• Journal of the Chungcheong Mathematical Society
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• v.28 no.4
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• pp.525-535
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• 2015

In this paper, we study on spinors with two hyperbolic components. Firstly, we express the hyperbolic spinor representation of a spacelike curve dened on an oriented (spacelike or time-like) surface in Minkowski space ${\mathbb{R}}^3_1$. Then, we obtain the relation between the hyperbolic spinor representation of the Frenet frame of the spacelike curve on oriented surface and Darboux frame of the surface on the same points. Finally, we give one example about these hyperbolic spinors.

• Journal of the Korean Mathematical Society
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• v.42 no.6
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• pp.1169-1186
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• 2005

We study the properties of the transversal Killing and twistor spinors for a Riemannian foliation with a transverse spin structure. And we investigate the relations between them. As an application, we give a new lower bound for the eigenvalues of the basic Dirac operator by using the transversal twistor operator.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.949-966
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• 2009

We proved in [10] that Friedrich's estimate [5] for the first eigenvalue of the Dirac operator can be improved when a Codazzi tensor exists. In the paper we further prove that his estimate can be improved as well via a well-chosen divergencefree symmetric tensor. We study the geometric implication of the new first eigenvalue estimates over Sasakian spin manifolds and show that some particular types of spinors appear as the limiting case.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1347-1370
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• 2016

On a closed eta-Einstein Sasakian spin manifold of dimension $2m+1{\geq}5$, $m{\equiv}0$ mod 2, we prove a new eigenvalue estimate for the Dirac operator. In dimension 5, the estimate is valid without the eta-Einstein condition. Moreover, we show that the limiting case of the estimate is attained if and only if there exists such a pair (${\varphi}_{{\frac{m}{2}}-1}$, ${\varphi}_{\frac{m}{2}}$) of spinor fields (called Sasakian duo, see Definition 2.1) that solves a special system of two differential equations.

• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1551-1567
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• 2019

In this paper, Seiberg-Witten-like equations are written down on 3-manifolds. Then, it has been proved that the $L^2$-solutions of these equations are trivial on ${\mathbb{R}}^3$. Finally, a global solution is obtained on the strictly pseudoconvex CR-3 manifolds for a given constant negative scalar curvature.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.669-681
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• 1998

In this paper, we prove that on the complete Kahler manifold, if ${\rho}(x){\geq}-\frac{1}{2}{\lambda}_0$ and either ${\rho}(x_0)>-\frac{1}{2}{lambda}_0$ at some point $x_0$ or Vol(M)=${\infty}$, then the Clifford $L^2$ cohomology group $L^2{\mathcal H}^{\ast}(M,S)$ is trivial, where $\rho(x)$ is the least eigenvalue of ${\mathcal R}_x + \bar{{\mathcal R}}(x)\;and\;{\lambda}_0$ is the infimum of the spectrum of the Laplacian acting on $L^2$-functions on M.

• Bulletin of the Korean Chemical Society
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• v.34 no.1
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• pp.179-187
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• 2013

We describe newly developed software named KPACK for relativistic electronic structure computation of molecules containing heavy elements that enables the two-component ab initio calculations in Kramers restricted and unrestricted formalisms in the framework of the relativistic effective core potential (RECP). The spin-orbit coupling as relativistic effect enters into the calculation at the Hartree-Fock (HF) stage and hence, is treated in a variational manner to generate two-component molecular spinors as one-electron wavefunctions for use in the correlated methods. As correlated methods, KPACK currently provides the two-component second-order M${\o}$ller-Plesset perturbation theory (MP2), configuration interaction (CI) and complete-active-space self-consistent field (CASSCF) methods. Test calculations were performed for the ground states of group-14 elements, for which the spin-orbit coupling greatly influences the determination of term symbols. A categorization of three procedures is suggested for the two-component methods on the basis of spin-orbit coupling manifested in the HF level.

• Bulletin of the Korean Chemical Society
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• v.12 no.6
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• pp.699-705
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• 1991

Procedures to perform reliable relativistic self-consistent-field (RSCF) calculations are described. Using light atoms and molecules, it is demonstrated that the present method always yields correct nonrelativistic limit by employing a sufficiently large value for the speed of light in RSCF calculations. Many problems associated with analytic expansions of the Dirac equations can be computationally avoided by kinetically balancing the basis sets for large and small component spinors. Results of RSCF calculations for Ne, Kr, $H_2$, and LiH indicate very small relativistic effects for these systems as expected. Trends found is these molecules, however, may be useful in understanding relativistic effects for molecules with similar valence electronic structures and heavier atoms.

• Bulletin of the Korean Chemical Society
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• v.16 no.6
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• pp.547-552
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• 1995

As an extension to the Kramers' restricted Hartree-Fock (KRHF) method [J. Comp. Chem., 13, 595 (1992)], we have implemented the Kramers' restricted configuration interaction (KRCI) program in order to calculate excited states as well as the ground state of polyatomic molecules containing heavy atoms. This KRCI is based on determinants composed of the two-component molecular spinors which are generated from KRHF calculations. The Hamiltonian employed in the KRHF and KRCI methods contains most of all the important relativistic effects including spin-orbit terms through the use of relativistic effective core potentials (REP). The present program which is limited to a small configuration space has been tested for a few atoms and molecules. Excitation energies of the group 14 and 16 elements calculated using the present KRCI program are in good accordance with the spectroscopic data. Calculated excitation energies for many Rydberg states of K and Cs indicate that spin-orbit terms in the REP, which are derived for the ground state, are also reliable for the description of highly excited states. The electronic states of the polyatomic molecule CH3I are probed from the molecular region to the dissociation limit. Test calculations demonstrate that the present KRCI is a useful method for the description of potential energy surface of polyatomic molecules containing heavy atoms.