On commuting differential operators of rank 2

We study examples of formally self-adjoint commuting ordinary differential operators of order 4 or 4g + 2 whose coefficients are analytic on ?. We prove that these operators do not commute with the operators of odd order, justifying rigorously that these operators are of rank 2.     

One-point commuting difference operators of rank 1

One-point commuting difference operators of rank 1 are considered. The coefficients in such operators depend on one functional parameter, and the degrees of shift operators in difference operators are positive. These operators are studied in the case of hyperelliptic spectral curves, where the base point coincides with a point of branching. Examples of operators with polynomial and trigonometric coefficients are constructed. Operators with polynomial coefficients are embedded in differential operators with polynomial coefficients. This construction provides a new method for constructing commutative subalgebras in the first Weyl algebra.     

On operators commuting with Toeplitz operators modulo the finite rank operators

Let X be a bounded linear operator on the Hardy space H² of the unit disk. We show that if is of finite rank for every inner function θ, then X=T_?+F for some Toeplitz operator T_? and some finite rank operator F on H². This solves a variant of an open question where the compactness replaces the finite rank conditions.     

Self-adjoint commuting ordinary differential operators

In this paper we study self-adjoint commuting ordinary differential operators of rank two. We find sufficient conditions when an operator of fourth order commuting with an operator of order 4g+2 is self-adjoint. We introduce an equation on potentials V(x),W(x) of the self-adjoint operator \(L=(\partial_{x}^{2}+V)^{2}+W\) and some additional data. With the help of this equation we find the first example of commuting differential operators of rank two corresponding to a spectral curve of higher genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.     

Identities for eigenfunctions of ordinary differential operators

The paper contains formulae for regularized k-sums of residues of the Green's function for an ordinary differential operator with regular boundary conditions.Translated from Trudy Seminara imeni I. G. Petrovskogo, Vol. 10, pp. 107–117, 1984.The author would like to express his gratitude to Victor Antonovich Sadovnichii for his interest in this work.     

Rank 2 commuting ordinary differential operators and Darboux conjugates of KdV

By considering the factorizations (flags) and associated (simultaneous) second order Darboux transformations of the square and cube of an arbitrary second order Schrödinger operator, we generate commuting ordinary differential operators of orders four and six with a singular elliptic spectrum. This procedure generates true rank 2 commutative algebras. Under the KdV flow, each such factorization (flag) leads to an integrable equation for which the corresponding Darboux transformation generates a Lax-type operator as one of a commuting pair of orders four and six with singular elliptic spectrum. Hence, these integrable equations are Darboux conjugates of KdV.     

Spectral theory of commuting self-adjoint partial differential operators

Let Ω denote a connected and open subset of $\text{R}$ⁿ. The existence of n commuting self-adjoint operators H₁,…, H_n on $\text{L}{}^2\text{(Ω)}$ such that each H_j is an extension of $\text{−}\text{i∂}\text{∂x}{}_{\mathrm{j}}$ (acting on $\text{C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(Ω))}$ is shown to be equivalent to the existence of a measure μ on $\text{R}$ⁿ such that f → $\text{}\text{̂}$tf (the Fourier transform of f) is unitary from $\text{L}{}^2\text{(Ω)}$ onto Ω. It is shown that the support of μ can be chosen as a subgroup of $\text{R}$ⁿ iff H₁,…, H_n can be chosen such that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$. This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of $\text{R}$ⁿ by some subgroup, i.e., iff Ω is essentially a fundamental domain.     

Quantum tops as examples of commuting differential operators

We study the quantum analogues of tops on the Lie algebras so(4) and e(3) represented by differential operators.     

On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

In this paper we study the class of differential operators with polynomial coefficients Q_j in one complex variable satisfying the condition degQ_jj with equality for at least one j. We show that if degQ_k<k then the root with the largest modulus of the nth degree eigenpolynomial p_n of T tends to infinity when n→∞, as opposed to the case when degQ_k=k, which we have treated previously in [T. Bergkvist, H. RullgÅrd, On polynomial eigenfunctions for a class of differential operators, Math. Res. Lett. 9 (2002) 153–171]. Moreover, we present an explicit conjecture and partial results on the growth of the largest modulus of the roots of p_n. Based on this conjecture we deduce the algebraic equation satisfied by the Cauchy transform of the asymptotic root measure of the appropriately scaled eigenpolynomials, for which the union of all roots is conjecturally contained in a compact set.     

Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions

Suppose is a sequence of polynomials orthogonal with respect to the moment functional τ=σ+ν, where σ is a classical moment functional (Jacobi, Laguerre, Hermite) and ν is a point mass distribution with finite support. In this paper, we develop a new method for constructing a differential equation having as eigenfunctions.     

An integral representation for eigenfunctions of linear ordinary differential operators

This paper generalizes an integral representation formula for eigenfunctions of Sturm-Liouville operators, known as the Volterra transformation operator in the theory of the inverse scattering problem, to higher-order differential operators. A specific fourth-order initial value problem is considered: $\text{Lφ = k}{}^4\text{φ, L =}\text{d}{}^4\text{dx}{}^4\text{+}\text{d}\text{dx}\text{(q}\text{d}\text{dx}\text{) + r}$φ(0) = 1, φ′(0) = 0, φ″(0) = ?k², φ? = 0 The solution for complex k is expressed as an inverse-Laplace-Borel transform. Jump formulae are obtained relating the representing kernel directly to the coefficients of L. The result admits obvious generalization to operators of arbitrary order.