Order Differential Operators Research Articles

Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D

It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.

Quantization of hyper-elliptic curves from isomonodromic systems and topological recursion

We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of spectral Darboux coordinates on the moduli space of meromorphic sl2-connections on P1 and argue that the topological recursion produces a 2g-parameter family of associated tau functions, where 2g is the dimension of the moduli space considered. We apply this procedure to the 6 Painlevé equations which correspond to g=1 and consider a g=2 example.

A Local-in-Time Theory for Singular SDEs with Applications to Fluid Models with Transport Noise

We establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in Hilbert spaces. The key requirement relies on an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. By establishing a cancellation estimate for certain differential operators of order one with suitable coefficients, we give the detailed constructions of such regular approximations for certain examples. In particular, we show novel local-in-time results for the stochastic two-component Camassa–Holm system and for the stochastic Córdoba–Córdoba–Fontelos model.

Closed-Form Solutions of Linear Ordinary Differential Equations with General Boundary Conditions

This paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear operators. The method is based on the theory of the extensions of linear operators in Banach spaces.

Bounds on the Phillips Calculus of Abstract First Order Differential Operators

For an operator generating a group on \(L^p\) spaces transference results give bounds on the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific group generators which are abstraction of first order differential operators and prove similar spectral multiplier estimates assuming only that the group is bounded on \(L^2\) rather than \(L^p\). We also prove an R-bounded Hörmander calculus result by assuming an abstract Sobolev embedding property and show that the square of a perturbed Hodge–Dirac operator has such calculus.

On the spectrum of fractional difference operator

This work mainly explores the idea of the spectrum of the fractional difference operator (of order ) over the Banach space ℓ1. The difference operator via the sequence x = (x k ) is defined by where (α) i denotes the Pochhammer symbol, and ν = (ν k ) is either a constant or strictly decreasing sequence of positive real numbers with some conditions. Sets comprising of the point, the residual and the continuous spectrum of the operator on the Banach space ℓ1 are computed.

Jacobi Manifolds, Contact Manifolds and Contactomorphism

Let M be a smooth manifold and let D(M) be the module of first order differential operators on M. In this work, we give a link between Jacobi manifolds and Contact manifolds. We also generalize the notion of contactomorphism on M and
 thus, we characterize the Contact diffeomorphisms.

Nevanlinna Theory for Jackson Difference Operators and Entire Solutions of q-Difference Equations

This paper establishes a version of Nevanlinna theory based on Jackson difference operator $D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z}$ for meromorphic functions of zero order in the complex plane $\mathbb{C}$. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem and five-value theorem in sense of Jackson $q$-difference operator. By using this theory, we investigate the growth of entire solutions of linear Jackson $q$-difference equations $D^{k}_{q}f(z)+A(z)f(z)=0$ with meromorphic coefficient $A,$ where $D^k_q$ is Jackson $k$-th order difference operator, and estimate the logarithmic order of some $q$-special functions.

Folded and contracted solutions of coupled classical dynamical Yang–Baxter and reflection equations

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space a satisfying three coupled classical dynamical Yang–Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on a, generalising asymptotic boundary Knizhnik–Zamolodchikov–Bernard (KZB) equations.The recipe involves folding and contracting a-invariant and θ-twisted symmetric classical dynamical r-matrices along an involutive automorphism θ. In case of the universal enveloping algebra of a simple Lie algebra g we determine the subclass of Schiffmann’s classical dynamical r-matrices which are a-invariant and θ-twisted.The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.

Holomorphic functions, relativistic sum, Blaschke products and superoscillations

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The notion of superoscillation is a particular case of that one of supershift. In the recent years, superoscillating functions, that appear for example in weak values in quantum mechanics, have become an interesting and independent field of research in complex analysis and in the theory of infinite order differential operators. The aim of this paper is to study some infinite order differential operators acting on entire functions which naturally arise in the study of superoscillating functions. Such operators are of particular interest because they are associated with the relativistic sum of the velocities and with the Blaschke products. To show that some sequences of functions preserve the superoscillatory behavior it is of crucial importance to prove that their associated infinite order differential operators act continuously on some spaces of entire functions with growth conditions.

Explicit inverse of near Toeplitz pentadiagonal matrices related to higher order difference operators

This paper analyzes the inverse of near Toeplitz pentadiagonal matrices, arising from a finite-difference approximation to the fourth-order nonlinear beam equation. Explicit non-recursive inverse matrix formulas and bounds of norms of the inverse matrix are derived for the clamped–free and clamped–clamped boundary conditions. The bound of norms is then used to construct a convergence bound for the fixed-point iteration of the form u=f(u) for solving the nonlinear equation. Numerical computations presented in this paper confirm the theoretical results.

