Jacobi Operator Research Articles

Real Hypersurfaces in Complex Two-Plane Grassmannians Whose Jacobi Operators Corresponding to -Directions are of Codazzi Type

We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose Jacobi operators corresponding to the directions in the distribution D  are of Codazzi type if they satisfy a further condition. We obtain that that they must be either of type (A) or of type (B) (see [1]), but no one of these satisfies our condition. As a consequence, we obtain the non-existence of Hopf real hypersurfaces in such ambient spaces whose Jacobi operators corresponding to D  -directions are parallel with the same further condition.

Spectral measures of Jacobi operators with random potentials

Let Hω be a self-adjoint Jacobi operator with a potential sequence {ω(n)}n of independently distributed random variables with continuous probability distributions and let μφ be the corresponding spectral measure generated by Hω and the vector φ. We consider sets A(ω) which depend on ω, but are independent of two consecutive given entries of the secuence ω, and prove that μφ(A(ω)) = 0 for almost every ω. This result is applied to show equivalence relations between spectral measures for random Jacobi matrices and to study the interplay of the eigenvalues of these matrices and their submatrices. This is based on joint work with Luis Silva.

Jacobi operators of quantum counterparts of three-dimensional real Lie algebras over the harmonic oscillator

Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of three-dimensional real Lie algebras. The Jacobi operators of these quantum algebras are explicitly calculated.

ON THE STRUCTURE JACOBI OPERATOR AND RICCI TENSOR OF REAL HYPERSURFACES IN NONFLAT COMPLEX SPACE FORMS

It is known that there are no real hypersurfaces with parallel structure Jacobi operator <TEX>$R_{\xi}$</TEX> (cf.[16], [17]). In this paper we investigate real hypersurfaces in a nonflat complex space form using some conditions of the structure Jacobi operator <TEX>$R_{\xi}$</TEX> which are weaker than <TEX>${\nabla}R_{\xi}$</TEX> = 0. Under further condition <TEX>$S\phi={\phi}S$</TEX> for the Ricci tensor S we characterize Hopf hypersurfaces in a complex space form.

Some geometric properties of hypersurfaces with constant $r$-mean curvature in Euclidean space

Let f: M → R m+1 be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors to analyze the stability of the differential operator L r associated with the rth Newton tensor of f. This appears in the Jacobi operator for the variational problem of minimizing the r-mean curvature H r . Two natural applications are found. The first one ensures that under a mild condition on the integral of H r over geodesic spheres, the Gauss map meets each equator of S m infinitely many times. The second one deals with hypersurfaces with zero (r + 1)-mean curvature. Under similar growth assumptions, we prove that the affine tangent spaces f * T p M, p ∈ M, fill the whole R m+1 .

Real hypersurfaces in a complex projective space with pseudo- % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv % 39gaiyaacqWFdcpraaa!47B8! $$ \mathbb{D} $$ -parallel structure Jacobi operator

We introduce the new notion of pseudo-\( \mathbb{D} \)-parallel real hypersurfaces in a complex projective space as real hypersurfaces satisfying a condition about the covariant derivative of the structure Jacobi operator in any direction of the maximal holomorphic distribution. This condition generalizes parallelness of the structure Jacobi operator. We classify this type of real hypersurfaces.

Non-Existence of Real Hypersurfaces Equipped with Recurrent Structure Jacobi Operator in Nonflat Complex Space Forms

The aim of the present paper is to prove the non-existence of real hypersurfaces equipped with recurrent structure Jacobi operator in a non-flat complex space form.

Hilbert transform and related topics associated with the differential Jacobi operator on (0, +∞)

We define Hilbert transform and conjugate Poisson integrals associated with the Jacobi differential operator on (0, +∞). We prove that these operators are bounded in the appropriate Lebesgue spaces L p , 1 < p < +∞. In this study, the tools used are the Littlewood–Paley g-functions associated with the Poisson semigroup and the supplementary Poisson semigroup which we introduce in this paper.

The structure vector field and structure Jacobi operator of real hypersurfaces in nonflat complex space forms

In this paper we determine the real hypersurfaces for which the structure Jacobi operator commutes over both the Ricci tensors and structure tensors (for a definition of the operator see Sect. 1). We prove that such hypersurfaces are homogneous real hypersurfaces of type (A) and are a special class of Hopf hypersurfaces.

Quantum counterparts of three-dimensional real Lie algebras over harmonic oscillator

Abstract Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of three-dimensional (3D) real Lie algebras in Bianchi classification. The Jacobi operators of the quantum algebras are found.

Conformally Osserman manifolds

An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every point is Osserman. We prove that a conformally Osserman manifold of dimension $n \ne 3, 4, 16$ is locally conformally equivalent either to a Euclidean space or to a rank-one symmetric space.

Geometry of D’Atri spaces of type k

A Riemannian n-dimensional manifold M is a D’Atri space of type k (or k-D’Atri space), 1 ≤ k ≤ n − 1, if the geodesic symmetries preserve the k-th elementary symmetric functions of the eigenvalues of the shape operators of all small geodesic spheres in M. Symmetric spaces are k-D’Atri spaces for all possible k ≥ 1 and the property 1-D’Atri is the D’Atri condition in the usual sense. In this article we study some aspects of the geometry of k-D’Atri spaces, in particular those related to properties of Jacobi operators along geodesics. We show that k-D’Atri spaces for all k = 1, . . ., l satisfy that \({{\rm{tr}}(R_{v}^{k})}\), v a unit vector in TM, is invariant under the geodesic flow for all k = 1, . . ., l. Further, if M is k-D’Atri for all k = 1, . . ., n − 1, then the eigenvalues of Jacobi operators are constant functions along geodesics. In the case of spaces of Iwasawa type, we show that k-D’Atri spaces for all k = 1, . . ., n − 1 are exactly the symmetric spaces of noncompact type. Moreover, in the class of Damek-Ricci spaces, the symmetric spaces of rank one are characterized as those that are 3-D’Atri.

A finite-sum representation for solutions for the Jacobi operator

We obtain a finite-sum representation for the general solution of the equation in terms of a non-vanishing solution corresponding to some fixed value of . Applications of this representation to some results on the boundedness of solutions, are given as well as illustrating examples.

On compact perturbations of finite-zone Jacobi operators

For a bounded Jacobi operator (a discrete analog of the Sturm–Liouville operator on the half-axis), the compactness of a perturbation is studied. The perturbation is produced by a change of the spectral measure (the essential spectrum remains unchanged). Bibliography: 21 titles.

Probabilistic Averages of Jacobi Operators

I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these can be reduced to questions about ergodic Jacobi operators. I use this to show that for finite gap Jacobi operators, regularity implies that they are in the Cesàro–Nevai class, proving a conjecture of Barry Simon. Furthermore, I use this to study Jacobi operators with coefficients a(n) = 1 and b(n) = f(n ρ (mod 1)) for ρ > 0 not an integer.

CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE A IN A COMPLEX SPACE FORM

Let M be a real hypersurface with almost contact metric structure <TEX>$(\phi,g,\xi,\eta)$</TEX> in a complex space form <TEX>$M_n(c)$</TEX>, <TEX>$c\neq0$</TEX>. In this paper we prove that if <TEX>$R_{\xi}L_{\xi}g=0$</TEX> holds on M, then M is a Hopf hypersurface in <TEX>$M_n(c)$</TEX>, where <TEX>$R_{\xi}$</TEX> and <TEX>$L_{\xi}$</TEX> denote the structure Jacobi operator and the operator of the Lie derivative with respect to the structure vector field <TEX>$\xi$</TEX> respectively. We characterize such Hopf hypersurfaces of <TEX>$M_n(c)$</TEX>.

RICCI OPERATORS AND STRUCTURAL JACOBI OPERATORS ON REAL HYPERSURFACES IN A COMPLEX SPACE FORM

We give a classification of real hypersurfaces in a non-flat complex space form, whose almost contact structure operator, induced from the complex structure of the complex space form, commutes with the Ricci operator and at the same time commutes with the structural Jacobi operator. In particular, we classify real hypersurfaces in 2-dimensional complex projective and hyperbolic spaces satisfying the first commutativity condition.

Local Jacobi Operators and Applications to thep-Version of Finite Element Method in Two Dimensions

Based on the Chebyshev projection-interpolation on each edge of elements and the Chebyshev projection on each element, we have designed the local Jacobi operators $\Pi_{\Omega_j}$ on the triangular or quadrilateral element $\Omega_j$, $1\leq j\leq J$ such that $\Pi_{\Omega_j}u$ is a polynomial of degree p on $\Omega_j$ which interpolates u at the vertices of $\Omega_j$, coincides with the Chebyshev projection-interpolation of u on the edges of $\Omega_j$, and possesses the best approximation to the smooth and singular functions u. By a simple assembly of $\Pi_{\Omega_j}u$, $1\leq j\leq J$ we construct a piecewise and globally continuous polynomial of degree p which has the best approximation error bound locally and globally for singular as well as smooth solutions on general quasi-uniform meshes and satisfies the homogeneous Dirichlet boundary conditions. An application of the local Jacobi operators to the p-version of the finite element method associated with general meshes composed of (curvilinear) triangular and quadrilateral elements for problems on polygonal domains leads to the optimal convergence, which has been an open problem for more than three decades.

Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator

Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator

Limit leaves of an $H$ lamination are stable

Suppose $L$ is a lamination of a Riemannian manifold by hypersurfaces with the same constant mean curvature $H$. We prove that every limit leaf of $L$ is stable for the Jacobi operator. A simple but important consequence of this result is that the set of stable leaves of $L$ has the structure of a lamination.