiv2024

Modified scattering operator for nonlinear Schrödinger equations with time-decaying harmonic potentials

This paper is concerned with nonlinear Schrödinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In [24] and [22], it is proved that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in the frameworks of both the final state problem and the initial value problem. Furthermore, a modified scattering operator has been established in the case without the potential in [15]. In this paper, we construct a modified scattering operator for our equation by utilizing a generator of the Galilean transformation. Moreover, we remove a restriction for the coefficient of the potential which is required in [22].

math.AP↗
arXiv2023

Nonexistence of wave operators via strong propagation estimates for Schrödinger operators with sub-quadratic repulsive potentials

Sub-quadratic repulsive potentials accelerate quantum particles and can relax the decay rate in the $x$ of the external potentials $V$ that guarantee the existence of the quantum wave operators. In the case where the sub-quadratic potential is $- |x|^α $ with $0< α< 2$ and the external potential satisfies $|V(x) | \leq C (1+|x|) ^{-(1- α/2) - \varepsilon} $ with $\varepsilon>0$, Bony et. al [3] determined the existence and completeness of the wave operators, and Itakura [12, 13, 14] then obtained their results using stationary scattering theory for more generalized external potentials. Based on their results, we naturally expect the following. If the decay power of the external potential $V$ is less than ${ -(1- α/2) } $, V is included in the short-range class. If the decay power is greater than or equal to ${ -(1- α/2) } $, $V$ is included in the long-range class. In this study, we first prove the new propagation estimates for the time propagator that can be applied to scattering theory. Second, we prove that the wave operators do not exist if the power is greater than or equal to $-(1- α/2)$ and that the threshold expectation of ${ -(1- α/2) } $ is true using the new propagation estimates.

math-ph↗
arXiv2022

Long-range scattering for a critical homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potentials

This paper is concerned with the final state problem for the homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potential. The nonlinearity has the critical order and is not necessarily the form of a polynomial. In the case of the gauge-invariant power-type nonlinearity, the first author proves that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in [22]. In this paper, we extend his result into the case with the general homogeneous nonlinearity by the technique due to the Fourier series expansion introduced by Masaki and the second author [26]. To adapt the argument in the aforementioned paper, we develop a factorization identity for the propagator and require a little stronger decay condition for the Fourier coefficients arising from the harmonic potential. Moreover, in two or three dimensions, we improve the regularity condition of the final data in [26, 28].

math.AP↗
arXiv2022

Strichartz estimates for quadratic repulsive potentials

Quadratic repulsive potentials $- τ^2 |x| ^2$ accelerate the quantum particle, increasing the velocity of the particle exponentially in $t$; this phenomenon yields fast decaying dispersive estimates. In this study, we consider the Strichartz estimates associated with this phenomenon. First, we consider the free repulsive Hamiltonian, and prove that the Strichartz estimates hold for every admissible pair $(q,r)$, which satisfies $1/q +n/(2r) \geq n/4$ with $q$, $r \geq 2$. Second, we consider the perturbed repulsive Hamiltonian with a slowly decaying potential, such that $|V(x)| \leq C(1+|x|)^{-δ}$ for some $δ>0$, and prove that the Strichartz estimate holds with the same admissible pairs for repulsive-admissible pairs.

math.AP↗
arXiv2022

Asymptotic behavior for nonlinear Schrödinger equations with critical time-decaying harmonic potential

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.

math.AP↗
arXiv2021

Asymptotic completeness of wave operators for Schrödinger operators with time-periodic magnetic fields

Under the effect of suitable time-periodic magnetic fields, the velocity of a charged particle grows exponentially in $t$; this phenomenon provides the asymptotic completeness for wave operators with slowly decaying potentials. These facts were shown under some restrictions for time-periodic magnetic fields and the range of wave operators. In this study, we relax these restrictions and finally obtain the asymptotic completeness of wave operators. Additionally, we show them under generalized conditions, which are truly optimal for time-periodic magnetic fields. Moreover, we provide a uniform resolvent estimate for the perturbed Floquet Hamiltonian.

math-ph↗
arXiv2021

Final state problem for nonlinear Schrödinger equations with time-decaying harmonic oscillators

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.

math.AP↗
arXiv2021

$L^2$-stableness for solution to linearized KdV equation

The linearized Korteweg-De Vries equation can be written as a Hamilton-like system. However, the Hamilton energy depends on the time, and is a nonsymmetric operator on $L^2({\bf R})$. By performing suitable unitary transforms on the Hamilton energy, we can reduce this operator into one that is not independent on the time but nonsymmetric. In this study, we consider the $L^2$-stability issues and smoothing estimates for this operator, and prove that it has no eigenvalues.

math.AP↗
arXiv2020

Absence of embedded eigenvalues for Hamiltonian with crossed magnetic and electric fields

In the presence of the homogeneous electric field ${\bf E}$ and the homogeneous perpendicular magnetic field ${\bf B}$, the classical trajectory of a quantum particle on ${\mathbb R}^2$ moves with drift velocity $α$ which is perpendicular to the electric and magnetic fields. For such Hamiltonians the absence of the embedded eigenvalues of perturbed Hamiltonian has been conjectured. In this paper one proves this conjecture for the perturbations $V(x, y)$ which have sufficiently small support in direction of drift velocity.

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