Order Ordinary Differential Operators Research Articles

Global regularity of second order ordinary differential operators with polynomial coefficients

Let Lu=u″+2Au′+Bu be an ordinary differential operator with A and B are polynomials of any degree. In this paper we show that L is globally regular, i.e. Lu∈S implies u∈S, for all tempered distribution u∈S′, if and only if the complex roots ξ±(x)=iA(x)±B(x)−A′(x)−(A(x))2 of the Weyl symbol σL(x,ξ)=−ξ2+2iA(x)ξ+B(x)−A′(x) satisfy the condition lim|x|→+∞⁡|xImξ±(x)|=+∞.

Direct and Inverse Problems of Spectral Analysis for Arbitrary-Order Differential Operators with Nonintegrable Regular Singularities

A short review is presented of results on the spectral theory of arbitrary order ordinary differential operators with non-integrable regular singularities. We establish properties of spectral characteristics, prove theorems on completeness of root functions in the corresponding spaces, prove expansion and equiconvergence theorems, and provide a solution of the inverse spectral problem for this class of operators.

Inverse Spectral Problems for Arbitrary-Order Differential Operators with Distribution Coefficients

In this paper, we propose an approach to inverse spectral problems for the n-th order (n≥2) ordinary differential operators with distribution coefficients. The inverse problems which consist in the reconstruction of the differential expression coefficients by the Weyl matrix and by several spectra are studied. We prove the uniqueness of solution for these inverse problems, by developing the method of spectral mappings. The results of this paper generalize the previously known results for the second-order differential operators with singular potentials and for the higher-order differential operators with regular coefficients. In the future, the approach of this paper can be used for constructive solution and for investigation of solvability of the considered inverse problems.

A trace formula for higher order ordinary differential operators

We obtain a first-order trace formula for a higher order differential operator on a closed interval in the case where the perturbation operator is the operator of multiplication by a finite complex-valued charge. For operators of even orders , the result contains a term of new type, previously unknown. Bibliography: 15 titles.

A trace formula for higher order ordinary differential operators

Получена формула следа первого порядка для дифференциального оператора высокого порядка на отрезке в случае, когда возмущающий оператор является оператором умножения на конечный комплекснозначный заряд. Для операторов четных порядков $n\geqslant4$ результат содержит слагаемое нового типа, не известное ранее. Библиография: 15 названий.

Riesz Inequality for the System of Root Functions of Second Order Ordinary Differential Operator

An ordinary differential operator of second order with coefficients is considered. The Riesz property of the system of root functions of the given operator is studied. The criterion of Bessel property in 2 L , of root functions system is established and use it to obtain sufficient conditions for the Riesz property of a system of normalized root functions of this operator in p L .

Global regularity of second order twisted differential operators

In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension 2. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree 2. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin.

Asymptotics of determinants of 4‐th order operators at zero

AbstractWe consider fourth order ordinary differential operators on the half‐line and on the line, where the perturbation has compactly supported coefficients. The Fredholm determinant for this operator is an analytic function in the whole complex plane without zero. We describe the determinant at zero. We show that in the generic case it has a pole of order 4 in the case of the line and of order 1 in the case of the half‐line.

Resonances of third order differential operators

We consider resonances for third order ordinary differential operators with compactly supported coefficients on the real line. Resonance are defined as zeros of a Fredholm determinant on a non-physical sheet of three sheeted Riemann surface. We determine upper bounds of the number of resonances in complex discs at large radius. We express the trace formula in terms of resonances only.

A Birman–Kreĭn–Vishik–Grubb Theory for Sectorial Operators

We consider densely defined sectorial operators $$A_\pm $$ that can be written in the form $$A_\pm =\pm iS+V$$ with $$\mathcal {D}(A_\pm )=\mathcal {D}(S)=\mathcal {D}(V)$$ , where both S and $$V\ge \varepsilon >0$$ are assumed to be symmetric. We develop an analog to the Birmin–Kreĭn–Vishik–Grubb (BKVG) theory of selfadjoint extensions of a given strictly positive symmetric operator, where we will construct all maximally accretive extensions $$A_D$$ of $$A_+$$ with the property that $$\overline{A_+}\subset A_D\subset A_-^*$$ . Here, D is an auxiliary operator from $$\ker (A_-^*)$$ to $$\ker (A_+^*)$$ that parametrizes the different extensions $$A_D$$ . After this, we will give a criterion for when the quadratic form $$\psi \mapsto {\text{ Re }}\langle \psi ,A_D\psi \rangle $$ is closable and show that the selfadjoint operator $$\widehat{V}$$ that corresponds to the closure is an extension of V. We will show how $$\widehat{V}$$ depends on D, which—using the classical BKVG-theory of selfadjoint extensions—will allow us to define a partial order on the real parts of $$A_D$$ depending on D. All of our results will be presented in a way that emphasizes their connection to the classical BKVG-theory and that shows how the BKVG formulas generalize when considering sectorial operators $$A_\pm =\pm iS+V$$ instead of just a strictly positive symmetric operator V. Applications to second order ordinary differential operators are discussed.

D-Orthogonal Analogs of Classical Orthogonal Polynomials

Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial systems, enjoy a number of properties which make them a natural analog of the classical orthogonal polynomials. In the present paper we continue their study. The most important new properties are their hypergeometric representations which allow us to derive their generating functions and in some cases also Mehler-Heine type formulas.

