Faculty Profiles - MIZUTANI Haruya

写真a

MIZUTANI Haruya

Organization

Graduate School of Mathematics Division of Mathematics Advanced Topology Professor

Graduate School

Graduate School of Mathematics

Undergraduate School

School of Science Department of Mathematics

Research Interests 4

散乱理論
偏微分方程式
スペクトル理論
シュレディンガー方程式

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Research Areas 2

Natural Science / Mathematical analysis / 偏微分方程式論
Natural Science / Basic analysis

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Research History 5

名古屋大学大学院多元数理科学研究科教授

2026.10
Osaka University Associate Professor

2017.4 - 2026.9
Osaka University Assistant Professor

2014.4 - 2017.3
Gakushuin University Faculty of Science Assistant Professor

2012.10 - 2014.3
Kyoto University Research Institute for Mathematical Sciences

2011.4 - 2012.9

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Education 1

The University of Tokyo Graduate School of Mathematical Sciences

2006.4 - 2011.3

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Awards 1

日本数学会賞建部賢弘賞特別賞

2017 日本数学会シュレディンガー方程式に対するストリッカーツ評価

水谷治哉

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Papers 30

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

Luca Fanelli, Haruya Mizutani, Luz Roncal, Nico Michele Schiavone

Journal d'Analyse Mathématique 2026.9

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Language：English Publishing type：Research paper (scientific journal) Publisher：Springer Science and Business Media LLC

DOI： 10.1007/s11854-026-0451-2

Other Link： https://link.springer.com/article/10.1007/s11854-026-0451-2
$L^p$-boundedness of wave operators for fourth order Schrödinger operators with zero resonances on R^3 Reviewed

Haruya Mizutani, Zijun Wan, Xiaohua Yao

Journal of Functional Analysis Vol. 289 ( 8 ) page： 111013 - 111013 2025.10

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Publishing type：Research paper (scientific journal) Publisher：Elsevier BV

DOI： 10.1016/j.jfa.2025.111013
Global Kato smoothing and Strichartz estimates for Schrödinger type equations with rough decay potentials Reviewed

Haruya Mizutani, Xiaohua Yao

Reviews in Mathematical Physics 2025.8

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Publishing type：Research paper (scientific journal)

DOI： 10.1142/S0129055X24500508
Modified scattering for the cubic nonlinear Schrödinger equation with long-range potentials in one space dimension Reviewed

Masaki Kawamoto, Haruya Mizutani

Proceedings of the American Mathematical Society 2025.6

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Publishing type：Research paper (scientific journal) Publisher：American Mathematical Society (AMS)

<p>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.</p>

DOI： 10.1090/proc/17268

Other Link： https://www.ams.org/proc/0000-000-00/S0002-9939-2025-17268-9/S0002-9939-2025-17268-9.pdf
Non-existence of Radial Eigenfunctions for the Perturbed Heisenberg Sublaplacian Reviewed Open Access

Luca Fanelli, Haruya Mizutani, Luz Roncal, Nico Michele Schiavone

Journal of Fourier Analysis and Applications Vol. 31 ( 3 ) 2025.5

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Publishing type：Research paper (scientific journal) Publisher：Springer Science and Business Media LLC

Abstract

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.

DOI： 10.1007/s00041-025-10167-6

Open Access

Other Link： https://link.springer.com/article/10.1007/s00041-025-10167-6/fulltext.html
Modified scattering for nonlinear Schrödinger equations with long-range potentials Reviewed

Masaki Kawamoto, Haruya Mizutani

Transactions of the American Mathematical Society 2025.2

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Publisher：American Mathematical Society (AMS)

<p>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 associated with the long-range part of the potential and the nonlinear modifier introduced by Ozawa. 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.</p>

DOI： 10.1090/tran/9369

arXiv

Other Link： https://www.ams.org/tran/0000-000-00/S0002-9947-2025-09369-3/S0002-9947-2025-09369-3.pdf
Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger operators in dimension three Reviewed Open Access

Haruya Mizutani, Zijun Wan, Xiaohua Yao

Journal of Spectral Theory Vol. 14 ( 4 ) page： 1409 - 1450 2024.9

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Publishing type：Research paper (scientific journal) Publisher：European Mathematical Society - EMS - Publishing House GmbH

