One-dimensional Operators Research Articles

Mathematical Modeling in the Study of the Ricci Operator on Four-Dimensional Locally Homogeneous (Pseudo)Riemannian Manifolds with Isotropic Weyl Tensor

It is known that a locally homogeneous manifold can be obtained from a locally conformally homogeneous (pseudo)Riemannian manifolds by a conformal deformation if the Weyl tensor (or the Schouten-Weyl tensor in the three-dimensional case) has a nonzero squared length. Thus, the problem arises of studying (pseudo)Riemannian locally homogeneous and locally conformally homogeneous manifolds, the Weyl tensor of which has zero squared length, and itself is not equal to zero (in this case, the Weyl tensor is called isotropic).
 One of the important aspects in the study of such manifolds is the study of the curvature operators on them, namely, the problem of restoring a (pseudo)Riemannian manifold from a given Ricci operator.
 The problem of the prescribed values of the Ricci operator on 3-dimensional locally homogeneous Riemannian manifolds has been solved by O. Kowalski and S. Nikcevic. Analogous results for the one-dimensional and sectional curvature operators were obtained by D.N. Oskorbin, E.D. Rodionov, and O.P Khromova.
 This paper is devoted to the description of an example of studying the problem of the prescribed Ricci operator for four-dimensional locally homogeneous (pseudo) Riemannian manifolds with a nontrivial isotropy subgroup and isotropic Weyl tensor.

Trace formulas for the one-dimensional Stark operator and integrals of motion for the cylindrical Korteweg–de Vries equation

Trace formulas for the one-dimensional Stark operator and integrals of motion for the cylindrical Korteweg–de Vries equation

Extension criteria for homogeneous Sobolev spaces of functions of one variable

For each p > 1 and each positive integer m , we give intrinsic characterizations of the restriction of the homogeneous Sobolev space L_{p}^{m}(\mathbb{R}) to an arbitrary closed subset E of the real line. We show that the classical one-dimensional Whitney extension operator is "universal" for the scale of L_{p}^{m}(\mathbb{R}) spaces in the following sense: For every p\in(1,\infty] , it provides almost optimal L^m_p -extensions of functions defined on E . The operator norm of this extension operator is bounded by a constant depending only on m . This enables us to prove several constructive L^m_p -extension criteria expressed in terms of m -th order divided differences of functions.

Approximation of one-dimensional relativistic point interactions by regular potentials revised

We show that the one-dimensional Dirac operator with quite general point interaction may be approximated in the norm resolvent sense by the Dirac operator with a scaled regular potential of the form $1/\varepsilon~h(x/\varepsilon)\otimes B$, where $B$ is a suitable $2\times 2$ matrix. Moreover, we prove that the limit does not depend on the particular choice of $h$ as long as it integrates to a constant value.

Spectral triangles of non-selfadjoint Hill and Dirac operators

This is a survey of results from the last 10 to 12 years about the structure of the spectra of Hill–Schrödinger and Dirac operators. Let be a Hill operator or a one-dimensional Dirac operator on the interval . If is considered with Dirichlet, periodic, or antiperiodic boundary conditions, then the corresponding spectra are discrete and, for sufficiently large , close to in the Hill case or close to in the Dirac case (). There is one Dirichlet eigenvalue and two periodic (if is even) or antiperiodic (if is odd) eigenvalues and (counted with multiplicity). Asymptotic estimates are given for the spectral gaps and the deviations in terms of the Fourier coefficients of the potentials. Moreover, precise asymptotic expressions for and are found for special potentials that are trigonometric polynomials. Bibliography: 45 titles.

Location of Eigenvalues of Non-self-adjoint Discrete Dirac Operators

We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex $\ell^p$-potentials for $1\leq p\leq\infty$. As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small $\ell^1$-potential. Further possible improvements and sharpness of the obtained spectral bounds are also discussed.

New Type Super Singular Integro-Differential Equation and Its Conjugate Equation

In this paper for a class of model partial integro-differential equation with super singularity in the kernels, is obtained an integral representation of manifold solutions by arbitrary constants. The conjugate equation for the above-mentioned type of equations is also investigated. Such types of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of theory of analytical functions. However, the method of analytical functions is not applicable for our case of super singular equations with integrals understanding in Riemann–Stieltjes sense. Here, we have used the method of representation the considering equation as a product of two one-dimensional singular first order integro-differential operators. Further, a complete integro-differential equation and its conjugate equation have been investigated. It is shown that in every cases of characteristic equation roots the homogeneous integro-differential equation can have a nontrivial solutions. Non-model equation is investigated by the regularization method. Regularization of non-model equation is based on selecting a model part of equation. On the basis of the analysis of a model part of equation the solution of non-model equation reduced to the solution of a second kind Volterra integral equations with super singular kernel. It is important to emphasize that in contrast to the usual theory of Volterra integral equations, the studied homogeneous integral equation has nontrivial solutions. It is easy to see that the presence of a non-model part in the equation does not affect to the general structure of the obtained solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important.

