arXiv subjects

Haruya Mizutani

Publications and source records attributed to Haruya Mizutani.

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

Scattering Theory For 3D Cubic Damped Magnetic Schrödinger Equation

We consider the three-dimensional defocusing cubic nonlinear Schrödinger equation with variable coefficients, a magnetic potential, and a non-negative localized damping term, \[ i\partial_tu+(\nabla-iA)\cdot G(\nabla-iA)u+ia(x)u=|u|^2u, \qquad t>0,\quad x\in\mathbb R^3. \] No non-trapping condition is imposed on the metric $G$. Instead, the variable-coefficient region is assumed to be contained in the effective damping region. Under a one-centre condition on the tangential magnetic field, we prove global well-posedness for initial data in $H^{1+\varepsilon}$, uniform mass and energy bounds, and show the local energy decay. To obtain scattering, we impose a support condition on the full magnetic field inside the damping region. Under these stronger assumptions, the solution scatters to a free Schrödinger evolution in $H^s$ for every $0\le s<1$. The appendix discusses a separate constant-damping framework for abstract Hamiltonians.

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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

$L^p$-boundedness of wave operators for fourth order Schrödinger operators with zero resonances on $\mathbb{R}^3$

Let $H = Δ^2 + V$ be the fourth-order Schrödinger operator on $\mathbb{R}^3$ with a real-valued fast-decaying potential $V$. If zero is neither a resonance nor an eigenvalue of $H$, then it was recently shown that the wave operators $W_\pm(H, Δ^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < \infty$ and unbounded at the endpoints $p=1$ and $p=\infty$. This paper is to further establish the $L^p$-boundedness of $W_\pm(H, Δ^2)$ that exhibit all types of singularities at the zero energy threshold. We first prove that $W_\pm(H, Δ^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < \infty$ in the first kind resonance case, and then proceed to establish for the second kind resonance case that they are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < 3$, but not if $3 \le p \le \infty$. In the third kind resonance case, we also show that $W_\pm(H, Δ^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1<p<3$ and generically unbounded on $L^p(\R^3)$ for any $3\le p\le\infty$. Moreover, it is also shown that $W_\pm(H, Δ^2)$ are bounded on $L^p(\R^3)$ for all $3\le p<\infty$ if in addition $H$ has the zero eigenvalue, but no $p$-wave zero resonances and all zero eigenfunctions are orthogonal to $x_ix_jx_kV$ in $L^2(\R^3)$ for all $i,j,k=1,2,3$ with $x=(x_1,x_2,x_3)\in \R^3$. These results describe precisely the validity of the $L^p$-boundedness of $W_\pm(H, Δ^2)$ in $\mathbb{R}^3$ for all types of singularities at the zero energy threshold with some exceptions for the endpoint cases $p=1,\infty$. As an application, $L^p$-$L^q$ decay estimates are also derived for the fourth-order Schrödinger equations and Beam equations with zero resonance singularities.

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

Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials

Let $ H = (-Δ)^m + V $ be a higher-order elliptic operator on $ L^2(\mathbb{R}^n) $, where $ V $ is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schrödinger-type equation associated with $ H $. In particular, we first establish sharp global Kato smoothing estimates for $ e^{itH} $, based on uniform resolvent estimates of Kato-Yajima type for the absolutely continuous part of $ H $. As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schrödinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig-Ruiz-Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates.

math.AP↗

arXiv2024

Uniform resolvent estimates, smoothing effects and spectral stability for the Heisenberg sublaplacian

We establish global bounds for solutions to stationary and time-dependent Schrödinger equations associated with the sublaplacian $\mathcal L$ on the Heisenberg group, as well as its pure fractional power $\mathcal L^s$ and conformally invariant fractional power $\mathcal L_s$. The main ingredient is a new abstract uniform weighted resolvent estimate which is proved by using the method of weakly conjugate operators -- a variant of Mourre's commutator method -- and Hardy's type inequalities on the Heisenberg group. As applications, we show Kato-type smoothing effects for the time-dependent Schrödinger equation, and spectral stability of the sublaplacian perturbed by complex-valued decaying potentials satisfying an explicit subordination condition. In the local case $s=1$, we obtain uniform estimates without any symmetry or derivative loss, which improve previous results.

