• 대한수학회지
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• 제47권2호
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• pp.425-443
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• 2010

We investigate the Schr$\ddot{o}$dinger type operator $H_2\;=\;(-\Delta_{\mathbb{H}^n})^2+V^2$ on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}^n}$ is the sublaplacian and the nonnegative potential V belongs to the reverse H$\ddot{o}$lder class $B_q$ for $q\geq\frac{Q}{2}$, where Q is the homogeneous dimension of $\mathbb{H}^n$. We shall establish the estimates of the fundamental solution for the operator $H_2$ and obtain the $L^p$ estimates for the operator $\nabla^4_{\mathbb{H}^n}H^{-1}_2$, where $\nabla_{\mathbb{H}^n}$ is the gradient operator on $\mathbb{H}^n$.

• 대한수학회보
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• 제56권5호
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• pp.1117-1127
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• 2019

Let ${\mathcal{L}}_2=(-{\Delta})^2+V^2$ be the $Schr{\ddot{o}}dinger$ type operator, where nonnegative potential V belongs to the reverse $H{\ddot{o}}lder$ class $RH_s$, s > n/2. In this paper, we consider the operator $T_{{\alpha},{\beta}}=V^{2{\alpha}}{\mathcal{L}}^{-{\beta}}_2$ and its conjugate $T^*_{{\alpha},{\beta}}$, where $0<{\alpha}{\leq}{\beta}{\leq}1$. We establish the $(L^p,\;L^q)$-boundedness of operator $T_{{\alpha},{\beta}}$ and $T^*_{{\alpha},{\beta}}$, respectively, we also show that $T_{{\alpha},{\beta}}$ is bounded from Hardy type space $H^1_{L_2}({\mathbb{R}}^n)$ into $L^{p_2}({\mathbb{R}}^n)$ and $T^*_{{\alpha},{\beta}}$ is bounded from $L^{p_1}({\mathbb{R}}^n)$ into BMO type space $BMO_{{\mathcal{L}}1}({\mathbb{R}}^n)$, where $p_1={\frac{n}{4({\beta}-{\alpha})}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})}}$.

• 충청수학회지
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• 제13권1호
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• pp.119-130
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• 2000

We give alternative proof of the existence theorem for certain elliptic systems describing competing interactions with nonlinear di usion. The existence of positive solution depends on the sign of the principal eigenvalue of suitable operators of Schr$\ddot{o}$dinger type. If the sign of such operators are both positive, then system has a positive solution. The main tool employed is the fixed point index of compact operator on positive cones.

• 대한수학회논문집
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• 제12권4호
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• pp.869-880
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• 1997

In this paper, we give sufficient conditions of certain elliptic systems involving competing iteractions with nonlinear diffusion rates. The existence of positive solution depends on the sign of the first eigenvalue of operators of Schr$\ddot{o}$dinger type. More precisely, if the sign of such operators are either both positive or both negative, then system has a positive solution. The main tool employed is the fixed point index of compact operator on positive cones.

• 대한수학회보
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• 제55권2호
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• pp.533-543
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• 2018

Under the symbol ${\Omega}$ is a combination of ${\phi}_i$ ($i=1,2,3,{\ldots},n$) which has a suitable growth condition, for dimension n = 2 and $n{\geq}3$, when the initial data f belongs to homogeneous Sobolev space, we obtain the global $L^q$ estimate for maximal operators generated by operators family $\{S_{t,{\Omega}}\}_{t{\in}{\mathbb{R}}}$ associated with solution to dispersive equations, which extend some results in [27].

• 대한수학회지
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• 제57권1호
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• pp.89-111
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• 2020

In the present paper, we give some characterization of the L₂ maximal estimate for the operator P^t_a,γf(Γ(x, t)) along curve with complex time, which is defined by $$P^t_{a,{\gamma}}f({\Gamma}(x,t))={\displaystyle\smashmargin{2}{\int\nolimits_{\mathbb{R}}}}\;e^{i{\Gamma}(x,t){\xi}}e^{it{\mid}{\xi}{\mid}^a}e^{-t^{\gamma}{\mid}{\xi}{\mid}^a}{\hat{f}}({\xi})d{\xi}$$, where t, γ > 0 and a ≥ 2, curve Γ is a function such that Γ : ℝ×[0, 1] → ℝ, and satisfies Hölder's condition of order σ and bilipschitz conditions. The authors extend the results of the Schrödinger type with complex time of Bailey [1] and Cho, Lee and Vargas [3] to Schrödinger operators along the curves.

• Journal of the Korean Physical Society
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• 제73권9호
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• pp.1269-1278
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• 2018

Using the thawed Gaussian wave packets [E. J. Heller, J. Chem. Phys. 62, 1544 (1975)] and the adaptive reinitialization technique employing the frame operator [L. M. Andersson et al., J. Phys. A: Math. Gen. 35, 7787 (2002)], a trajectory-based Gaussian wave packet method is introduced that can be applied to scattering and time-dependent problems. This method does not require either the numerical multidimensional integrals for potential operators or the inversion of nearly-singular matrices representing the overlap of overcomplete Gaussian basis functions. We demonstrate a possibility that the method can be a promising candidate for the time-dependent $Schr{\ddot{o}}dinger$ equation solver by applying to tunneling, high-order harmonic generation, and above-threshold ionization problems in one-dimensional model systems. Although the efficiency of the method is confirmed in one-dimensional systems, it can be easily extended to higher dimensional systems.

• 대한수학회보
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• 제58권1호
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• pp.235-251
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• 2021

We consider the Schrödinger type operator ��k = (-∆)k+Vk on ℝn(n ≥ 2k + 1), where k = 1, 2 and the nonnegative potential V belongs to the reverse Hölder class RHs with n/2 < s < n. In this paper, we establish the (Lp, Lq)-boundedness of the higher order Riesz transform T��,�� = V2��∇2��-��2 (0 ≤ �� ≤ 1/2 < �� ≤ 1, �� - �� ≥ 1/2) and its adjoint operator T∗��,�� respectively. We show that T��,�� is bounded from Hardy type space $H^1_{\mathcal{L}_2}({\mathbb{R}}_n)$ into Lp2 (ℝn) and T∗��,�� is bounded from ��p1 (ℝn) into BMO type space $BMO_{\mathcal{L}_1}$ (ℝn) when �� - �� > 1/2, where $p_1={\frac{n}{4({\beta}-{\alpha})-2}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})+2}}$. Moreover, we prove that T��,�� is bounded from $BMO_{\mathcal{L}_1}({\mathbb{R}}_n)$ to itself when �� - �� = 1/2.