This paper is dedicated to investigating the L^{p} -bounds of wave operators W_{\pm}(H,\Delta^{2}) associated with fourth-order Schrödinger operators H=\Delta^{2}+V on \mathbb{R}^{3} with real potentials satisfying |V(x)|\lesssim \langle x\rangle^{-\mu} for some \mu>0 . A recent work by Goldberg and Green (2021) has demonstrated that wave operators W_{\pm}(H,\Delta^{2}) are bounded on L^{p}(\mathbb{R}^{3}) for all 1<p<\infty under the condition that \mu>9 and zero is a regular point of H . In the paper, we aim to further establish endpoint estimates for W_{\pm}(H,\Delta^{2}) in two significant ways. First, we provide counterexamples to illustrate the unboundedness of W_{\pm}(H,\Delta^{2}) on the endpoint spaces L^{1}(\mathbb{R}^{3}) and L^{\infty}(\mathbb{R}^{3}) for non-zero compactly supported potentials V . Second, we establish weak (1,1) estimates for the wave operators W_{\pm}(H,\Delta^{2}) and their dual operators W_{\pm}(H,\Delta^{2})^{*} in the case where zero is a regular point and \mu>11 . These estimates depend critically on the singular integral theory of Calderón–Zygmund on a homogeneous space (X,d\omega) with a doubling measure d\omega .

DOI： 10.4171/jst/522

Open Access
Global in time Strichartz inequalities on asymptotically flat manifolds with temperate trapping Reviewed

Jean-Marc BOUCLET, Haruya MIZUTANI

Mémoires de la Société mathématique de France Vol. 182 2024.9

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Publishing type：Research paper (scientific journal) Publisher：Societe Mathematique de France

DOI： 10.24033/msmf.490
$L^p$-boundedness of wave operators for bi-Schrödinger operators on the line Reviewed

Haruya Mizutani, Zijun Wan, Xiaohua Yao

Advances in Mathematics Vol. 451 page： 109806 2024.8

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Language：English Publishing type：Research paper (scientific journal) Publisher：Elsevier BV

DOI： 10.1016/j.aim.2024.109806
Asymptotic behavior for the long-range nonlinear Schrödinger equation on the star graph with the Kirchhoff boundary condition Reviewed Open Access

Kazuki Aoki, Takahisa Inui, Hayato Miyazaki, Haruya Mizutani, Kota Uriya

Pure and Applied Analysis Vol. 4 ( 2 ) page： 287 - 311 2022.10

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Language：English Publishing type：Research paper (scientific journal) Publisher：Mathematical Sciences Publishers

DOI： 10.2140/paa.2022.4.287
Spectral enclosures for Dirac operators perturbed by rigid potentials Reviewed Open Access

Haruya Mizutani, Nico M. Schiavone

Reviews in Mathematical Physics Vol. 34 ( 8 ) page： 250023 2022.5

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Publishing type：Research paper (scientific journal)

DOI： 10.1142/S0129055X22500234
Kato Smoothing, Strichartz and Uniform Sobolev Estimates for Fractional Operators With Sharp Hardy Potentials Reviewed Open Access

Haruya Mizutani, Xiaohua Yao

Communications in Mathematical Physics Vol. 388 ( 1 ) page： 581 - 623 2021.11

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Publishing type：Research paper (scientific journal) Publisher：Springer Science and Business Media LLC

DOI： 10.1007/s00220-021-04229-1

Other Link： https://link.springer.com/article/10.1007/s00220-021-04229-1/fulltext.html
Remarks on asymptotic order for the linear wave equation with the scale-invariant damping and mass with $L^r$-data Reviewed

Takahisa Inui, Haruya Mizutani

Proceedings of the American Mathematical Society Vol. 149 ( 8 ) page： 3473 - 3484 2021.5

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Language：English Publishing type：Research paper (scientific journal) Publisher：American Mathematical Society ({AMS})

DOI： 10.1090/proc/15481

Other Link： https://www.ams.org/proc/earlyview/#proc15481/.pdf
Scattering and asymptotic order for the wave equations with the scale-invariant damping and mass Reviewed

Takahisa Inui, Haruya Mizutani

Nonlinear Differential Equations and Applications NoDEA Vol. 28 ( 1 ) 2021.1

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Language：English Publishing type：Research paper (scientific journal) Publisher：Springer Science and Business Media {LLC}

