• Journal of the Korean Mathematical Society
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• v.55 no.2
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• pp.373-390
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• 2018

In this paper we study the small-data scattering of the d dimensional fractional $Schr{\ddot{o}}dinger$ equations with d = 2, 3, $L{\acute{e}}vy$ index 1 < ${\alpha}$ < 2 and Hartree type nonlinearity $F(u)={\mu}({\mid}x{\mid}^{-{\gamma}}{\ast}{\mid}u{\mid}^2)u$ with max(${\alpha}$, ${\frac{2d}{2d-1}}$) < ${\gamma}{\leq}2$, ${\gamma}$ < d. This equation is scaling-critical in ${\dot{H}}^{s_c}$, $s_c={\frac{{\gamma}-{\alpha}}{2}}$. We show that the solution scatters in $H^{s,1}$ for any s > $s_c$, where $H^{s,1}$ is a space of Sobolev type taking in angular regularity with norm defined by ${\parallel}{\varphi}{\parallel}_{H^{s,1}}={\parallel}{\varphi}{\parallel}_{H^s}+{\parallel}{\nabla}_{{\mathbb{S}}{\varphi}}{\parallel}_{H^s}$. For this purpose we use the recently developed Strichartz estimate which is $L^2$-averaged on the unit sphere ${\mathbb{S}}^{d-1}$ and utilize $U^p-V^p$ space argument.

• Bulletin of the Korean Chemical Society
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• v.31 no.12
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• pp.3573-3578
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• 2010

Recently it is suggested that it may be possible to obtain the approximate or exact bound state solutions of nonrelativistic Schr$\ddot{o}$dinger equation from relativistic Klein-Gordon equation, which seems to be counter-intuitive. But the suggestion is further elaborated to propose a more detailed method for obtaining nonrelativistic solutions from relativistic solutions. We demonstrate the feasibility of the proposed method with the Morse potential as an example. This work shows that exact relativistic solutions can be a good starting point for obtaining nonrelativistic solutions even though a rigorous algebraic method is not found yet.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.825-844
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• 2020

We obtain infinitely many solutions for a class of fractional Schrödinger equation, where the nonlinearity is superquadratic or involves a combination of superquadratic and subquadratic terms at infinity. By using some weaker conditions, our results extend and improve some existing results in the literature.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.175-190
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• 1996

In this paper we establish the existence of the conditional Feynman integral of certain functions which are not in the Banach algebra S of functions on Wiener space which are a kind of stochastic Fourier transform of complex Borel measures on $L^2[a, b]$. This result is used to provide the fundamental solution for the Schr$\ddot{o}$dinger equation for the forced harmonic potential.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.615-640
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• 2016

We consider the fourth-order $Schr{\ddot{o}}dinger$ equation with focusing inhomogeneous nonlinearity ($-{\mid}x{\mid}^{-2}{\mid}u{\mid}^{\frac{4}{n}}u$) in high space dimensions. Extending Glassey's virial argument, we show the finite time blowup of solutions when the energy is negative.

• Journal of the Korean Physical Society
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• v.73 no.9
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• pp.1211-1218
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• 2018

Within the framework of the modified factorization method, we solve the $Schr{\ddot{o}}dinger$ equation with the modified Yukawa potential. The energy spectrum is obtained using the Pekeris approximation scheme for the centrifugal term. The thermodynamic properties, including the vibrational partition function, vibrational mean energy, vibrational mean free energy, vibrational specific heat capacity and vibrational entropy, are calculated. As a special case, we compare our result with that work of Dong [Int. J. Quant. Chem. 107, 366 (2007)] and find good agreement.

• ETRI Journal
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• v.20 no.4
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• pp.361-379
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• 1998

We present an improved transfer matrix algorithm which can be used in solving general n-band effective-mass $Schr{\ddot{o}}dinger$ equation for quantum well structures with arbitrary shaped potential profiles(where n specifies the number of bands explicitly included in the effective-mass equation). In the proposed algorithm, specific formulas are presented for the three-band (the conduction band and the two heavy- and light-hole bands) and two-band (the heavy- and light-hole bands) effective-mass eigensystems. Advantages of the present method can be taken in its simple and unified treatment for general $n{\times}n$ matrix envelope-function equations, which requires relatively smaller computation efforts as compared with existing methods of similar kind. As an illustration of application of the method, numerical computations are performed for a single GaAs/AlGaAs quantum well using both the two-band and three-band formulas. The results are compared with those obtained by the conventional variational procedure to assess the validity of the method.

• Journal of the Korean Physical Society
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• v.73 no.9
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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.

• Applied Science and Convergence Technology
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• v.24 no.5
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• pp.156-161
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• 2015

Application of magnetic fields is important to characterize the carrier dynamics in semiconductor quantum structures. We performed photoluminescence (PL) measurements from an InGaP-AlInGaP single quantum well under pulsed magnetic fields to 50 T. The zero field interband PL transition energy matches well with the self-consistent Poisson-$Schr{\ddot{o}}dinger$ equation. We attempted to analyze the dimensionality of the quantum well by using the diamagnetic shift of the magnetoexciton. The real quantum well has finite thickness that causes the quasi-two-dimensional behavior of the exciton diamagnetic shift. The PL intensity diminishes with increasing magnetic field because of the exciton motion in the presence of magnetic field.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1347-1372
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• 2020

We study bifurcation for the following fractional Schrödinger equation $$\{\left.\begin{eqnarray}(-{\Delta})^su+V(x)u&=&{\lambda}f(u)&&{\text{in}}\;{\Omega}\\u&>&0&&{\text{in}}\;{\Omega}\\u&=&0&&{\hspace{32}}{\text{in}}\;{\mathbb{R}}^n{\backslash}{\Omega}\end{eqnarray}\right$$ where 0 < s < 1, n > 2s, Ω is a bounded smooth domain of ℝⁿ, (-∆)^s is the fractional Laplacian of order s, V is the potential energy satisfying suitable assumptions and λ is a positive real parameter. The nonlinear term f is a positive nondecreasing convex function, asymptotically linear that is $\lim_{t{\rightarrow}+{\infty}}\;{\frac{f(t)}{t}}=a{\in}(0,+{\infty})$. We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schrödinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.