Jacobi Operator Research Articles

HOMOGENEOUS RIEMANNIAN MANIFOLDS WITH NON-TRIVIAL NULLITY

We develop a general theory for irreducible homogeneous spaces M = G/H, in relation to the nullity distribution ν of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e., where the nullity distribution is not parallel. Moreover, we construct irreducible examples of conullity k = 3, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that H is trivial and G is solvable if k = 3. Another of our main results is that the leaves, i.e., the integral manifolds, of the nullity are closed (we used a rather delicate argument). This implies that M is a Euclidean affine bundle over the quotient by the leaves of ν. Moreover, we prove that ν⊥ defines a metric connection on this bundle with transitive holonomy or, equivalently, ν⊥ is completely non-integrable (this is not in general true for an arbitrary autoparallel and at invariant distribution). We also found some general obstruction for the existence of non-trivial nullity: e.g., if G is reductive (in particular, if M is compact), or if G is two-step nilpotent.

Real hypersurfaces in [formula omitted] with constant Reeb sectional curvature

We prove that the structure Jacobi operator Rξ on a real hypersurface M in the complex projective space CP2 satisfies Rξ+κϕ2=0 with κ∈R⁎ (or equivalently, constant Reeb sectional curvature) if and only if either M is of type (A) or M is locally congruent to a non-homogeneous Hopf hypersurface with vanishing Hopf principal curvature.

Non-local hydrodynamics as a slow manifold for the one-dimensional kinetic equation

We prove an explicit, non-local hydrodynamic closure for the linear one-dimensional kinetic equation independent on the size of the relaxation time. We compare this dynamical equation to the local approximations obtained from the Chapman--Enskog expansion for small relaxation times. Our results rely on the spectral theory of Jacobi operators with rank-one perturbations.

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators

In this paper, we consider the Schr\"odinger equation, \begin{equation*} Hu=-u^{\prime\prime}+(V(x)+V_0(x))u=Eu, \end{equation*} where $V_0(x)$ is 1-periodic and $V (x)$ is a decaying perturbation. By Floquet theory, the spectrum of $H_0=-\nabla^2+V_0$ is purely absolutely continuous and consists of a union of closed intervals (often referred to as spectral bands). Given any finite set of points $\{ E_j\}_{j=1}^N$ in any spectral band of $H_0$ obeying a mild non-resonance condition, we construct smooth functions $V(x)=\frac{O(1)}{1+|x|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$. Given any countable set of points $\{ E_j\}$ in any spectral band of $H_0$ obeying the same non-resonance condition, and any function $h(x)>0$ going to infinity arbitrarily slowly, we construct smooth functions $|V(x)|\leq \frac{h(x)}{1+|x|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$. On the other hand, we show that there is no eigenvalue of $H=H_0+V$ embedded in the spectral bands if $V(x)=\frac{o(1)}{1+|x|}$ as $x$ goes to infinity. We prove also an analogous result for Jacobi operators.

New Conditions on Normal Jacobi Operator of Real Hypersurfaces in the Complex Quadric

On a real hypersurface M of a complex quadric we have an almost contact metric structure induced by the Kahlerian structure of the ambient space. Therefore, on M we have the Levi-Civita connection $$\nabla $$ and, for any non-null real number k, the so called kth generalized Tanaka Webster connection $${\hat{\nabla }}^{(k)}$$ . We introduce the notions of $$({\hat{\nabla }}^{(k)},\nabla )$$ -Codazzi and $$({\hat{\nabla }}^{(k)},\nabla )$$ -Killing normal Jacobi operator on such a real hypersurface and classify Hopf real hypersurface in a complex quadric whose normal Jacobi operators satisfy any of both conditions.