DOI： 10.1007/s00030-020-00671-7

Other Link： http://link.springer.com/article/10.1007/s00030-020-00671-7/fulltext.html
Strichartz estimates for Schrödinger equations with slowly decaying potentials Reviewed Open Access

Haruya Mizutani

Journal of Functional Analysis Vol. 279 ( 12 ) page： 108789 - 108789 2020.12

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Publishing type：Research paper (scientific journal) Publisher：Elsevier {BV}

DOI： 10.1016/j.jfa.2020.108789
Scattering theory in homogeneous Sobolev spaces for Schrödinger and wave equations with rough potentials Reviewed

Haruya Mizutani

Journal of Mathematical Physics Vol. 61 ( 9 ) page： 091505 - 091505 2020.9

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Publishing type：Research paper (scientific journal) Publisher：AIP Publishing

DOI： 10.1063/5.0019682
Failure of scattering to standing waves for a Schrödinger equation with long-range nonlinearity on star graph Reviewed

Aoki, K., Inui, T., Mizutani, H.

Journal of Evolution Equations Vol. 21 page： 297 - 312 2020

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Publishing type：Research paper (scientific journal) Publisher：Springer Science and Business Media LLC

DOI： 10.1007/s00028-020-00579-w

Scopus
Wave operators on Sobolev spaces Reviewed

Mizutani, H.

Proceedings of the American Mathematical Society Vol. 148 ( 4 ) page： 1645 - 1652 2020

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.1090/proc/14838

Scopus
Uniform Sobolev estimates for Schrodinger operators with scaling-critical potentials and applications Reviewed Open Access

Mizutani, H.

Analysis and PDE Vol. 13 ( 5 ) page： 1333 - 1369 2020

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Publishing type：Research paper (scientific journal) Publisher：Mathematical Sciences Publishers

DOI： 10.2140/apde.2020.13.1333

Scopus
Uniform resolvent estimates for Schrödinger operator with an inverse-square potential Reviewed Open Access

Mizutani, H., Zhang, J., Zheng, J.

Journal of Functional Analysis Vol. 278 ( 4 ) page： 108350 2020

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.1016/j.jfa.2019.108350

Scopus
Eigenvalue bounds for non-self-adjoint Schrödinger operators with the inverse-square potential Reviewed

Haruya Mizutani

Journal of Spectral Theory Vol. 9 ( 2 ) page： 677 - 709 2018.10

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Publishing type：Research paper (scientific journal)

DOI： 10.4171/jst/260

Web of Science

Scopus
Global-in-time smoothing effects for schrödinger equations with inverse-square potentials Reviewed Open Access

Mizutani, H.

Proceedings of the American Mathematical Society Vol. 146 ( 1 ) page： 295 - 307 2018

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.1090/proc/13729

Web of Science

Scopus
Uniform resolvent and strichartz estimates for Schrödinger equations with critical singularities Reviewed

Bouclet, J.-M., Mizutani, H.

Transactions of the American Mathematical Society Vol. 370 ( 10 ) page： 7293 - 7333 2018

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Publishing type：Research paper (scientific journal)

DOI： 10.1090/tran/7243

Web of Science

Scopus
Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential Reviewed Open Access

Mizutani, H.

Journal of Differential Equations Vol. 263 ( 7 ) page： 3832 - 3853 2017

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.1016/j.jde.2017.05.006

Web of Science

Scopus
Strichartz Estimates for Non-Elliptic Schrödinger Equations on Compact Manifolds Reviewed

Mizutani, H., Tzvetkov, N.

Communications in Partial Differential Equations Vol. 40 ( 6 ) page： 1182 - 1195 2015

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.1080/03605302.2015.1010211

Web of Science

Scopus
Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials Reviewed

Mizutani, H.

Communications on Pure and Applied Analysis Vol. 13 ( 6 ) page： 2177 - 2210 2014

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.3934/cpaa.2014.13.2177

Web of Science

Scopus
Strichartz estimates for Schrödinger equations with variable coefficients and potentials at most linear at spatial infinity Reviewed Open Access

Mizutani, H.

Journal of the Mathematical Society of Japan Vol. 65 ( 3 ) page： 687 - 721 2013

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.2969/jmsj/06530687

Web of Science

Scopus
Strichartz estimates for schrödinger equations with variable coefficients and unbounded potentials Reviewed Open Access

Mizutani, H.