math.AP↗

arXiv2024

Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger operators in dimension three

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,Δ^2)$ associated with fourth-order Schrödinger operators $H=Δ^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim \langle x\rangle^{-μ}$ for some $μ>0$. A recent work by Goldberg and Green \cite{GoGr21} has demonstrated that wave operators $W_\pm(H,Δ^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 9$, and zero is a regular point of $H$. In this paper, we aim to further establish endpoint estimates for $W_\pm(H,Δ^2)$ in two significant ways. First, we provide counterexamples that illustrate the unboundedness of $W_\pm(H,Δ^2)$ on the endpoint spaces $L^1(\mathbb{R}^3)$ and $L^\infty(\mathbb{R}^3)$, even for non-zero compactly supported potentials $V$. Second, we establish weak (1,1) estimates for the wave operators $W_\pm(H,Δ^2)$ and their dual operators $W_\pm(H,Δ^2)^*$ in the case where zero is a regular point and $μ>11$. These estimates depend critically on the singular integral theory of Calderón-Zygmund on a homogeneous space $(X,dω)$ with a doubling measure $dω$.

math.AP↗

arXiv2024

Limiting absorption principle and absence of eigenvalues for massless Klein-Gordon operators on perturbations of the Minkowski spacetime

We prove a uniform weighted resolvent estimate for the massless Klein-Gordon operator on a curved spacetime which is sufficiently close to the Minkowski spacetime. This particularly implies the existence and Hölder continuity of the limiting resolvents at all energies, as well as the absolute continuity, of the massless Klein-Gordon operator. The proof is based on a simple version of Mourre's commutator method and does not rely on microlocal analysis. We also prove the absence of eigenvalues via the Virial theorem under an ellipticity condition on the commutator of the massless Klein-Gordon operator against the generator of a wick-rotated dilation, which is weaker than the smallness condition for the metric perturbation.

math-ph↗

arXiv2024

$L^p$-boundedness of wave operators for bi-Schrödinger operators on the line

This paper is devoted to establishing several types of $L^p$-boundedness of wave operators $W_\pm=W_\pm(H, Δ^2)$ associated with the bi-Schrödinger operators $H=Δ^{2}+V(x)$ on the line $\mathbb{R}$. Given suitable decay potentials $V$, we firstly prove that the wave and dual wave operators are bounded on $L^p(\mathbb{R})$ for all $1<p<\infty$: $$ \|W_\pm f\|_{L^p(\mathbb{R})}+\|W_\pm^* f\|_{L^p(\mathbb{R})}\lesssim \|f\|_{L^p(\mathbb{R})},$$ which are further extended to the $L^p$-boundedness on the weighted spaces $L^p(\mathbb{R},w)$ with general even $A_p$-weights $w$ and to the boundedness on the Sobolev spaces $W^{s,p}(\mathbb{R})$. For the limiting case, we prove that $W_\pm$ are bounded from $L^1(\R)$ to $L^{1,\infty}(\R)$ as well as bounded from the Hardy space $\H^1(\R)$ to $L^1(\R)$. These results especially hold whatever the zero energy is a regular point or a resonance of $H$. We also obtain that $W_\pm$ are bounded from $L^\infty(\R)$ to $\BMO(\R)$ if zero is a regular point or a first kind resonance of $H$. Next, we show that $W_\pm$ are neither bounded on $L^1(\mathbb{R})$ nor on $L^\infty(\mathbb{R})$ even if zero is a regular point of $H$. Moreover, if zero is a second kind resonance of $H$, then $W_\pm$ are shown to be even not bounded from $L^\infty(\R)$ to $\BMO(\R)$ in general. In particular, we remark that our results give a complete picture of the validity of $L^p$-boundedness of the wave operators for all $1\le p\le \infty$ in the regular case. Finally, as applications, we deduce the $L^p$-$L^q$ decay estimates for the propagator $e^{-itH}P_{\mathrm{ac}}(H)$ with pairs $(1/p,1/q)$ belonging to a certain region of $\mathbb{R}^2$, as well as establish the Hörmander-type $L^p$-boundedness theorem for the spectral multiplier $f(H)$.

