• Journal of the Chungcheong Mathematical Society
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• v.15 no.2
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• pp.25-34
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• 2003

We prove a local unique continuation for Schr$\ddot{o}$dinger equations with time independent coefficients. The method of proof combines a technique of Fourier-Gauss transformation and a Carleman inequality for parabolic operator.

• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.547-572
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• 2010

We study existence of positive solutions of the classical nonlinear Schr$\ddot{o}$dinger equation $-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$. In fact, we consider the following more general quasi-linear Schr$\ddot{o}$odinger equation $-div(|{\nabla}u|^{m-2}{\nabla}u)\;+\;V(x)u^{m-1}$ $-f(x,\;u)\;-\;H(x)u^{m^*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$, where m $\in$ (1, n) is a positive number and $m^*\;:=\;\frac{mn}{n-m}\;>\;0$, is the corresponding critical Sobolev embedding number in $\mathbb{R}^n$. Under appropriate conditions on the functions V(x), f(x, u) and H(x), existence and non-existence results of positive solutions have been established.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.15 no.5
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• pp.3107-3113
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• 2014

We present a quantitative analysis of a dipole antenna and its characteristics from the viewpoint of quantum mechanics. The method makes use of a Maxwell equation used in an existing antenna propagation formula. This includes radiation resistance, input reactance, and antenna efficiency as functions of frequency and antenna length. Particular attention is paid to the Schr$\ddot{o}$odinger equation. We accomplish E-field and H-field analyses of a dipole antenna by combining the Maxwell and Schr$\ddot{o}$odinger wave equations. When comparing the existing Maxwell wave equation with the Schr$\ddot{o}$odinger wave equation, quantum-electric movement is more accurate than using the Maxwell wave equation alone.