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Kenji Yajima

Publications and source records attributed to Kenji Yajima.

18 recordsLinked to original sources
arXiv2025

The $L^p$-boundedness of wave operators for 4-th order Schr\"odinger operators on $\mathbb{R}^2$, I. Regular case

We prove that wave operators of scattering theory for fourth order Schr\"odinger operators $H = \Delta^2 + V (x)$ on $\mathbb{R}^2$ with real potentials $V(x)$ such that $\langle x \rangle^3 V(x) \in L^{\frac43}(\mathbb{R}^2)$ and $\langle x \rangle^{10+\varepsilon} V(x) \in L^1 (\mathbb{R}^2)$ for an $\varepsilon>0$, $\langle x \rangle=(1+|x|^2)^{\frac12}$, are bounded in $L^p (\mathbb{R}^2)$ for all $1<p<\infty$ if $H$ is regular at zero in the sense that there are no non-trivial solutions to $(\Delta^2 + V(x))u(x)=0$ such that $\langle x \rangle^{-1} u(x) \in L^\infty(\mathbb{R}^2)$ and if positive eigenvalues are absent from $H$. This reduces $L^p$-mapping properties of functions $f(H)$ of $H$ to those of Fourier multipliers $f(\Delta^2)$.

math-ph↗
arXiv2024

Boundedness of energy for N-body Schr\"odinger equations with time dependent small potentials

We prove that Sobolev norms of solutions to time dependent Schr\"odinger equations for $d$-dimensional $N$-partcles interacting via time dependent two body potentials are bounded in time if certain Lebesgue norms of the potentials are small uniformly in time. The proof uses the scattering theory in the extended phase space which proves that all particles scatter freely in the remote past and far future.

math.AP↗
arXiv2022

The $L^p$-boundedness of wave operators for four dimensional Schr\"odinger operators with threshold resonances

We prove that the low energy parts of the wave operators $W_\pm$ for Schr\"odinger operators $H = -\lap + V(x)$ on $\R^4$ are bounded in $ L^p(\R^4)$ for $1<p\leq 2$ and are unbounded for $2<p\leq \infty$ if $H$ has resonances at the threshold. If $H$ has eigenfunctions only at the threshold, it has recently been proved that they are bounded in $L^p(\R^4)$ for $1\leq p<4$ in general and for $1\leq p<\infty$ if all threshold eigenfunctions $\ph$ satisfy $\int_{\R^4}x_j V(x) \ph(x)dx=0$ for $1\leq j\leq 4$. We prove in this case that they are unbounded in $L^p(\R^4)$ for $4<p<\infty$ unless the latter condition is satisfied. It is long known that the high energy parts are bounded in $L^p(\R^4)$ for all $1\leq p\leq \infty$ and that the same holds for $W_\pm$ if $H$ has no eigenfunctions nor resonances at the threshold.

math-ph↗
arXiv2021

The $L^p$-boundedness of wave operators for two dimensional Schrödinger operators with threshold singularities

We generalize the recent result of Erdo{\u g}an, Goldberg and Green on the $L^p$-boundedness of wave operators for two dimensional Schrödinger operators and prove that they are bounded in $L^p(\R^2)$ for all $1<p<\infty$ if and only if the Schrödinger operator possesses no $p$-wave threshold resonances, viz. Schrödinger equation $(-\lap + V(x))u(x)=0$ possesses no solutions which satisfy $u(x)= (a_1x_1+a_2 x_2)|x|^{-2}+ o(|x|^{-1})$ as $|x|\to \infty$ for an $(a_1, a_2) \in \R^2\setminus \{(0,0)\}$ and, otherwise, they are bounded in $L^p(\R^2)$ for $1<p\leq 2$ and unbounded for $2<p<\infty$. We present also a new proof for the known part of the result.