math.AP↗

arXiv2023

Non-existence of radial eigenfunctions for the perturbed Heisenberg sublaplacian

We prove uniform resolvent estimates in weighted $L^2$-spaces for radial solutions of the sublaplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^d$. The proofs are based on the multipliers methods, and strongly rely on the use of suitable multipliers and of the associated Hardy inequalities. The constants in our inequalities are explicit and depend only on the dimension $d$. As application of the method, we obtain some suitable smallness and repulsivity conditions on a complex radial potential $V$ on $\mathbb{H}^d$ such that $\mathcal{L}+V$ has no radial eigenfunctions.

math.AP↗

arXiv2021

Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials

Let $0<σ -C_{σ,n}$, we first prove {\it uniform resolvent estimates} of Kato--Yajima type for all $0<σ 1/2$ and {\it uniform Sobolev estimates} of Kenig--Ruiz--Sogge type for $σ\ge n/(n+1)$. These extend the same properties for the Schrödinger operator with the inverse-square potential to the higher-order and fractional cases. Moreover, we also obtain {\it improved Strichartz estimates with a gain of regularities} for general initial data if $1<σ<n/2$ and for radially symmetric data if $n/(2n-1)<σ\le1$, which extends the corresponding results for the free evolution to the case with Hardy potentials. These arguments can be further applied to a large class of higher-order inhomogeneous elliptic operators and even to certain long-range metric perturbations of the Laplace operator. Finally, in the critical coupling constant case (i.e. $a=-C_{σ,n}$), we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions.

math.AP↗

arXiv2021

Keller-type bounds for Dirac operators perturbed by rigid potentials

In this paper we are interested in generalizing Keller-type eigenvalue estimates for the non-selfadjoint Schrödinger operator to the Dirac operator, imposing some suitable rigidity conditions on the matricial structure of the potential, without necessarily requiring the smallness of its norm.

math.SP↗

arXiv2021

Modified scattering for inhomogeneous nonlinear Schrödinger equations with and without inverse-square potential

We consider the final state problem for the inhomogeneous nonlinear Schrödinger equation with a critical long-range nonlinearity. Given a prescribed asymptotic profile, which has a logarithmic phase correction compared with the free evolution, we construct a unique global solution which converges to the profile. As a consequence, the existence of modified wave operators for localized small scattering data is obtained. We also study the same problem for the case with the critical inverse-square potential under the radial symmetry. In particular, we construct the modified wave operators for the long-range nonlinear Schrödinger equation with the critical inverse-square potential in three space dimensions, under the radial symmetry.

math.AP↗

arXiv2020

Strichartz estimates for Schrödinger equations with slowly decaying potentials

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.

math.AP↗

arXiv2020

Scattering theory in homogeneous Sobolev spaces for Schrödinger and wave equations with rough potentials

We study the scattering theory for the Schrödinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of the paper concerns with an inverse-square potential in both of subcritical and critical constant cases, which is a particular model of scaling-critical singular perturbations. In the subcritical case, the existence of the wave and inverse wave operators defined on a range of homogeneous Sobolev spaces is obtained. In particular, we have the scattering to a free solution in the homogeneous energy space for both of the Schrödinger and wave equations. In the critical case, it is shown that the solution is asymptotically a sum of a $n$-dimensional free wave and a rescaled two-dimensional free wave. The second half of the paper is concerned with a generalization to a class of strongly singular decaying potentials. We provides a simple criterion in an abstract framework to deduce the existence of wave operators defined on a homogeneous Sobolev space from the existence of the standard ones defined on a base Hilbert space.

math.AP↗

arXiv2020

Asymptotic behavior for the long-range nonlinear Schrödinger equation on star graph with the Kirchhoff boundary condition

We consider the cubic nonlinear Schrödinger equation on the star graph with the Kirchhoff boundary condition. We prove modified scattering for the final state problem and the initial value problem. Moreover, we also consider the failure of scattering for the Schrödinger equation with power-type long-range nonlinearities. These results are extension of the results for NLS on the one dimensional Euclidean space.

math.AP↗