Analysis and PDE Vol. 6 ( 8 ) page： 1857 - 1898 2013

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.2140/apde.2013.6.1857

Web of Science

Scopus
Strichartz Estimates for Schrödinger Equations on Scattering Manifolds Reviewed Open Access

Mizutani, H.

Communications in Partial Differential Equations Vol. 37 ( 2 ) page： 169 - 224 2012

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.1080/03605302.2011.593017

Web of Science

Scopus
Dispersive estimates and asymptotic expansions for Schrödinger equations in dimension one Reviewed Open Access

Mizutani, H.

Journal of the Mathematical Society of Japan Vol. 63 ( 1 ) page： 239 - 261 2011

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Language：English Publishing type：Research paper (scientific journal)

DOI： 10.2969/jmsj/06310239

Web of Science

Scopus

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MISC 8

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

Masaki Kawamoto, Haruya Mizutani

2026.9

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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.

arXiv

Other Link： https://arxiv.org/pdf/2609.12518v1
Scattering Theory For 3D Cubic Damped Magnetic Schrödinger Equation

Luca Fanelli, Haruya Mizutani, Yilin Song, Ying Wang, Jiqiang Zheng

2026.8

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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.

arXiv

Other Link： https://arxiv.org/pdf/2608.11834v1
Modified wave operators for the defocusing cubic nonlinear Schrödinger equation in one space dimension with large scattering data

Masaki Kawamoto, Haruya Mizutani

2025.6

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In the present paper, we construct modified wave operators for the defocusing
cubic nonlinear Schr\"odinger 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
Schr\"odinger 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 smoothness of nonlinearity, and also works
for short-range perturbations of the cubic nonlinearity.

arXiv

Other Link： http://arxiv.org/pdf/2506.01871v1
Limiting absorption principle and absence of eigenvalues for massless Klein-Gordon operators on perturbations of the Minkowski spacetime

Haruya Mizutani

2023.12

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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 H\"{o}lder 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.

arXiv

Other Link： http://arxiv.org/pdf/2312.15675v1
Modified scattering for inhomogeneous nonlinear Schrödinger equations with and without inverse-square potential

Kazuki Aoki, Takahisa Inui, Hayato Miyazaki, Haruya Mizutani, Kota Uriya

2021.1

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We consider the final state problem for the inhomogeneous nonlinear
Schr\"odinger 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 Schr\"odinger equation with the critical
inverse-square potential in three space dimensions, under the radial symmetry.

arXiv

Other Link： http://arxiv.org/pdf/2101.09423v1
Maciej Zworski: Semiclassical Analysis, Grad. Stud. Math., 138, Amer. Math. Soc., 2012年，xii+431ページ．

水谷治哉

数学 Vol. 68 ( 4 ) page： 417 - 420 2016

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Language：Japanese Publisher：一般社団法人日本数学会

CiNii Research
REMARKS ON STRICHARTZ ESTIMATES FOR SCHRODINGER EQUATIONS WITH POTENTIALS SUPERQUADRATIC AT INFINITY (Spectral and Scattering Theory and Related Topics)

MIZUTANI HARUYA

RIMS Kokyuroku Vol. 1891 page： 170 - 182 2014.4

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Language：English Publisher：Kyoto University

CiNii Research

Other Link： http://hdl.handle.net/2433/195789
Remarks on Strichartz estimates for Schrodinger equations on manifolds with ends (Harmonic Analysis and Nonlinear Partial Differential Equations)

MIZUTANI Haruya

RIMS Kokyuroku Bessatsu Vol. 42 page： 33 - 49 2013.8

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Language：English Publisher：Kyoto University

CiNii Research

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Presentations 38

Large data modified wave operators for the 1D defocusing cubic NLS Invited

Haruya Mizutani

Mathematical Aspects of Physics with Non-Self-Adjoint Operators 2026.4.22

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Event date： 2026.4

Presentation type：Oral presentation (invited, special)
Modified wave operators for nonlinear Schrodinger equations with long-range potentials Invited

Haruya Mizutani

Harmonic Analysis and its Interactions @ Nagoya 2026 2026.2.17

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Event date： 2026.2

Presentation type：Oral presentation (invited, special)
Modified wave operators for the defocusing cubic nonlinear Schrödinger equation in one space dimension with large scattering data Invited