On the geometry of Petrov type II spacetimes

In general, geometries of Petrov type II do not admit symmetries in terms of Killing vectors or spinors. We introduce a weaker form of Killing equations which do admit solutions. In particular, there is an analog of the Penrose–Walker Killing spinor. Some of its properties, including associated conservation laws, are discussed. Perturbations of Petrov type II Einstein geometries in terms of a complex scalar Debye potential yield complex solutions to the linearized Einstein equations. The complex linearized Weyl tensor is shown to be half Petrov type N. The remaining curvature component on the algebraically special side is reduced to a first order differential operator acting on the potential.

M\xfcller\u2013Zhang truncation for general linear constraints with first or second order potential

Let mathcal {B} be a homogeneous differential operator of order l=1 or l=2. We show that a sequence of functions of the form (mathcal {B}u_j)_j converging in the L^1-sense to a compact, convex set K can be modified into a sequence converging uniformly to this set provided that the derivatives of order l are uniformly bounded. We prove versions of our result on the whole space, an open domain, and for K varying uniformly continuously on an open, bounded domain. This is a conditional generalization of a theorem proved by S. Müller for sequences of gradients. Moreover, a potential of order two for the linearized isentropic Euler system is constructed.

Spectral analysis of one class perturbed first order differential operators

We use the method of similar operators to study a mixed problem for a first order differential equation with a fractional integration operator. The differential operator defined by the equation is transformed into a similar operator that is an orthogonal direct sum of finite-rank operators. The estimates of eigenvalues, eigenvectors are obtained. The result is used to construct an operator group.

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.

Solvability of a Class of First Order Differential Operators on the Torus

This paper deals with Gevrey global solvability on the N-dimensional torus ( $${\mathbb {T}}^{N}\simeq {\mathbb {R}}^{N}/2\pi {\mathbb {Z}}^{N}$$ ) to a class of nonlinear first order partial differential equations in the form $$Lu-au-b{\overline{u}}=f$$ , where a, b, and f are Gevrey functions on $${\mathbb {T}}^{N}$$ and L is a complex vector field defined on $${\mathbb {T}}^{N}$$ . Diophantine properties of the coefficients of L appear in a natural way in our results. Also, we present results in $$C^\infty $$ context.

Heat kernel upper bounds for symmetric Markov semigroups

It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of [6] for Nash-type inequalities with power order considerably and also extend that of [26] for second order differential operators on a complete non-compact manifold.

A Second Regularized Trace Formula for a Fourth Order Differential Operator

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.

Higher order difference operators and uniqueness of meromorphic functions

It is a known uniqueness result that if f is a transcendental meromorphic function of finite order with two Borel excetional values $$a \ne \infty , b$$ and its first order difference operator $${\triangle }_c f\not \equiv 0$$ , for some complex number c, and if f and $${\triangle }_c f$$ share a, b CM, then $$a=0, b=\infty $$ and $$f=\exp (Az+B)$$ , where $$A\ne 0, B \in {\mathbb {C}} $$ . This type of results has its origin dating back to Csillag-Tumura’s uniqueness theorems. In this paper, by using completely different methods, we shall show that the result holds for arbitrary higher order difference operators. Examples are provided to show that this result is not valid for meromorphic functions with infinite order, which also shows a distinction between the derivatives and the difference operators of meromorphic functions, in view of Csillag-Tumura type uniqueness theorems.

DNT preconditioner for one-sided space fractional diffusion equations

In this paper, we consider diagonal-times-nonsymmetric-Toeplitz (DNT) preconditioner for Toeplitz-like linear systems arising from one-sided space fractional diffusion equations (OSFDE) with variable coefficients. We first correct the result of Lemma 3.3 in Lin et al. (BIT Numer Math 58, 729–748, 2018) and prove the corrected result. We then extend the DNT preconditioner to Toeplitz-like linear systems arising from OSFDEs, where high order difference operators are applied to discretize the fractional derivative. Theoretically, we prove that the condition number of the preconditioned matrix is uniformly bounded by a constant under mild assumptions and verify that several discretization schemes from the literature satisfy the required assumptions. Numerical results are reported to show the efficiency of the DNT preconditioner.