Linear Ordinary Differential Operators Research Articles

Maximum, anti-maximum principles and monotone methods for boundary value problems for Riemann-Liouville fractional differential equations in neighborhoods of simple eigenvalues

It has been shown that, under suitable hypotheses, boundary value problems of the form, $Ly+\lambda y=f,$ $BC y =0$ where $L$ is a linear ordinary or partial differential operator and $BC$ denotes a linear boundary operator, then there exists $\Lambda >0$ such that $f\ge 0$ implies $\lambda y \ge 0$ for $\lambda\in [-\Lambda ,\Lambda ]\setminus\{0\},$ where $y$ is the unique solution of $Ly+\lambda y=f,$ $BC y =0$. So, the boundary value problem satisfies a maximum principle for $\lambda\in [-\Lambda ,0)$ and the boundary value problem satisfies an anti-maximum principle for $\lambda\in (0, \Lambda ]$. In an abstract result, we shall provide suitable hypotheses such that boundary value problems of the form, $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y=f,$ $BC y =0$ where $D_{0}^{\alpha}$ is a Riemann-Liouville fractional differentiable operator of order $\alpha$, $1<\alpha \le 2$, and $BC$ denotes a linear boundary operator, then there exists $\mathcal{B} >0$ such that $f\ge 0$ implies $\beta D_{0}^{\alpha -1}y \ge 0$ for $\beta \in [-\mathcal{B} ,\mathcal{B} ]\setminus\{0\},$ where $y$ is the unique solution of $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y =f,$ $BC y =0$. Two examples are provided in which the hypotheses of the abstract theorem are satisfied to obtain the sign property of $\beta D_{0}^{\alpha -1}y.$ The boundary conditions are chosen so that with further analysis a sign property of $\beta y$ is also obtained. One application of monotone methods is developed to illustrate the utility of the abstract result.

Financial Interpretation of Feller’s Factorization

The infinitesimal generator of a time-homogeneous univariate diffusion process is a second-order linear ordinary differential operator. Feller (1952) famously factorized this generator into successive differentiations with respect to scale and speed measure. Later, Feller (1957) also factored an extended generator that loads also on the identity operator in a particular way. We provide a novel financial interpretation of these factorization results and show that they produce an operator representation of a conditionally linear risk-return tradeoff when the conditioning variable evolves like a one-dimensional diffusion process.

Construction of an Infinite Cyclic Group Formed by Artificial Differential Neurons

Infinite cyclic groups created by various objects belong to the class to the class basic algebraic structures. In this paper, we construct the infinite cyclic group of differential neurons which are modifications of artificial neurons in analogy to linear ordinary differential operators of the n-th order. We also describe some of their basic properties.

Sequences of Groups, Hypergroups and Automata of Linear Ordinary Differential Operators

The main objective of our paper is to focus on the study of sequences (finite or countable) of groups and hypergroups of linear differential operators of decreasing orders. By using a suitable ordering or preordering of groups linear differential operators we construct hypercompositional structures of linear differential operators. Moreover, we construct actions of groups of differential operators on rings of polynomials of one real variable including diagrams of actions–considered as special automata. Finally, we obtain sequences of hypergroups and automata. The examples, we choose to explain our theoretical results with, fall within the theory of artificial neurons and infinite cyclic groups.

Gamma functions, monodromy and Frobenius constants

In their paper on the gamma conjecture in mirror symmetry, Golyshev and Zagier introduce what we refer to as Frobenius constants associated to an ordinary linear differential operator L with a reflection type singularity. These numbers describe the variation around the reflection point of Frobenius solutions to L defined near other singular points. Golyshev and Zagier show that in certain geometric cases Frobenius constants are periods, and they raise the question quite generally how to describe these numbers motivically. In this paper we give a relation between Frobenius constants and Taylor coefficients of generalized gamma functions, from which it follows that Frobenius constants of Picard--Fuchs differential operators are periods. We also study the relation between these constants and periods of limiting Hodge structures. This is a major revision of the previous version of the manuscript. The notion of Frobenius constants and our main result are extended to the general case of regular singularities with any sets of local exponents. In addition, the generating function of Frobenius constants is given explicitly for all hypergeometric connections.

Minimality of eigenfunctions and associated functions of ordinary differential operators

Let E, F, H be Banach spaces, $$A,B\in L(E,F)$$ such that $$A+\lambda B$$ and its adjoint $$A^*+\lambda B^*$$ have a biorthogonal system of eigenvectors and associated vectors $$(y_i)_{i=1}^\infty$$ and $$(v_i)_{i=1}^\infty$$ . It is assumed that a closed finite codimensional subspace $$E_0$$ of E exists such that $$E_0$$ is dense in H and such that the restriction of B to $$E_0$$ has a continuous extension to H. It is shown that a subsystem $$(y_i)_{i=k+1}^\infty$$ is minimal in H if $$(v_i)_{i=1}^k$$ has a certain basis property. This abstract result can be applied to regular ordinary linear differential operators in Sobolev spaces whose boundary conditions may depend linearly on the eigenvalue parameter for proving minimality in $$L_p$$ . Explicit examples for Sturm–Liouville problems with one or both boundary conditions depending on $$\lambda$$ are considered.

On a spectral theorem of Weyl

We give a new proof of a theorem of Weyl on the continuous part of the spectrum of Sturm–Liouville operators on the half-line with asymptotically constant coefficients. Earlier arguments, due to Weyl and Kodaira, depended on particular features of Green’s functions for linear ordinary differential operators. We use a concept of asymptotic containment of C∗-algebra representations that has geometric origins. We apply the concept elsewhere to the Plancherel formula for spherical functions on reductive groups.