12th East Asian Conference in Harmonic Analysis and Applications 2025.8.11

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Event date： 2025.8

Presentation type：Oral presentation (invited, special)
Uniform Sobolev estimates for Schrödinger operators Invited

水谷治哉

TOPICS IN HARMONIC ANALYSIS - Intensive Lectures Series

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Event date： 2017.11
Uniform Sobolev estimates for the resolvent of scaling-critical Schrödinger operators and applications Invited

水谷治哉

The 34th Kyushu Symposium on Partial Differential Equations 2017.1.30
Uniform Sobolev estimates for Schrödinger operators with scaling-critical potentials, Invited

水谷治哉

微分方程式の総合的研究 2017.12.17
Uniform Sobolev estimates for Schrödinger operators with scaling-critical potentials Invited

水谷治哉

Spectral and Scattering Theory and Related Topics 2017.12.6
Uniform Sobolev estimates for Schrödinger operators with scaling-critical potentials Invited

水谷治哉

2017 Taiwan Mathematical Society Annual Meeting 2017.12.9
Uniform resolvent estimates for Schrödinger operators with singular potentials and their applications Invited

水谷治哉

Seminaire: Problemes spectraux en physique mathematique 2016.6.6
Uniform resolvent estimates for a few non-elliptic operators Invited

Haruya Mizutani

2024.6.25

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Language：English
Uniform resolvent and smoothing estimates for the Heisenberg sublaplacian Invited

Himeji Conference on Partial Differential Equations 2025.3.5
Strichartz estimates on manifolds with polynomially growing ends Invited

水谷治哉

Seminaire de lequipe MIP, Toulouse III University 2013.2.19
Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials Invited

水谷治哉

微分方程式の総合的研究 2011.12.18
Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials Invited

水谷治哉

Seminaire Geometrie, EDP et Physique Mathematique 2013.2.13
Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials Invited

水谷治哉

Scattering Theory, An International Workshop in Tokyo 2011.6.22
Strichartz estimates for Schrödinger equations with slowly decaying potentials Invited

2020.12.2
Strichartz estimates for Schrödinger equations with slowly decaying potentials Invited

水谷治哉

Critical exponent and nonlinear evolution equations 2019 2019.2.22
Strichartz estimates for Schrödinger equations with slowly decaying positive poten- tials Invited

水谷治哉

2019 Zhuhai PDE Workshop 2019.3.11
Strichartz estimates for Schrödinger equations with long-range potentials Invited

2023.8.29

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Presentation type：Oral presentation (general)
Strichartz estimates for Schrödinger equations on manifolds with ends Invited

水谷治哉

Lectures on Semi-Classical Analysis 2014.7.4
Remarks on Strichartz estimates for Schrödinger equations with potentials superquadratic at infinity Invited

水谷治哉

Spectral and Scattering Theory and Related Topics 2012.12.14
Remarks on Strichartz estimates for Schrödinger equations on manifolds with ends Invited

水谷治哉

Harmonic Analysis and Nonlinear Partial Differential Equations 2012.7.3
Remarks on endpoint Strichartz estimates for Schrödinger equations with inverse-square potentials Invited

水谷治哉

Nonlinear Partial Differential Equations for Future Applications “Hyperbolic and Dispersive PDE” 2017.7.24
On Strichartz estimates for the Schrödinger equation with decaying potentials Invited

水谷治哉

2018 Taiwan-Japan Workshop on Scattering, Dispersion, Traveling Waves, and Inverse Problems 2018.6.17
Modified wave operators for the defocusing cubic nonlinear Schrödinger equation in one space dimension with large scattering data Invited

Haruya Mizutani

International Workshop on “Fundamental Problems in Mathematical and Theoretical Physics” 2025.7.26
Modified scattering for nonlinear Schrödinger equations with linear long-range potentials Invited

水谷治哉

The 49th Sapporo Symposium on Partial Differential Equations 2024.8.20
Kato–Yajima and Strichartz estimates for fractional and higher-order Schrödinger equations with Hardy potential Invited

Harmonic Analysis and Wave Phenomena 2021.8.10
Global-in-time Strichartz estimates for Schrödinger equations on long-range asymptotically conic manifolds Invited

