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Kouichi Taira

Publications and source records attributed to Kouichi Taira.

At least 19 recordsLinked to original sources
arXiv2026

Dispersive estimates for Schrödinger operators with negative Coulomb-like potentials in one dimension

In this paper, we consider the dispersive estimates for Schrödinger operators with Coulomb-like decaying potentials, such as $V(x)=-c|x|^{-μ}$ for $|x|\gg 1$ with $0<μ<2$, in one dimension. As an application, we establish both the standard and orthonormal Strichartz estimates for this model. One of the difficulties here is that perturbation arguments, which are typically applicable to rapidly decaying potentials, are not available. To overcome this, we derive a WKB expression for the spectral density and use a variant of the degenerate stationary phase formula to exploit its oscillatory behavior in the low-energy regime.

math.AP↗
arXiv2025

A microlocal Cauchy problem through a crossing point of Hamiltonian flows

In this paper, we consider $2\times 2$ matrix-valued pseudodifferential equations in which the two characteristic sets intersect with finite contact order. We show that the asymptotic behavior of its solution changes dramatically before and after the crossing point, and provide a precise asymptotic formula. This is a generalization of the previous results for matrix-valued Schrödinger operators and Landau-Zener models. The proof relies on a normal form reduction and a detailed analysis of a simple first-order system.

math.AP↗
arXiv2025

Dispersive estimates and optimality for Schrödinger equations on product cones

In this paper, we study time decay estimates for the Schrödinger propagator on the product cone $(X,g)$, where $X=C(ρ\mathbb{S}^{n-1})=(0,\infty)\times ρ\mathbb{S}^{n-1}$. We prove that the usual dispersive estimate holds when the radius $ρ$ is greater than or equal to 1 and fails otherwise. A part of the former result was already established in a recent paper by Jia-Zhang. The method used here relies purely on harmonic analysis, whereas Jia-Zhang employed microlocal analysis to capture the precise asymptotic behavior of the propagator.

math.AP↗
arXiv2025

Strichartz Estimates for the $(k,a)$-Generalized Laguerre Operators

In this paper, we prove Strichartz estimates for the $(k,a)$-generalized Laguerre operators $a^{-1}\bigl(-|x|^{2-a}Δ_k+|x|^a\bigr)$ which were introduced by Ben Sa\"ıd-Kobayashi-Orsted, and for the operators $|x|^{2-a}Δ_k$. Here $k$ denotes a non-negative multiplicity function for the Dunkl Laplacian $Δ_k$ and $a$ denotes a positive real number satisfying certain conditions. The cases $a=1,2$ were studied previously. We consider more general cases here. The proof depends on symbol-type estimates of special functions and a discrete analog of the stationary phase theorem inspired by the work of Ionescu-Jerison.

math.AP↗
arXiv2024

Equivalence of classical and quantum completeness for real principal type operators on the circle

In this article, we prove that the completeness of the Hamilton flow and essential self-adjointness are equivalent for real principal type operators on the circle. Moreover, we study spectral properties of these operators. The proof is based on the construction of eigenfunctions with non-real eigenvalues which is well-known in scattering theory. Moreover, the relationship between scattering theory and the essential self-adjointness is explained.

math.AP↗
arXiv2024

Local time decay for fractional Schrödinger operators with slowly decaying potentials and a weaker Agmon type estimate in a classically forbidden region

A local time decay estimate of fractional Schrödinger operators with slowly decaying positive potentials are studied. It is shown that its resolvent is smooth near zero and the time propagator has fast local time decay which is very different from very short-range cases. The key element of the proof is to establish a weaker Agmon estimate for a classically forbidden region using exotic symbol calculus. As a byproduct, we prove that the Riesz operator is a pseudodifferential operator with an exotic symbol.

math.AP↗
arXiv2022

Smoothness of the fundamental solution of Schrödinger equations with mild trapping

In this short note, smoothness of the fundamental solution of Schrödinger equations on a complete manifold is studied. It is shown that (1) the fundamental solution is smooth under "mild" trapping conditions; (2) there is a Riemannian manifold which is equal to Euclidean space outside a compact set such that the fundamental solution is not smooth.

math.AP↗
arXiv2022

Essential self-adjointness for the Klein-Gordon type operators on asymptotically static spacetime

Let $X=\mathbb{R}\times M$ be the spacetime, where $M$ is a closed manifold equipped with a Riemannian metric $g$, and we consider a symmetric Klein-Gordon type operator $P$ on $X$, which is asymptotically converges to $\partial_t^2-\triangle_g$ as $|t|\to\infty$, where $\triangle_g$ is the Laplace-Beltrami operator on $M$. We prove the essential self-adjointness of $P$ on $C_0^\infty(X)$. The idea of the proof is closely related to a recent paper by the authors on the essential self-adjointness for Klein-Gordon operators on asymptotically flat spaces.

math-ph↗
arXiv2022

A remark on the essential self-adjointness for Klein-Gordon type operators

Here we discuss a new simplified proof of the essential self-adjointness for formally self-adjoint differential operators of real principal type, previously proved by Vasy (2020) and Nakamura-Taira (2021). For simplicity, here we discuss the second order cases, i.e., Klein-Gordon type operators only.

math-ph↗
arXiv2021

Remarks on the geodesically completeness and the smoothing effect on asymptotically Minkowski spacetimes

In this note, we study a geometric property of asymptotically Minkowski spacetimes and an analytic property of the Klein-Gordon operator. Precisely, our first main results show that asymptotically Minkowski spacetimes are geodesically complete under a null non-trapping condition. Secondly, we prove that Sobolev index of a real principal type estimate used in the previous work is actually optimal.

math-ph↗
arXiv2021

Limiting absorption principle and equivalence of Feynman propagators on asymptotically Minkowski spacetimes

In this paper, we shall show that the limiting absorption principle for the wave operator on the asymptotically Minkowski spacetime. This problem was previously considered by [A. Vasy, J. Spect. Theory, 10,439-461 , (2020)]. Here, we employ a more transparent tool, the Mourre theory and removes an additional condition which is imposed in his paper. Moreover, we also prove that the anti-Feynman propagator defined by Gérard and Wrochna coincides with the outgoing resolvent.

math-ph↗
arXiv2020

Limiting absorption principle on $L^p$-spaces and scattering theory

In this paper, we study the mapping property form $L^p$ to $L^q$ of the resolvent of the Fourier multiplier operators and scattering theory of generalized Schrödinger operators. Though the first half of the subject is studied in [4], we extend their result to away from the duality line and we also study the Hölder continuity of the resolvent.

math.AP↗
arXiv2019

Essential self-adjointness of real principal type operators

We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential self-adjointness under the null non-trapping condition.

math.AP↗
arXiv2019

Uniform bounds of discrete Birman-Schwinger operators

In this note, uniform bounds of the Birman-Schwinger operators in the discrete setting are studied. For uniformly decaying potentials, we obtain the same bound as in the continuous setting. However, for non-uniformly decaying potential, our results are weaker than in the continuous setting. As an application, we obtain unitary equivalence between the discrete Laplacian and the weakly coupled systems.

math-ph↗