Space versus energy oscillations of Prufer phases for matrix Sturm-Liouville and Jacobi operators

This note considers Sturm oscillation theory for regular matrix Sturm-Liouville operators on finite intervals and for matrix Jacobi operators. The number of space oscillations of the eigenvalues of the matrix Prufer phases at a given energy, defined by a suitable lift in the Jacobi case, is shown to be equal to the number of eigenvalues below that energy. This results from a positivity property of the Prufer phases, namely they cannot cross -1 in the negative direction, and is also shown to be closely linked to the positivity of the matrix Prufer phase in the energy variable. The theory is illustrated by numerical calculations for an explicit example.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/76/abstr.html

Harmonic Spaces and Density Functions

We use the density function of a harmonic space to obtain estimates for the eigenvalues of the Jacobi operator; when these estimates are sharp, then the harmonic space is a symmetric Osserman space.

Semiclassical asymptotic behavior of orthogonal polynomials

Our goal is to find asymptotic formulas for orthonormal polynomials $$P_{n}(z)$$ with the recurrence coefficients slowly stabilizing as $$n\rightarrow \infty $$ . To that end, we develop scattering theory of Jacobi operators with long-range coefficients and study the corresponding second-order difference equation. We introduce the Jost solutions $$f_{n}(z)$$ of this equation by a condition for $$n\rightarrow \infty $$ and suggest an Ansatz for them playing the role of the semiclassical Liouville–Green Ansatz for the corresponding solutions of the Schrodinger equation. This allows us to study Jacobi operators and their eigenfunctions $$P_{n}(z)$$ by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for $$P_{n}(z)$$ as $$\rightarrow \infty $$ in terms of the Wronskian of the solutions $$\{P_{n} (z)\}$$ and $$\{f_{n} (z)\}$$ .

Derivatives of normal Jacobi operator on real hypersurfaces in the complex quadric

In \cite{S 2017}, Suh gave a non-existence theorem for Hopf real hypersurfaces in the complex quadric with parallel normal Jacobi operator. Motivated by this result, in this paper, we introduce some generalized conditions named $\mathcal C$-parallel or Reeb parallel normal Jacobi operators. By using such weaker parallelisms of normal Jacobi operator, first we can assert a non-existence theorem of Hopf real hypersurfaces with $\mathcal C$-parallel normal Jacobi operator in the complex quadric $Q^{m}$, $m \geq 3$. Next, we prove that a Hopf real hypersurface has Reeb parallel normal Jacobi operator if and only if it has an $\mathfrak A$-isotropic singular normal vector field.

Asymptotic behavior of orthogonal polynomials without the Carleman condition

Our goal is to find an asymptotic behavior as n→∞ of orthogonal polynomials Pn(z) defined by the Jacobi recurrence coefficients an,bn. We suppose that the off-diagonal coefficients an grow so rapidly that the series ∑an−1 converges, that is, the Carleman condition is violated. With respect to diagonal coefficients bn we assume that −bn(anan−1)−1/2→2β∞ for some β∞≠±1. The asymptotic formulas obtained for Pn(z) are quite different from the case ∑an−1=∞ when the Carleman condition is satisfied. In particular, if ∑an−1<∞, then the phase factors in these formulas do not depend on the spectral parameter z∈C. The asymptotic formulas obtained in the cases |β∞|<1 and |β∞|>1 are also qualitatively different from each other. As an application of these results, we find necessary and sufficient conditions for the essential self-adjointness of the corresponding minimal Jacobi operator.

Real Hypersurfaces in $$Q^m$$ with Commuting Structure Jacobi Operator

In this paper, we study real hypersurfaces in the complex quadric space $$Q^m$$ whose structure Jacobi operator commutes with their structure tensor field. When the normal vector field is $$\mathfrak {A}$$ -principal we show that the Reeb curvature $$\alpha $$ is non-vanishing and determine principal curvatures of the hypersurface. In the case of $$\mathfrak {A}$$ -isotropic normal vector field, we prove that the hypersurface is Hopf if it has vanishing Reeb curvature or commuting shape operator. We also consider Reeb flat hypersurfaces, namely when the Reeb curvature is zero. We see that this family of hypersurfaces is non-empty and among other results we prove that if the Ricci tensor of a Reeb flat Hopf hypersurfaces is Killing, then the Ricci tensor is parallel.