水谷治哉

Tokyo–Berkeley Mathematics Workshop “Partial Differential Equations and Mathematical Physics” 2017.1.9
Existence of wave operators on Sobolev spaces Invited

水谷治哉

第15回非線型の諸問題 2019.9.12
Eigenvalue bounds for non-self-adjoint Schrödinger operators with the inverse square potential Invited

水谷治哉

Mathematical Aspects of Physics with Non-Self-Adjoint Operators, CIRM 2017.6.3
1次元4 階シュレディンガー方程式に対する波動作用素のL^p有界性 Invited

水谷治哉

偏微分方程式の解の特異性とその周辺 2023.1.12
$L^p$-boundedness of wave operators for fourth-order Schrodinger operators in one space dimension, Invited

Harmonic Analysis and Nonlinear Partial Differential Equations 2022.7.11

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Presentation type：Oral presentation (invited, special)
Uniform Sobolev Estimates for the Resolvent of Scaling-Critical Schrödinger Operators and Applications Invited

水谷治哉

Himeji Conference on Partial Differential Equations 2017.3.6
変数係数シュレディンガー方程式に対するストリッカーツ評価について Invited

水谷治哉

日本数学会年会, 函数方程式論分科会特別講演 2014.3.15
ポテンシャルを伴うシュレディンガー方程式のストリッカーツ評価について Invited

水谷治哉

RIMS共同研究（グループ型A）「量子散乱における順問題と逆問題の新展開」 2022.9.20

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Presentation type：Oral presentation (invited, special)
ポテンシャルを伴うシュレディンガー方程式に対する時間大域的ストリッカーツ評価 Invited

水谷治哉

日本数学会秋季総合分科会函数解析学分科会 2021.9.14

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Presentation type：Oral presentation (invited, special)
Wave operator on Sobolev space Invited

2019.7.26
Uniform Sobolev inequalities for Schrödinger operators and applications Invited

水谷治哉

Interna- tional Conference for the 70th Anniversary of Korean Mathematical Society 2016.10.22

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Research Project for Joint Research, Competitive Funding, etc. 4

一般化シュレディンガー方程式に対する実解析的評価式と非線形分散型方程式への応用

Grant number：21K03325 2021.4 - 2026.3

科学研究費助成事業基盤研究(C) 基盤研究(C)

水谷治哉

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Grant amount：\4030000 （ Direct Cost: \3100000 、 Indirect Cost：\930000 ）

研究計画に基づき、以下の2点について研究を行なった。
1. 分数階および高階のシュレディンガー作用素に臨界なハーディー型ポテンシャルを摂動した自己共役作用素に対する幾つかの大域的性質を証明した。まず、Mourre 理論を用いて加藤-谷島型レゾルベント評価を証明し、これと滑らかな摂動論を組み合わせることで加藤型平滑化作用およびストリッカーツ評価を示した。さらに、レゾルベント方程式を用いて一様ソボレフ評価等も証明した。これは通常のシュレディンガー作用素に対する先行研究を大幅に拡張、統一化する結果である。さらに、今後の研究計画で予定している非線形問題においても本質的な役割を果たすと考えている。この結果は国際論文雑誌「Communications in Mathematical Physics」に掲載済みである。
2. 4階シュレディンガー作用素を空間1次元において考察し、ゼロレゾナンスの存在・非存在に関わらず端点指数を除いて波動作用素がLp有界であることを示した。さらに、一般の対称なMuckenhoupt重みに対する重みつき有界性や端点指数における有界性の破綻および弱い意味での有界性など、既存研究では扱われていないような、より精密な性質も証明した。さらに、ゼロレゾナンスがない場合に、波動作用素の有界性・非有界性について完全な分類を得た。これは今後の研究計画に対して基礎となる結果である。この結果は論文にまとめ現在投稿中である。
Global dynamics of nonlinear wave equations

Grant number：17H02854 2017.4 - 2022.3

Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

Nakanishi Kenji

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Authorship：Coinvestigator(s)

