Jacobi Operator Research Articles

Eigenvalue Estimates on Weighted Manifolds

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for f-minimal hypersurfaces and the spectrum of the Hodge Laplacian.

Derivatives of structure Jacobi operator on real hypersurfaces in complex Grassmannians of rank two

Derivatives of structure Jacobi operator on real hypersurfaces in complex Grassmannians of rank two

Anderson localization for block Jacobi operators with quasi‐periodic meromorphic potential

In this paper, we study block Jacobi operators on with quasi‐periodic meromorphic potential. We prove the nonperturbative Anderson localization for such operators in the large coupling regime.

Inverse source problem for the pseudoparabolic equation associated with the Jacobi operator

In this paper, we investigate direct and inverse problems for time-fractional pseudoparabolic equations associated with the Jacobi operator. The existence and uniqueness of the solutions are proven. Also, the stability result of the inverse source problem (ISP) is established.

Pointwise modulus of continuity of the Lyapunov exponent and integrated density of states for analytic multi-frequency quasi-periodic M(2,C) cocycles

It is known that the Lyapunov exponent for multifrequency analytic cocycles is weak-Hölder continuous in cocycle for certain Diophantine frequencies, and that this implies certain regularity of the integrated density of states in energy for Jacobi operators. In this paper, we establish the pointwise modulus of continuity in both cocycle and frequency and obtain analogous regularity of the integrated density of states in energy, potential, and frequency.

The absence of singular continuous spectrum for perturbed Jacobi operators

The absence of singular continuous spectrum for perturbed Jacobi operators

Stability and Deformation Criteria in Free Boundary CMC Immersions

Let ∑n and M n+1 be smooth manifolds with smooth boundary. Given a free boundary constant mean curvature (CMC) immersion φ: ∑ → M, we found results related to the existence and uniqueness of a deformation family of φ, {φt}t ∈I , composed by free boundary CMC immersions. In addition, we give to some criteria of stability and unstability for this type of deformations. These results are obtained from properties of the eigenvalues and eigenfunctions of the Jacobi operator Jφ associated to φ and establishing conditions for this operator such as Dim(Ker(Jφ)) = 0, or if Dim(Ker(Jφ)) = 1 and, for f ∈ Ker(Jφ); f ≠ 0, ∫∑ volφ*(g) ≠ 0. The deformation family is unique up to diffeomorphisms.

Cyclic Parallel Structure Jacobi Operator for Real Hypersurfaces in the Complex Quadric

Cyclic Parallel Structure Jacobi Operator for Real Hypersurfaces in the Complex Quadric

Spectral decimation of piecewise centrosymmetric Jacobi operators on graphs

We study the spectral theory of a class of piecewise centrosymmetric Jacobi operators defined on an associated family of substitution graphs. Given a finite centrosymmetric matrix viewed as a weight matrix on a finite directed path graph and a probabilistic Laplacian viewed as a weight matrix on a locally finite strongly connected graph, we construct a new graph and a new operator by edge substitution. Our main result proves that the spectral theory of the piecewise centrosymmetric Jacobi operator can be explicitly related to the spectral theory of the probabilistic Laplacian using certain orthogonal polynomials. Our main tools involve the so-called spectral decimation, known from the analysis on fractals, and the classical Schur complement. We include several examples of self-similar Jacobi matrices that fit into our framework.

Variation and oscillation for semigroups associated with discrete Jacobi operators

In this paper we prove weighted ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^p$$\\end{document}-inequalities for variation and oscillation operators defined by semigroups of operators associated with discrete Jacobi operators. Also, we establish that certain maximal operators involving sums of differences of discrete Jacobi semigroups are bounded on weighted ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^p$$\\end{document}-spaces. ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^p$$\\end{document}-boundedness properties for the considered operators provide information about the convergence of the semigroup of operators defining them.

Characteristic Jacobi Operator on Almost Kenmotsu $3$-manifolds

The Ricci tensor field, $\varphi$-Ricci tensor field and the characteristic Jacobi operator on almost Kenmotsu $3$-manifolds are investigated. We give a classification of locally symmetric almost Kenmotsu $3$-manifolds.

Geometry of static perfect fluid space-time

In this article, we investigate the geometry of static perfect fluid space-time (SPFST) on compact manifolds with boundary. We use the generalized Reilly’s formula to establish a geometric inequality for a SPFST involving the area of the boundary and its volume. Moreover, we obtain new boundary estimates for SPFST. One of the boundary estimates is obtained in terms of the Brown–York mass and another one related to the first eigenvalue of the Jacobi operator. In addition, we provide a new (simply connected) counterexample to the Cosmic no-hair conjecture for arbitrary dimension

A multiplicity estimate for the Jacobi operator of a nonflat Yang–Mills field over mathbb {S}^m

Here we provide refinements of the stability results of Simons and Xin, concerning the stability of Yang–Mills fields and harmonic maps respectively. The result also implies the earlier Morse index estimates for both cases.

Contact 3-Manifolds with Pseudo-parallel Characteristic Jacobi Operator

Contact 3-Manifolds with Pseudo-parallel Characteristic Jacobi Operator

GTW parallel structure Jacobi operator of real hypersurfaces in nonflat complex space forms

We classify real hypersurfaces in nonflat complex space forms of complex dimension two whose structure Jacobi operator is parallel with respect to the generalized Tanaka-Webster connection. We classify these hypersurfaces when the dimension is more than two under the Hopf assumption.

On the Splitting Tensor of the Weak f-Contact Structure

A weak f-contact structure, introduced in our recent works, generalizes the classical f-contact structure on a smooth manifold, and its characteristic distribution defines a totally geodesic foliation with flat leaves. We find the splitting tensor of this foliation and use it to show positive definiteness of the Jacobi operators in the characteristic directions and to obtain a topological obstruction (including the Adams number) to the existence of weak f-K-contact manifolds, and prove integral formulas for a compact weak f-contact manifold. Based on applications of the weak f-contact structure in Riemannian contact geometry considered in the article, we expect that this structure will also be fruitful in theoretical physics, e.g., in QFT.

On p-harmonic self-maps of spheres

In this manuscript we study rotationally p-harmonic maps between spheres. We prove that for (p) given, there exist infinitely many p-harmonic self-maps of mathbb {S}^m for each min mathbb {N} with p<m< 2+p+2sqrt{p}. In the case of the identity map of mathbb {S}^m we explicitly determine the spectrum of the corresponding Jacobi operator and show that for p>m, the identity map of mathbb {S}^m is equivariantly stable when interpreted as a p-harmonic self-map of mathbb {S}^m.

Equiconvergence for perturbed Jacobi polynomial expansions

We show asymptotic expansions of the eigenfunctions of certain perturbations of the Jacobi operator in a bounded interval, deducing equiconvergence results between expansions with respect to the associated orthonormal basis and expansions with respect to the cosine basis. Several results for pointwise convergence then follow.

Two-Root Riemannian Manifolds

Osserman manifolds are a generalization of locally two-point homogeneous spaces. We introduce k-root manifolds in which the reduced Jacobi operator has exactly k eigenvalues. We investigate one-root and two-root manifolds as another generalization of locally two-point homogeneous spaces. We prove that there is no two-root Riemannian manifold of odd dimension. In twice an odd dimension, we describe all two-root Riemannian algebraic curvature tensors and give additional conditions for two-root Riemannian manifolds.

On the existence of a curvature tensor for given Jacobi operators

It is well known that the Jacobi operators completely determine the curvature tensor. The question of existence of a curvature tensor for given Jacobi operators naturally arises, which is considered and solved in the previous work. Unfortunately, although the published theorem is correct, its proof is incomplete because it contains some omissions, and the aim of this paper is to present a complete and accurate proof. We also generalize the main theorem to the case of indefinite scalar product space. Accordingly, we generalize the proportionality principle for Osserman algebraic curvature tensors.