Invertible linear ordinary differential operators and their generalizations

In this paper, the previously obtained description of invertible linear ordinary differential operators is generalized to invertible linear discrete-time operators, unimodular matrices, and transformations of control systems. The description is based on assigning a model in the form of a set of squares on the plane to each generalized invertible operator. The number of squares of the model is invariant and can be used to classify generalized invertible operators and, in particular, transformations of control systems.

Formal Factorization of Higher Order Irregular Linear Differential Operators

We study the problem of formal decomposition (non-commutative factorization) of linear ordinary differential operators over the field $${{\mathbb {C}}}(\!(t)\!)$$ of formal Laurent series at an irregular singular point corresponding to $$t=0$$ . The solution (given in terms of the Newton diagram and the respective characteristic numbers) is known for quite some time, though the proofs are rather involved. We suggest a process of reduction of the non-commutative problem to its commutative analog, the problem of factorization of pseudopolynomials, which is known since Newton invented his method of rotating ruler. It turns out that there is an “automatic translation” which allows to obtain the results for formal factorization in the Weyl algebra from well known results in local analytic geometry. In addition, we draw some (apparently unnoticed) parallels between the formal factorization of linear operators and formal diagonalization of systems of linear first order differential equations.

Invertible Linear Ordinary Differential Operators Represented as a Composition of the Simplest Operators

This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.

Invertible linear ordinary differential operators

We consider invertible linear ordinary differential operators whose inversions are also differential operators. To each such operator we assign a numerical table. These tables are described in the elementary geometrical language. The table does not uniquely determine the operator. To define this operator uniquely some additional information should be added, as it is described in detail in this paper. The possibility of generalization of these results to partial differential operators is also discussed.

Characterization of self-adjoint domains for differential operators with interior singular points

We characterize the self-adjoint domains of general even order linear ordinary differential operators which have finite interior singular points in terms of real-parameter solutions of the differential equation. For the purpose we constructed a direct sum space. By the theory of direct sum space and the decomposition of the corresponding maximal domain, we give this complete and analytic characterization in terms of limit-circle solutions. This is for endpoints which are regular or singular and for arbitrary deficiency index.

Linear differential equations on [formula omitted] and root systems

In this paper, we study the Euler transform on linear ordinary differential operators on P1. The spectral type is the tuple of integers which count the multiplicities of local formal solutions with the same leading terms. We compute the changes of spectral types under the action of the Euler transform and show that the changes of spectral types generate a transformation group of a Z-lattice which is isomorphic to a quotient lattice of a Kac–Moody root lattice with the Weyl group as the transformation group.

Transmutations, L-bases and complete families of solutions of the stationary Schrödinger equation in the plane

An L-basis associated to a linear second-order ordinary differential operator L is an infinite sequence of functions {φk}k=0∞ such that Lφk=0 for k=0,1, Lφk=k(k−1)φk−2, for k=2,3,… and all φk satisfy certain prescribed initial conditions. We study the transmutation operators related to L in terms of the transformation of powers of the independent variable {(x−x0)k}k=0∞ to the elements of the L-basis and establish a precise form of the transmutation operator realizing this transformation. We use this transmutation operator to establish a completeness of an infinite system of solutions of the stationary Schrödinger equation from a certain class. The system of solutions is obtained as an application of the theory of bicomplex pseudoanalytic functions and its completeness was a long sought result. Its use for constructing reproducing kernels and solving boundary and eigenvalue problems has been considered even without the required completeness justification. The obtained result on the completeness opens the way for further development and application of the tools of pseudoanalytic function theory.

Fast computation of common left multiples of linear ordinary differential operators

We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coef- ficients. We analyse the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM compu- tation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output.

An algorithm for the decomposition of differential polynomials in the general case

The theory of decomposition of differential polynomials (DPs) is considered. For a given differential algebraic equation $P(x,y(x), y'(x), \ldots ,y^{(n)} (x)) = 0,$ the possibility to represent the differential polynomial P as the composition P = Q(x, R, R?, ..., R (q)), R = R(x, y(x), ..., y (r)(x)) of DPs Q and R is studied. It is shown that the decomposition problem is reduced to the factorization of linear ordinary differential operators (LODOs). A generic decomposition algorithm for DPs is described based on the Vandermonde differential theorem.

On the exact WKB analysis of higher order simple-pole type operators

Higher order simple-pole type operators, that is, higher order linear ordinary differential operators with a large parameter η whose coefficients have simple poles at the origin, are discussed from the viewpoint of the exact WKB analysis. Making use of the technique of microdifferential operators, we clarify the singularity structure of the Borel transform of their WKB solutions.

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.

Characterization of Domains of Self-Adjoint Ordinary Differential Operators II

We characterize the self-adjoint domains of general even order linear ordinary differential operators in terms of real-parameter solutions of the differential equation. This for endpoints which are regular or singular and for arbitrary deficiency index. This characterization is obtained from a new decomposition of the maximal domain in terms of limit-circle solutions. These are the solutions which contribute to the self-adjoint domains in analogy with the celebrated Weyl limit-circle solutions in the second order Sturm–Liouville case. As a special case we obtain the previously known characterizations when one or both endpoints are regular.

Galoisian approach to integrability of Schrödinger equation

In this paper, we examine the nonrelativistic stationary Schrodinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second-order ordinary linear differential operators, so as to achieve rational function coefficients (“algebrization”), and Kovacic's algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schrodinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible.