Commuting Differential Operators of Rank 2 with Rational Coefficients

In this paper we find new pairs of self-adjoint commuting differential operators of rank 2 with rational coefficients and prove that any curve of genus 2 written as a hyperelliptic curve is the spectral curve of a pair of commuting differential operators with rational coefficients. We also study the case where curves of genus 3 are the spectral curves of pairs of commuting differential operators of rank 2 with rational coefficients.     

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.     

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.     

Commuting Differential Operators of Rank 2 with Polynomial Coefficients

Self-adjoint commuting differential operators with polynomial coefficients are considered. These operators form a commutative subalgebra of the first Weyl algebra. New examples of commuting differential operators of rank 2 are found.     

Alternative Proof of Mironov’s Results on Commuting Self-Adjoint Operators of Rank 2

We give an alternative proof of Mironov’s results on commuting self-adjoint operators of rank 2. Mironov’s proof is based on Krichever’s complicated theory of the existence of a high-rank Baker–Akhiezer function. In contrast to Mironov’s proof, our proof is simpler but the results are slightly weaker. Note that the method of this article can be extended to matrix operators. Using the method, we can construct the first explicit examples of matrix commuting differential operators of rank 2 and arbitrary genus.     

The Dressing Chain and One-Point Commuting Difference Operators of Rank 1

We construct solutions to the difference-differential equation that are associated with onepoint commuting difference operators of rank 1 in the case of spectral curves of genus 1.     

Commuting Krichever–Novikov differential operators with polynomial coefficients

Under study are some commuting rank 2 differential operators with polynomial coefficients. We prove that, for every spectral curve of the form w² = z³+c₂z2+c₁z+c₀ with arbitrary coefficients c_i, there exist commuting nonselfadjoint operators of orders 4 and 6 with polynomial coefficients of arbitrary degree.     

Commuting Rank 2 Differential Operators Corresponding to a Curve of Genus 2

We construct commuting rank 2 formally self-adjoint ordinary differential operators corresponding to a curve of genus 2.     

One-Point Commuting Difference Operators of Rank One and Their Relation with Finite-Gap Schrödinger Operators

One-point commuting difference operators of rank one in the case of hyperelliptic spectral curves are studied. A relationship between such operators and one-dimensional finite-gap Schrödinger operators is investigated. In particular, a discretization of finite-gap Lamé operators is obtained.     

Nonisothermal filtration and seismic acoustics in porous soil: Thermoviscoelastic equations and Lamé equations

We study applications of a new class of infinite-dimensional Lie algebras, called Lax operator algebras, which goes back to the works by I. Krichever and S. Novikov on finite-zone integration related to holomorphic vector bundles and on Lie algebras on Riemann surfaces. Lax operator algebras are almost graded Lie algebras of current type. They were introduced by I. Krichever and the author as a development of the theory of Lax operators on Riemann surfaces due to I. Krichever, and further investigated in a joint paper by M. Schlichenmaier and the author. In this article we construct integrable hierarchies of Lax equations of that type. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 216–226. Dedicated to S.P. Novikov on the occasion of his 70th birthday     

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.     

Symmetric Fock space and orthogonal symmetric polynomials associated with the Calogero model

Using a similarity transformation that maps the Calogero model into N decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their simultaneous eigenfunctions.The simultaneous eigenfunction is a deformation of the symmetrized number state (bosonic state) and forms an orthogonal basis of the Hilbert (Fock) space of the model. This orthogonal basis is different from the known one that is a variant of the Jack polynomial, i.e., the Hi-Jack polynomial. This fact shows that the conserved operators derived by the similarity transformation and those derived by the Dunkl operator formulation do not commute. Thus we conclude that the Calogero model has two, algebraically inequivalent sets of mutually commuting conserved operators, as is the case with the hydrogen atom. We also confirm the same story for the B_N-Calogero model.     

On stable operator polynomials

We attempt to generalize the classical theorem of Wall on the stability of an ordinary numerical polynomial to the operator-valued case. It is shown that such a generalization is admissible if the coefficients of the operator polynomial are uniformly positive commuting operators in Hilbert space. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 144–147.     

Well-Boundedness of Sums and Products of Operators

A sufficient condition is given under which the sum, productand indeed any polynomial combination of a well-bounded operatorand a commuting real scalar-type spectral operator is well-bounded.This generalizes a result of Gillespie for Hilbert space operators.It is shown in particular that if X is a UMD space, then thesum of finitely many commuting real scalar-type spectral operatorsacting on X is a well-bounded operator (a result which failson general reflexive Banach spaces).     

Necessary Covariance Conditions for a One-Field Lax Pair

We study the covariance with respect to Darboux transformations of polynomial differential and difference operators with coefficients given by functions of one basic field. In the scalar (Abelian) case, the functional dependence is established by equating the Frechet differential (the first term of the Taylor series on the prolonged space) to the Darboux transform; a Lax pair for the Boussinesq equation is considered. For a pair of generalized Zakharov-Shabat problems (with differential and shift operators) with operator coefficients, we construct a set of integrable nonlinear equations together with explicit dressing formulas. Non-Abelian special functions are fixed as the fields of the covariant pairs. We introduce a difference Lax pair, a combined gauge-Darboux transformation, and solutions of the Nahm equations.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 122–132, July, 2005.     

Remarks on the classification of a pair of commuting semilinear operators

Gelfand and Ponomarev [I.M. Gelfand, V.A. Ponomarev, Remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space, Funct. Anal. Appl. 3 (1969) 325–326] proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k-tuples of linear operators for any k. We prove an analogous statement for semilinear operators.     

Symbolic Computation of Newton Sum Rules for the Zeros of Polynomial Eigenfunctions of Linear Differential Operators

A symbolic algorithm based on the generalized Lucas polynomials of first kind is used in order to compute the Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators with polynomial coefficients.     

New results on the Bochner condition about classical orthogonal polynomials

The classical polynomials (Hermite, Laguerre, Bessel and Jacobi) are the only orthogonal polynomial sequences (OPS) whose elements are eigenfunctions of the Bochner second-order differential operator F (Bochner, 1929 [3]). In Loureiro, Maroni and da Rocha (2006) [18] these polynomials were described as eigenfunctions of an even order differential operator Fk with polynomial coefficients defined by a recursive relation. Here, an explicit expression of Fk for any positive integer k is given. The main aim of this work is to explicitly establish sums relating any power of F with Fk, k?1, in other words, to bring a pair of inverse relations between these two operators. This goal is accomplished with the introduction of a new sequence of numbers: the so-called A-modified Stirling numbers, which could be also called as Bessel or Jacobi-Stirling numbers, depending on the context and the values of the complex parameter A.     

Halving for the 2-Sylow subgroup of genus 2 curves over binary fields

We give a deterministic polynomial time algorithm to find the structure of the 2-Sylow subgroup of the Jacobian of a genus 2 curve over a finite field of characteristic 2. Our procedure starts with the points of order 2 and then performs a chain of successive halvings while such an operation makes sense. The stopping condition is triggered when certain polynomials fail to have roots in the base field, as previously shown by I. Kitamura, M. Katagi and T. Takagi. The structure of our algorithm is similar to the already known case of genus 1 and odd characteristic.     

Linear Operators that Preserve Commuting Pairs of Nonnegative Real Matrices

A pair of n×n matrices (A, B) is called a commuting pair if AB=BA. We characterize the linear operators that preserve the set of commuting pairs of matrices over a subsemiring of nonnegative real numbers.