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Ladder operator
About: Ladder operator is a research topic. Over the lifetime, 3019 publications have been published within this topic receiving 63439 citations.
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272 citations
TL;DR: In this paper, a lower bound for the square of the eigenvalues of the Dirac operator by the smallest eigenvalue of the conformal Laplacian (the Yamabe operator) was given.
Abstract: On a Riemannian spin manifold, we give a lower bound for the square of the eigenvalues of the Dirac operator by the smallest eigenvalue of the conformal Laplacian (the Yamabe operator). We prove, in the limiting case, that the eigenspinor field is a killing spinor, i.e., parallel with respect to a natural connection. In particular, if the scalar curvature is positive, the eigenspinor field is annihilated by harmonic forms and the metric is Einstein.
272 citations
TL;DR: In this article, the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates.
267 citations
TL;DR: In this article, a parametrization of the density operator, a coherence vector representation, which uses a basis of orthogonal, traceless, Hermitian matrices is discussed.
Abstract: A parametrization of the density operator, a coherence vector representation, which uses a basis of orthogonal, traceless, Hermitian matrices is discussed. Using this parametrization we find the region of permissible vectors which represent a density operator. The inequalities which specify the region are shown to involve the Casimir invariants of the group. In particular cases, this allows the determination of degeneracies in the spectrum of the operator. The identification of the Casimir invariants also provides a method of constructing quantities which are invariant under local unitary operations. Several examples are given which illustrate the constraints provided by the positivity requirements and the utility of the coherence vector parametrization.
256 citations
TL;DR: In this article, the authors present different data-driven dimension reduction techniques for dynamical systems that are based on transfer operator theory as well as methods to approximate transfer operators and their eigenvalues, eigenfunctions, and eigenmodes.
Abstract: In this review paper, we will present different data-driven dimension reduction techniques for dynamical systems that are based on transfer operator theory as well as methods to approximate transfer operators and their eigenvalues, eigenfunctions, and eigenmodes The goal is to point out similarities and differences between methods developed independently by the dynamical systems, fluid dynamics, and molecular dynamics communities such as time-lagged independent component analysis, dynamic mode decomposition, and their respective generalizations As a result, extensions and best practices developed for one particular method can be carried over to other related methods
255 citations