math.AP↗
arXiv2020

A solvable model of the breakdown of the adiabatic approximation

Let $L\geq0$ and $0<\varepsilon\ll1$. Consider the following time-dependent family of $1D$ Schrödinger equations with scaled and translated harmonic oscillator potentials $ i\varepsilon\partial_t u_{\varepsilon}=-\tfrac12\partial_x^2u_{\varepsilon}+V(t,x)u_{\varepsilon}$, $u_{\varepsilon}(-L-1,x)=π^{-1/4}\exp(-x^2/2) $, where $ V(t,x)= (t+L)^2x^2/2$, $t<-L$, $ V(t,x)= 0$, $-L\leq t \leq L$, and $ V(t,x)=(t-L)^2x^2/2$, $t>L$. The initial value problem is explicitly solvable in terms of Bessel functions. Using the explicit solutions we show that the adiabatic theorem breaks down as $\varepsilon\to 0$. For the case $L=0$ complete results are obtained. The survival probability of the ground state $π^{-1/4}\exp(-x^2/2)$ at microscopic time $t=1/\varepsilon$ is $1/\sqrt{2}+O(\varepsilon)$. For $L>0$ the framework for further computations and preliminary results are given.

math-ph↗
arXiv2020

$L^p$-boundedness of wave operators for 2D Schrödinger operators with point interactions

For two dimensional Schrödinger operator $H$ with point interactions, We prove that wave operators of scattering for the pair $(H,H_0)$, $H_0$ being the free Schrödinger operator, are bounded in the Lebesgue space $L^p(\R^2)$ for $1<p<\infty$ if and only if there are no generalized eigenfunctions of $Hu(x)=0$ which satisfy $u(x)= C|x|^{-1}+ o(|x|^{-1})$ as $|x|\to \infty$, $C\not=0$. Otherwise they are bounded for $1<p\leq 2$ and unbounded for $2<p<\infty$.

math-ph↗
arXiv2018

Instability of resonances under Stark perturbations

Let $H^{\varepsilon}=-\frac{d^2}{dx^2}+\varepsilon x +V$, $\varepsilon\geq0$, on $L^2(\mathbf{R})$. Let $V=\sum_{k=1}^Nc_k|ψ_k\rangle\langleψ_k|$ be a rank $N$ operator, where the $ψ_k\in L^2(\mathbf{R})$ are real, compactly supported, and even. Resonances are defined using analytic scattering theory. The main result is that if $ζ_n$, ${\rm Im}ζ_n<0$, are resonances of $H^{\varepsilon_n}$ for a sequence $\varepsilon_n\downarrow0$ as $n\to\infty$ and $ζ_n\toζ_0$ as $n\to\infty$, ${\rm Im}ζ_0<0$, then $ζ_0$ is \emph{not} a resonance of $H^0$.

math-ph↗
arXiv2018

Two dimensional Schr{\" o}dinger operators with point interactions: threshold expansions, zero modes and $L^p$-boundedness of wave operators

We study the threshold behaviour of two dimensional Schr{\" o}dinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the operator is of regular type, or it has singularities associated with $s$ or p-wave resonances or even with an embedded eigenvalue at zero, for whose existence we give necessary and sufficient conditions. An embedded eigenvalue at zero may appear only if we have at least three centres. When the operator is of regular type we prove that the wave operators are bounded in $L^p(\R^2)$ for all $1<p<\infty$. With a single center we always are in the regular type case.

math.SP↗
arXiv2016

On wave operators for Schrödinger operators with threshold singuralities in three dimensions

We show that wave operators for three dimensional Schrödinger operators $H=-Δ+ V$ with threshold singularities are bounded in $L^1({\mathbb R}^3)$ if and only if zero energy resonances are absent from $H$ and the existence of zero energy eigenfunctions does not destroy the $L^1$-boundedness of wave operators for $H$ with the regular threshold behavior. We also show in this case that they are bounded in $L^p({\mathbb R}^3)$ for all $1\leq p \leq \infty$ if all zero energy eigenfunctions $ϕ(x)$ have vanishing first three moments: $\int_{{\mathbb R}^3} x^αV(x)ϕ(x)dx=0$, $|α|=0,1,2$.

math-ph↗
arXiv2016

Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities

We consider the continuity property in Lebesgue spaces $L^p(\R^m)$ of wave operators $W_\pm$ of scattering theory for Schrödinger operator $H=-\lap + V$ on $\R^m$, $|V(x)|\leq C\ax^{-δ}$ for some $δ>2$ when $H$ is of exceptional type, i.e. $\Ng=\{u \in \ax^{-s} L^2(\R^m) \colon (1+ (-\lap)^{-1}V)u=0 \}\not=\{0\}$ for some $1/2 m/2$ more precise and prove in particular that these conditions are also necessary for the stated properties of $W_\pm$. We also prove that, for $m=3$, $W_\pm$ are bounded in $L^p(\R^3)$ for $1<p<3$ and that the same holds for $1<p<\infty$ if and only if all $\f\in \Ng$ satisfy $\int_{\R^3}V\f dx=0$ and $\int_{\R^3} x_i V\f dx=0$, $i=1, 2, 3$, simultaneously.

math-ph↗
arXiv2015

Existence and regularity of propagators for multi-particle Schrödinger equations in external fields

We prove that the Schrödinger equation for N number of particles in the time dependent electro-magnetic field generates a unique unitary propagator on the state space under the condition that the field is smooth and moderately but almost critically increases at the spatial infinity such that propagator for every single particle in the field enjoys the time local Strichartz estimates and that the time dependent inter-particle potentials are almost critically singular for Hamiltonians to have a unique selfadjoint realization at every fixed time. We also show that the domain of definition of the quantum harmonic oscillator is invariant under the propagator and, for initial states in that space, solutions are continuously differentiable function of time variable with values in the state space under the additional assumption that the time derivative of inter-particle potentials exists almost everywhere and it increases the spatial singularities by at most the inverse power of the inter-particle distances. New estimates of Strichartz type for the propagator for N independent particles in the field are proved and used for the proof.

math-ph↗
arXiv2015

Wave Operators for Schrödinger Operators with Threshold Singuralities, Revisited

The continuity property in the Sobolev space $W^{k,p}({\bf R}^m)$ of wave operators of scattering theory for $m$-dimensional single-body Schrödinger operator is considered when the resolvent of the operator has singularities at the bottom of the continuous spectrum. It is shown that they are continuous in $W^{k,p}({\bf R}^m)$, $0\leq k \leq 2$, for $1 3$ if $m=3$ and, for $1 m/2$ if $m\geq 5$. This extends the previously known interval of $p$ for the continuity, $3/2<p<3$ for $m=3$ and $m/(m-2)<p<m/2$ for $m\geq 5$. The formula which represents the integral kernel of the resolvent of the even dimensional free Schödinger operator as the superposition of exponential-polynomial like functions substantially simplifies the proof of the previous paper when $m \geq 6$ is even.

math-ph↗
arXiv2013

Schrödinger equations with time-dependent strong magnetic fields

We consider d-dimensional time dependent Schrödinger equations on the Hilbert space of square integrable functions. We assume magnetic and scalar potentials are almost critically singular with respect to spatial variables both locally and at infinity for the fixed time Schrödinger operator H(t) to be essentially self-adjoint on the compactly supported smooth functions. In particular, if magnetic field B(t,x) is very strong at infinity, the scalar potential can explode to negative infinity faster than quadratic functions. We show that equations uniquely generate unitary propagators under suitable conditions on the size and singularities of time derivatives of potentials. Basic tools are Kato's abstract theory for evolution equations, Iwatsuka's identity which rewrites H(t) to an elliptic differential operator in which B(t,x) appears explicitly, and a new diamagnetic like inequality.

math-ph↗
arXiv2006

The $L^p$ boundedness of wave operators for Schrödinger operators with threshold singularities II. Even dimensional case

In this paper we consider the wave operators $W_{\pm}$ for a Schrödinger operator $H$ in ${\bf{R}}^n$ with $n\geq 4$ even and we discuss the $L^p$ boundedness of $W_{\pm}$ assuming a suitable decay at infinity of the potential $V$. The analysis heavily depends on the singularities of the resolvent for small energy, that is if 0-energy eigenstates exist. If such eigenstates do not exist $W_{\pm}: L^p \to L^p$ are bounded for $1 \leq p \leq \infty$ otherwise this is true for $ \frac{n}{n-2} < p < \frac{n}{2} $. The extension to Sobolev space is discussed.

math-ph↗