arXiv subjects

Artbazar Galtbayar

Publications and source records attributed to Artbazar Galtbayar.

4 recordsLinked to original sources

arXiv2026

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

We prove that wave operators of scattering theory for fourth order Schrödinger operators $H = Δ^2 + V (x)$ 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 $(Δ^2 + V(x))u(x)=0$ 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 $f(Δ^2)$.

math-ph↗

arXiv2023

The $L^p$-boundedness of wave operators for fourth order Schrödinger operators on ${\mathbb R}^4$

We prove that the wave operators of scattering theory for the fourth order Schrödinger operators $Δ^2 + V(x)$ in ${\mathbb R}^4$ are bounded in $L^p({\mathbb R}^4)$ for the set of $p$'s of $(1,\infty)$ depending on the kind of spectral singularities of $H$ at zero which can be described by the space of bounded solutions of $(Δ^2 + V(x))u(x)=0$.

math-ph↗

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↗

arXiv2019

On the approximation by regular potentials of Schrödinger operators with point interactions

We prove that the wave operators for Schrödinger operators with multi-center local point interactions are the scaling limits of the ones for Schrödinger operators with regular potentials. We simultaneously present a proof of the corresponding well-known result for the resolvent which substantially simplifies the one by Albeverio et al.

math-ph↗