Sufficient Conditions for the Basicity in Lp and Equiconvergence with the Trigonometric Series of Spectral Decompositions for the Second Order Ordinary Differential Operator

Theorems about sufficient conditions for the basicity for second order ordinary differential operator are proved.

Generation of analytic semigroups with generalized Wentzell boundary condition

In this paper we study second order ordinary and partial differential equations with generalized Wentzell boundary condition. We prove, by a perturbation method, that certain second order ordinary differential operators generate an analytic semigroup in \(W^{1,p}([0,1])\) and the same result has been extended for the degenerate operator \(Au(x):=x(1-x)u''(x)\). Finally, we prove that certain linear partial differential operators of the second order generate analytic semigroups in the space of continuous functions.

Unitary Equivalence of Proper Extensions of a Symmetric Operator and the Weyl Function

Let A be a densely defined simple symmetric operator in \({\mathfrak{H}}\), let \({\Pi=\{\mathcal{H},\Gamma_0, \Gamma_1}\}\) be a boundary triplet for A * and let M(·) be the corresponding Weyl function. It is known that the Weyl function M(·) determines the boundary triplet Π, in particular, the pair {A, A 0}, uniquely up to the unitary similarity. Here \({A_0 := A^* \upharpoonright \text{ker}\, \Gamma_0 ( = A^*_0)}\). At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to the weak similarity. We consider a symmetric dual pair {A, A} with symmetric \({A \subset A^*}\) and a special boundary triplet \({\widetilde{\Pi}}\) for{A, A} such that the corresponding Weyl function is \({\widetilde{M}(z) = K^*(B-M(z))^{-1} K}\), where B is a non-self-adjoint bounded operator in \({\mathcal{H}}\). We are interested in the problem whether the result on the unitary similarity remains valid for \({\widetilde{M}(\cdot)}\) in place of M(·). We indicate some sufficient conditions in terms of the operators A 0 and \({A_B= A^* \upharpoonright \text{ker}\, (\Gamma_1-B \Gamma_0)}\), which guaranty an affirmative answer to this problem. Applying the abstract results to the minimal symmetric 2nth order ordinary differential operator A in \({L^2(\mathbb{R}_+)}\), we show that \({\widetilde{M}(\cdot)}\) defined in \({\Omega_+ \subset \mathbb{C}_+}\)determines the Dirichlet and Neumann realizations uniquely up to the unitary equivalence. At the same time similar result for realizations of Dirac operator fails. We obtain also some negative abstract results demonstrating that in general the Weyl function \({\widetilde{M}(\cdot)}\) does not determine A B even up to the similarity.

Sobolev type inequalities of time-periodic boundary value problems for Heaviside and Thomson cables

We consider a time-periodic boundary value problem of n th order ordinary differential operator which appears typically in Heaviside cable and Thomson cable theory. We calculate the best constant and a family of the best functions for a Sobolev type inequality by using the Green function and apply its results to the cable theory. Physical meaning of a Sobolev type inequality is that we can estimate the square of maximum of the absolute value of AC output voltage from above by the power of input voltage. MSC:46E35, 41A44, 34B27.

The spectrum of differential operators and square-integrable solutions

We give a comprehensive account of the relationship between the square-integrable solutions for real values of the spectral parameter λ and the spectrum of self-adjoint even order ordinary differential operators with real coefficients and arbitrary deficiency index d and we solve an open problem stated by Weidmann in his well-known 1987 monograph. According to a well-known result, if one endpoint is regular and for some real value of the spectral parameter λ the number of linearly independent square-integrable solutions is less than d, then λ is in the essential spectrum of every self-adjoint realization of the equation. Weidmann extends this result to the two singular endpoint case provided an additional condition is satisfied. Here we prove this result without the additional condition.

On stable self-similar blowup for equivariant wave maps

We consider corotational wave maps from (3 + 1) Minkowski space into the 3-sphere. This is an energy supercritical model that is known to exhibit finite-time blowup via self-similar solutions. The ground state self-similar solution f0 is known in closed form, and according to numerics, it describes the generic blowup behavior of the system. We prove that the blowup via f0 is stable under the assumption that f0 does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blowup to a linear ODE spectral problem. Although we are unable at the moment to verify the mode stability of f0 rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of f0. © 2011 Wiley Periodicals, Inc.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

Resolvent Estimate in L_p for Discretisations of Second Order Ordinary Differential Operators on Variable Grids

Abstract In this paper, a resolvent estimate is proved for discretisations of second order ordinary differential operators subject to Dirichlet boundary conditions on a finite or infinite interval. As a discretisation method, a fully discrete Galerkin method using continuous splines of order r > 2 on a locally quasi-uniform grid is considered. As a byproduct, an a priori estimate for the discretised differential operator is obtained.

The Spectrum of Differential Operators of Order 2n with Almost Constant Coefficients

We discuss the spectral properties of higher order ordinary differential operators. If the coefficients differ from constants by small perturbations, then the spectral properties are preserved. In this context, “small perturbations” are either short range (i.e., integrable) or long range, but slowly varying. This generalizes classical results on second order operators. Our approach relies on an analysis of the associated differential equations with the help of uniform asymptotic integration techniques.