arXiv subjects

Seunghun Hong

Publications and source records attributed to Seunghun Hong.

7 recordsLinked to original sources

arXiv2018

Volume of the set of locally diagonalizable bipartite states

The purpose of this article is to investigate the geometry of the set of locally diagonalizable bipartite quantum states. We have the following new results: the Hilbert-Schmidt volume of all locally diagonalizable states, and a necessary and sufficient condition for local diagonalizability in the qubit-qubit case. Besides, we partition the set of all locally diagonalizable states as local unitary orbits (or coadjoint orbits) of diagonal forms. It is well-known that the Riemannian volume of a coadjoint orbit for a regular point in a specified Weyl chamber can be calculated by Harish-Chandra's volume formula. By modifying Harish-Chandra's volume formula, we give, for the first time, a specific formula for the Riemannian volume of a local unitary orbit of a regular point in a specified Weyl chamber. Several open questions are presented as well.

quant-ph↗

arXiv2017

Duflo Isomorphism and Chern-Weil Theory

We explain how the distributional index of a transversally elliptic operator on a principal G-manifold P that is obtained by lifting a Dirac operator on P/G can serve as a link between the Duflo isomorphism and Chern-Weil forms.

math.DG↗

arXiv2016

A Lie-algebraic approach to the local index theorem on compact homogeneous spaces

Using a K-theory point of view, Bott related the Atiyah-Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This article explains how to prove the local index theorem for compact homogenous spaces using Lie algebra methods. The method follows in outline the proof of the local index theorem due to Berline and Vergne. But the use of Kostant's cubic Dirac operator in place of the Riemannian Dirac operator leads to substantial simplifications. An important role is also played by the quantum Weil algebra of Alekseev and Meinrenken.

math.FA↗

arXiv2014

Borel-Weil-Bott Theorem via Equivariant McKean-Singer Formula

After reviewing how the Borel-Weil-Bott theorem can be interpreted as an index theorem, we present a proof using Kostant's cubic Dirac operator and the equivariant McKean-Singer formula.

math.RT↗

arXiv2013

Harish-Chandra's volume formula via Weyl's Law and Euler-Maclaurin formula

Harish-Chandra's volume formula shows that the volume of a flag manifold $G/T$, where the measure is induced by an invariant inner product on the Lie algebra of $G$, is determined up to a scalar by the algebraic properties of $G$. This article explains how to deduce Harish-Chandra's formula from Weyl's law by utilizing the Euler-Maclaurin formula. This approach leads to a mystery that lies under the Atiyah-Singer index theorem.

math.RT↗

arXiv2011

The asymptotic expansion of the heat kernel on a compact Lie group

Let $G$ be a compact connected Lie group equipped with a bi-invariant metric. We calculate the asymptotic expansion of the heat kernel of the laplacian on $G$ and the heat trace using Lie algebra methods. The Duflo isomorphism plays a key role.

math.FA↗

arXiv1998

Negative Differential Resistance in the Scanning Tunneling Spectroscopy of Organic molecules

The conductance-voltage spectrum of molecular nanostructures measured by scanning tunneling spectroscopy (STS) is generally assumed to reflect the local density of states of the molecule. This excludes the possibility of observing negative differential resistance (NDR). We report here the observation of NDR in the scanning tunneling microscope (STM) current-voltage (I-V) characteristics of self-assembled monolayer (SAM) of 4-p-Terphenylthiol molecules on gold substrate measured using a platinum probe. We argue that the NDR arises from narrow structures in the local density of states at the tip apex atom and show that depending on the electrostatic potential profile across the system, NDR could be observed in one or both bias directions.

cond-mat.mes-hall↗