Properties and decompositions of domains for powers of the Jacobi differential operator

We set out to build a framework for self-adjoint extension theory for powers of the Jacobi differential operator that does not make use of classical deficiency elements. Instead, we rely on simpler functions that capture the impact of these elements on extensions but are defined by boundary asymptotics. This new perspective makes calculations much more accessible and allows for a more nuanced analysis of the associated domains.The maximal domain for n-th composition of the Jacobi operator is characterized in terms of a smoothness condition for each derivative, and the endpoint behavior of functions in the underlying Hilbert space can then be classified, for j∈N0, by (1−x)j, (1+x)j, (1−x)−α+j and (1+x)β+j. Most of these behaviors can only occur when functions are in the associated minimal domain, and this leads to a formulation of the defect spaces with a convenient basis. Self-adjoint extensions, including the important left-definite domain, are then given in terms of the new basis functions for the defect spaces using GKN theory. Comments are made for the Laguerre operator as well.

Real Hypersurfaces in the Complex Hyperbolic Quadric with Reeb Parallel Structure Jacobi Operator

We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.

English

We study the first eigenvalue of the Jacobi operator on closed hypersurfaces with constant mean curvature in non-flat Riemannian space forms. Under an appropriate constraint on the totally umbilical tensor of the hypersurfaces and following Melendez’s ideas in J. Melendez (2014) we obtain a new sharp upper bound of the first eigenvalue of the Jacobi operator.

CR -hypersurfaces of conformal Kenmotsu manifolds with ξ-parallel normal Jacobi operator

CR -hypersurfaces of conformal Kenmotsu manifolds with ξ-parallel normal Jacobi operator

CR -hypersurfaces of conformal Kenmotsu manifolds with ξ-parallel normal Jacobi operator

In this paper, we study CR-hypersurfaces of a conformal Kenmotsu manifold with a ξ-parallel or Lie ξ-parallel normal Jacobi operator.

Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency

In this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-Russmann shift on the Torus. Then, we apply it to prove the large deviation theorems for the finite scale Dirichlet determinants of quasi-periodic analytic Jacobi operators with this frequency. It shows that the Brjuno-Russmann function, which reflects the irrationality of the frequency, plays the key role in these theorems via the smallest deviation. At last, as an application, we obtain a distribution of the eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which also depends on the smallest deviation, essentially on the irrationality of the frequency.

Osserman Conditions in Lightlike Warped Product Geometry

In this paper, we consider Osserman conditions on lightlike warped product (sub-)manifolds with respect to the Jacobi Operator. We define the Jacobi operator for lightlike warped product manifold and introduce a study of lightlike warped product Osserman manifolds. For the coisotropic case with totally degenerates first factor, we prove that this class consists of Einstein and locally Osserman lightlike warped product.

Real Hypersurfaces with Invariant Normal Jacobi Operator in the Complex Hyperbolic Quadric

Real Hypersurfaces with Invariant Normal Jacobi Operator in the Complex Hyperbolic Quadric

Multivariate orthogonal polynomials: Quantum decomposition, deficiency rank and support of measure

In this paper we investigate the multivariate orthogonal polynomials based on the theory of interacting Fock spaces. Our framework is on the same stream line of the recent paper by Accardi, Barhoumi, and Dhahri [1]. The (classical) coordinate variables are decomposed into non-commuting (quantum) operators called creation, annihilation, and preservation operators, in the interacting Fock spaces. Getting the commutation relations, which follow from the commuting property of the coordinate variables between themselves, we can develop the reconstruction theory of the measure, namely the Favard's theorem. We then further develop some related problems including the marginal distributions and the rank theory of the Jacobi operators. We will see that the deficiency rank of the Jacobi operator implies that the underlying measure is supported on some algebraic surface and vice versa. We will provide with some examples.