Covariant differential operators

Supported in part by the Louisiana Education Quality Support Fund 86-LBR-016-04     

Commuting differential operators

We consider a model problem, related to Bessel functions, in the theory of commutative rings of linear differential operators with one independent variable. We construct new series of such commutative rings and give simple examples of rings with three generators.     

Traces of differential operators

The present survey is dedicated to the history of the state of the art in the theory of regularized traces of linear differential operators. Attention is mainly paid to operators with discrete spectrum.     

Hypoelliptic ordinary differential operators

Necessary and sufficient conditions for the hypoellipticity of an ordinary differential operator withC ^∞ coefficients in a neighborhood of a zero of finite order of the leading term are given. A sufficient condition for such an operator to be in a certain Hörmander class is also given.     

Self-adjoint abstract differential operators

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H₁ whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula AE, B(x)0, and the operator B(x) is bounded for all x. An operator L₀ is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) H₁ according to the formula: L₀y=–y + Q(x)y (–0 is a self-adjoint operator in H₁ under the given assumptions.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 65–72, July, 1969.     

Hankel operators that commute with second-order differential operators

Suppose that Γ is a continuous and self-adjoint Hankel operator on L²(0,∞) with kernel ?(x+y) and that with a(0)=0. If a and b are both quadratic, hyperbolic or trigonometric functions, and ? satisfies a suitable form of Gauss's hypergeometric differential equation, or the confluent hypergeometric equation, then ΓL=LΓ. The paper catalogues the commuting pairs Γ and L, including important cases in random matrix theory. There are also results proving rapid decay of the singular numbers of Hankel integral operators with kernels that are analytic and of exponential decay in the right half-plane.     

Rings of h-deformed differential operators

We describe the center of the ring Diff _h (n) of h-deformed differential operators of type A. We establish an isomorphism between certain localizations of Diff _h (n) and the Weyl algebra W _n , extended by n indeterminates.     

Spectrum of quasielliptic differential operators

Translated from Matematicheskie Zametiki, Vol. 53, No. 6, pp. 145–148, June, 1993.     

Skew fields of differential operators

We show in this note that two curves are defined up to birational equivalence by the skew fields of differential operators on these curves. The project was started when this author was visiting Wayne State University. This author is supported by an NSF grant.     

Supercyclic sequences of differential operators

Summary Several necessary and sufficient conditions for a sequence of infinite order differential linear operators on spaces of holomorphic functions on a domain of the complex plane to be supercyclic or c-hypercyclic are given in this paper, so completing earlier work of the authors on hypercyclicity, which in turn extended Birkhoff--MacLane--Godefroy--Shapiro's theorems. A new, general eigenvalue criterium for supercyclicity is also provided.     

Comparison of linear differential operators

In this paper, we study the question of the existence of the inequality $$\left\| {Q(D)f} \right\|_{L_q } \leqslant \gamma _0 \left\| {P(D)f} \right\|_{L_p } $$ , where P and Q are algebraic polynomials, D = d/dx, and γ ₀ is independent of the function f. We obtain criteria (necessary and simultaneously sufficient conditions) for the existence of such inequalities for functions on the circle, on the whole line, and on the semiaxis. Besides, for the semiaxis, we obtain an inequality for q = ∞ and any p ≥ 1 with the smallest constant γ ₀.     

Complexes of Dirac operators in Clifford algebras

In this paper we study complexes of k Dirac operators (or variations of Dirac operators) in the real or complex Clifford algebras i.e. complexes in which the first map is induced by the matrix where is the Dirac operator with respect to the variable . In particular we prove that, if , the complex in the case of 3 operators can be described in terms of relations coming from the so called radial algebra. Moreover we show that if the dimension m is less than 2k-1, then the Fischer decomposition does not hold. Received: 2 February 2000; in final form: 20 June 2000 / Published online: 25 June 2001