Grant amount：\12090000 （ Direct Cost: \9300000 、 Indirect Cost：\2790000 ）

For solutions of several equations describing nonlinear waves, we gave characterizations on various time-global behavior generated by strong competitions between wave dispersion and nonlinear interactions, and also classifications on solutions by initial data, which corresponds to predictions for the phenomena. Moreover, we developed methods for those mathematical analysis. Specifically, we extended the case where nonlinear interaction remains forever on dispersive waves such that the equation and/or the solution itself may include solitons etc. We also developed a method for the classification, deriving and using uniform and global space-time estimates for the linear equation where one of the interacting waves is regarded as a potential. For a dissipative equation, we gave a complete classification of global behavior for solutions near two solitons.
Quantitative analysis of global behaviors for scaling-critical Schr\"{o}dinger equations

Grant number：17K14218 2017.4 - 2021.3

Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B) Grant-in-Aid for Young Scientists (B)

Mizutani Haruya

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Authorship：Principal investigator Grant type：Competitive

Grant amount：\4030000 （ Direct Cost: \3100000 、 Indirect Cost：\930000 ）

It is widely recognized that invariant properties under the parabolic scaling are very important for the study of global dynamics for the solutions to the Schr\"odinger equation. In this study, we have proved several uniform resolvent estimates for the stationary problem for the Schr\"odinger equation with scaling-critical potentials. As applications, we have also studied global-in-time Strichartz estimates, scattering theory for the associated nonlinear Schr\"odinger equation. Moreover, the uniform resolvent estimates have also been applied to obtain some semiclassical inequalities for the spectral radius of the eigenvalues for non-self-adjoint Schr\"odinger operators with non-symmetric potentials.
Qualitative and quantitative analysis of solution to Schrodinger equations

Grant number：25800083 2013.4 - 2018.3

Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B) Grant-in-Aid for Young Scientists (B)

Mizutani Haruya

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Authorship：Principal investigator Grant type：Competitive

Grant amount：\4030000 （ Direct Cost: \3100000 、 Indirect Cost：\930000 ）

A mathematical analysis of Schrodinger equation, describing the motion of particles in quantum mechanics, has been developed. We have studied effects of the behavior of external electromagnetic potentials and Hamilton flows to the behavior of the solution of Schrodinger equation, through several space-time norm inequalities. In particular, we have constructed a unified approach to deal with both the asymptotically flat metric and unbounded electromagnetic potentials. As a consequence, Strichartz estimates without loss of derivatives have been shown for such a mixed case.

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KAKENHI (Grants-in-Aid for Scientific Research) 3

Space-time estimates for Schroedinger equations with zero resonances

Grant number：26K22272 2026.6 - 2029.3

Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Research (Exploratory)

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Authorship：Principal investigator

Grant amount：\6110000 （ Direct Cost: \4700000 、 Indirect Cost：\1410000 ）
Long-range scattering theory for nonlinear dispersive and wave equations in many body Quantum mechanics

Grant number：26K00612 2026.4 - 2030.3

Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

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Grant amount：\18460000 （ Direct Cost: \14200000 、 Indirect Cost：\4260000 ）
Research on balance and cooperation phenomena of dispersion effect and nonlinear effect in nonlinear dispersive equations

Grant number：24K00529 2024.4 - 2028.3

Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

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Authorship：Coinvestigator(s)

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Academic Activities 15

The 22nd Linear and Nonlinear Waves

Role(s)：Planning, management, etc.

2025.11

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Japan–Spain Forum for PDEs and their applications -Asymptotic analysis and blow-up phenomena-

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Luca Fanelli, Satoshi Masaki, Haruya Mizutani, Nico Michele Schiavone 2025.8

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RIMS Symposia (open), Spectral and Scattering Theory and Related Topics

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2024.12

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The 21st Linear and Nonlinear Waves

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2024.11

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第45 回発展方程式若手セミナー

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水谷治哉 2024.8

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The 20th Linear and Nonlinear Waves

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2023.11

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スペクトル・散乱待兼山シンポジウム

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2023.1

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The 19th Linear and Nonlinear Waves

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2022.11

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The 18th Linear and Nonlinear Waves

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2021.11

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オンラインワークショップ「分散型方程式と実解析」

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2020.12

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The 17th Linear and Nonlinear Waves

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2019.10

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The 16th Linear and Nonlinear Waves

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2018.10 - 2018.11

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RIMS Workshop「Scaling invariant structures for spectral analysis and perturbation theory」

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2018.9

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The 15th Linear and Nonlinear Waves

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2017.11

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京都大学数理解析研究所共同研究「線形および非線形分散型方程式の研究」

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2015.5

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