• Title, Summary, Keyword: Dirac operator

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RIQUIER AND DIRICHLET BOUNDARY VALUE PROBLEMS FOR SLICE DIRAC OPERATORS

  • Yuan, Hongfen
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.149-163
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    • 2018
  • In recent years, the study of slice Dirac operators has attracted more and more attention in the literature. In this paper, Almansitype decompositions for null solutions to the iterated slice Dirac operator and the generalized slice Dirac operator are obtained without a star-like domain centered at the origin. As applications, we investigate Riquier type problems and Dirichlet type problems in the theory of slice monogenic functions.

INVERSE PROBLEM FOR INTERIOR SPECTRAL DATA OF THE DIRAC OPERATOR

  • Mochizuki, Kiyoshi;Trooshin, Igor
    • Communications of the Korean Mathematical Society
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    • v.16 no.3
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    • pp.437-443
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    • 2001
  • In this paper the inverse problems for the Dirac Operator are studied. A set of values of eigenfunctions in some internal point and spectrum are taken as a data. Uniqueness theorems are obtained. The approach that was used in the investigation of inverse problems for interior spectral data of the Sturm-Liouville operator is employed.

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THE SYMMETRY OF spin DIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS

  • HONG, KYUSIK;SUNG, CHANYOUNG
    • Journal of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1037-1049
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    • 2015
  • It is well-known that the spectrum of a $spin^{\mathbb{C}}$ Dirac operator on a closed Riemannian $spin^{\mathbb{C}}$ manifold $M^{2k}$ of dimension 2k for $k{\in}{\mathbb{N}}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $M^{2p}_1{\times}M^{2q+1}_2$ with a product $spin^{\mathbb{C}}$ structure for $p{\geq}1$, $q{\geq}0$, the spectrum of a $spin^{\mathbb{C}}$ Dirac operator given by a product connection is symmetric if and only if either the $spin^{\mathbb{C}}$ Dirac spectrum of $M^{2q+1}_2$ is symmetric or $(e^{{\frac{1}{2}}c_1(L_1)}{\hat{A}}(M_1))[M_1]=0$, where $L_1$ is the associated line bundle for the given $spin^{\mathbb{C}}$ structure of $M_1$.

Eigenvalues of Type r of the Basic Dirac Operator on Kähler Foliations

  • Jung, Seoung Dal
    • Kyungpook Mathematical Journal
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    • v.53 no.3
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    • pp.333-340
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    • 2013
  • In this paper, we prove that on a K$\ddot{a}$hler spin foliatoin of codimension $q=2n$, any eigenvalue ${\lambda}$ of type $r(r{\in}\{1,{\cdots},[\frac{n+1}{2}]\})$ of the basic Dirac operator $D_b$ satisfies the inequality ${\lambda}^2{\geq}\frac{r}{4r-2}\;{\inf}_M{\sigma}^{\nabla}$, where ${\sigma}^{\nabla}$ is the transversal scalar curvature of $\mathcal{F}$.

A NOTE ON GENERALIZED DIRAC EIGENVALUES FOR SPLIT HOLONOMY AND TORSION

  • Agricola, Ilka;Kim, Hwajeong
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.6
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    • pp.1579-1589
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    • 2014
  • We study the Dirac spectrum on compact Riemannian spin manifolds M equipped with a metric connection ${\nabla}$ with skew torsion $T{\in}{\Lambda}^3M$ in the situation where the tangent bundle splits under the holonomy of ${\nabla}$ and the torsion of ${\nabla}$ is of 'split' type. We prove an optimal lower bound for the first eigenvalue of the Dirac operator with torsion that generalizes Friedrich's classical Riemannian estimate.

DIRAC EIGENVALUES ESTIMATES IN TERMS OF DIVERGENCEFREE SYMMETRIC TENSORS

  • Kim, Eui-Chul
    • 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.

SASAKIAN TWISTOR SPINORS AND THE FIRST DIRAC EIGENVALUE

  • Kim, Eui Chul
    • 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.

THE PROPERTIES OF THE TRANSVERSAL KILLING SPINOR AND TRANSVERSAL TWISTOR SPINOR FOR RIEMANNIAN FOLIATIONS

  • Jung, Seoung-Dal;Moon, Yeong-Bong
    • 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.

THE FIRST POSITIVE EIGENVALUE OF THE DIRAC OPERATOR ON 3-DIMENSIONAL SASAKIAN MANIFOLDS

  • Kim, Eui Chul
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.2
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    • pp.431-440
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    • 2013
  • Let ($M^3$, $g$) be a 3-dimensional closed Sasakian spin manifold. Let $S_{min}$ denote the minimum of the scalar curvature of ($M^3$, $g$). Let ${\lambda}^+_1$ > 0 be the first positive eigenvalue of the Dirac operator of ($M^3$, $g$). We proved in [13] that if ${\lambda}^+_1$ belongs to the interval ${\lambda}^+_1{\in}({\frac{1}{2}},\;{\frac{5}{2}})$, then ${\lambda}^+_1$ satisfies ${\lambda}^+_1{\geq}{\frac{S_{min}+6}{8}}$. In this paper, we remove the restriction "if ${\lambda}^+_1$ belongs to the interval ${\lambda}^+_1{\in}({\frac{1}{2}},\;{\frac{5}{2}})$" and prove $${\lambda}^+_1{\geq}\;\{\frac{S_{min}+6}{8}\;for\;-\frac{3}{2}<S_{min}{\leq}30, \\{\frac{1+\sqrt{2S_{min}}+4}{2}}\;for\;S_{min}{\geq}30$$.