One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics

We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type $n^{-\alpha}$ for $\alpha>0$. We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region $\alpha>\frac12$; a transition from pure point to singular continuous spectrum in the critical region $\alpha=\frac12$; and pure point spectrum in the sub-critical region $\alpha<\frac12$. From the dynamical point of view, delocalization in the super-critical region follows from the RAGE theorem. In the critical region, we exhibit a simple argument based on lower bounds on eigenfunctions showing that no dynamical localization can occur even in the presence of point spectrum. Finally, we show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions.

Maximal determinants of Schrödinger operators on bounded intervals

We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schro dinger operator defined on a bounded interval with Dirichlet boundary conditions under an Lq-norm res- triction (q ≥ 1). This is done by first extending the definition of the functional determinant to the case of Lq potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all q ≥ 1, providing a complete characterization of extremal potentials in the case where q is one (a pulse) and two (Weierstrass’s ℘ function).

On Spectral Properties of the One-Dimensional Stark Operator on the Semiaxis

We consider a one-dimensional Stark operator on the semiaxis with Dirichlet boundary condition at the origin. The asymptotic behavior of eigenvalues at infinity is analyzed.

The Essential Spectrum of One-Dimensional Dirac Operators with Delta Interactions

We describe the location of the essential spectrum of one-dimensional Dirac operators with singular potentials supported on discrete sets in ℝ in terms of limit operators.

On the existence of an operator group generated by the one-dimensional Dirac system

We study the one-dimensional Dirac operator with a complex-valued summable potential. The possibility of constructing an operator group generated by this operator is investigated in various spaces. Estimates are established for the growth of the group constructed.

Dirac Operators and Domain Walls

We study the eigenvalue problem for a one-dimensional Dirac operator with a spatially varying “mass” term. It is well-known that when the mass function has the form of a kink, or domain wall, transitioning between strictly positive and strictly negative asymptotic mass, $\pm\kappa_\infty$, at $\pm\infty$, the Dirac operator has a simple eigenvalue of zero energy (geometric multiplicity equal to one) within a gap in the continuous spectrum, with corresponding exponentially localized zero mode. We consider the eigenvalue problem for the one-dimensional Dirac operator with mass function defined by “gluing” together $n$ domain wall--type transitions, assuming that the distance between transitions, $2 \delta$, is sufficiently large, focusing on the illustrative cases $n = 2$ and $3$. When $n = 2$ we prove that the Dirac operator has two real simple eigenvalues of opposite sign and of order $e^{- 2 |\kappa_\infty| \delta}$. The associated eigenfunctions are, up to $L^2$ error of order $e^{- 2 |\kappa_\infty| \delta}$, linear combinations of shifted copies of the single domain wall zero mode. For the case $n = 3$, we prove the Dirac operator has two nonzero simple eigenvalues as in the two domain wall case and a simple eigenvalue at energy zero. The associated eigenfunctions of these eigenvalues can again, up to small error, be expressed as linear combinations of shifted copies of the single domain wall zero mode. When $n > 3$ no new technical difficulty arises and the result is similar. Our methods are based on a Lyapunov--Schmidt reduction/Schur complement strategy, which maps the Dirac operator eigenvalue problem for eigenstates with near-zero energies to the problem of determining the kernel of an $n \times n$ matrix reduction, which depends nonlinearly on the eigenvalue parameter. The class of Dirac operators we consider controls the bifurcation of topologically protected “edge states” from Dirac points (linear band crossings) for classes of Schrödinger operators with domain wall modulated periodic potentials in one and two space dimensions. The present results may be used to construct a rich class of defect modes in periodic structures modulated by multiple domain walls.

ИССЛЕДОВАНИЕ НЕКОТОРЫХ КЛАССОВ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ В ЧАСТНЫХ ПРОИЗВОДНЫХ ВТОРОГО ПОРЯДКА СО СТЕПЕННО-ЛОГАРИФМИЧЕСКОЙ ОСОБЕННОСТЬЮ В ЯДРЕ

For a class of second-order partial integro-differential equations with a power singularity and logarithmic singularity in the kernel, integral representations of the solution manifold in terms of arbitrary constants are obtained in the class of functions vanishing with a certain asymptotic behavior. Although the kernel of the given equation is not a Fredholm type kernel, the solution of the studied equation in a class of vanishing functions is found in an explicit form. We represent a second-order integro-differential equation as a product of two first-order integro-differential operators. For these one-dimensional integro-differential operators, in the cases when the roots of the corresponding characteristic equations are real and different, real and equal and complex and conjugate, the inverse operators are found. It is found that the presence of power singularity and logarithmic singularity in the kernel affects the number of arbitrary constants in the general solution. This number, depending on the roots of the corresponding characteristic equations, can reach nine. Also, the cases when the given integro-differential equation has a unique solution are found. The correctness of the obtained results with the help of the detailed solutions of concrete examples are shown. The method of solving the given problem can be used for solving model and nonmodel integro-differential equations with a higher order power singularity and logarithmic singularity in the kernel.

Essential spectrum of one-dimensional Dirac operators with delta-interactions

В заметке дается описание существенного спектра одномерных операторов Дирака с сингулярными потенциалами, имеющими носители на дискретных множествах в $\mathbb{R}$, в терминах предельных операторов.

Instability of unidirectional flows for the 2D &lt;i&gt;α&lt;/i&gt;-Euler equations

We study stability of unidirectional flows for the linearized 2D $\alpha$-Euler equations on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a vector $\mathbf p \in \mathbb Z^{2}$. We linearize the $\alpha$-Euler equation and write the linearized operator $L_{B} $ in $\ell^{2}(\mathbb Z^{2})$ as a direct sum of one-dimensional difference operators $L_{B,\mathbf q}$ in $\ell^{2}(\mathbb Z)$ parametrized by some vectors $\mathbf q\in\mathbb Z^2$ such that the set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ covers the entire grid $\mathbb Z^{2}$. The set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ can have zero, one, or two points inside the disk of radius $\|\mathbf p\|$. We consider the case where the set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ has exactly one point in the open disc of radius $\mathbf p$. We show that unidirectional flows that satisfy this condition are linearly unstable. Our main result is an instability theorem that provides a necessary and sufficient condition for the existence of a positive eigenvalue to the operator $L_{B,\mathbf q}$ in terms of equations involving certain continued fractions. Moreover, we are also able to provide a complete characterization of the corresponding eigenvector. The proof is based on the use of continued fractions techniques expanding upon the ideas of Friedlander and Howard.

Quantum strips in higher dimensions

We consider the Dirichlet Laplacian in unbounded strips on ruled surfaces in any space dimension. We locate the essential spectrum under the condition that the strip is asymptotically flat. If the Gauss curvature of the strip equals zero, we establish the existence of discrete spectrum under the condition that the curve along which the strip is built is not a geodesic. On the other hand, if it is a geodesic and the Gauss curvature is not identically equal to zero, we prove the existence of Hardy-type inequalities. We also derive an effective operator for thin strips, which enables one to replace the spectral problem for the Laplace-Beltrami operator on the two-dimensional surface by a one-dimensional Schroedinger operator whose potential is expressed in terms of curvatures. In the appendix, we establish a purely geometric fact about the existence of relatively parallel adapted frames for any curve under minimal regularity hypotheses.

On the Decomposition of the Laplacian on Metric Graphs

We study the Laplacian on family preserving metric graphs. These are graphs that have a certain symmetry that, as we show, allows for a decomposition into a direct sum of one-dimensional operators whose properties are explicitly related to the structure of the graph. Such decompositions have been extremely useful in the study of Schr\"odinger operators on metric trees. We show that the tree structure is not essential, and moreover, obtain a direct and simple correspondence between such decompositions in the discrete and continuum case.

On families of optimal Hardy-weights for linear second-order elliptic operators

We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterize all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on (a,b), ∞≤a<b≤∞, and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. As an application, we prove a new Rellich inequality.

Discrete spectrum of Schrödinger operators with potentials concentrated near conical surfaces

In this paper we study spectral properties of a three-dimensional Schr\"odinger operator $-\Delta+V$ with a potential $V$ given, modulo rapidly decaying terms, by a function of the distance of $x \in \mathbb{R}^3$ to an infinite conical hypersurface with a smooth cross-section. As a main result we show that there are infinitely many discrete eigenvalues accumulating at the bottom of the essential spectrum which itself is identified as the ground-state energy of a certain one-dimensional operator. Most importantly, based on a result of Kirsch and Simon we are able to establish the asymptotic behavior of the eigenvalue counting function using an explicit spectral-geometric quantity associated with the cross-section. This shows a universal character of some previous results on conical layers and $\delta$-potentials created by conical surfaces.