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.     

Little Hankel operators on the half plane

In this paper we consider a class of weighted integral operators onL ² (0, ) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.     

The quantum differential equation of the Hilbert scheme of points in the plane

We discuss here basic properties of the quantum differential equation of the Hilbert scheme of points in the plane. Our emphasis is on intertwining operators (which shift equivariant parameters) and their applications. In particular, we obtain an exact solution to the connection problem from the Donaldson-Thomas point q = 0 to the Gromov-Witten point q = -1.     

Directional operators and radial functions on the plane

LetEçS ¹ be a set with Minkowski dimensiond(E)1. We consider the Hardy-Littlewood maximal function, the Hilbert transform and the maximal Hilbert transform along the directions ofE. The main result of this paper shows that these operators are bounded onL _rad ^p (R²) forp>1+d(E) and unbounded whenp<1+d(E). We also give some end-point results.     

Composition operators on Bergman spaces over the punctured plane

We study the spaces of holomorphic functions on the punctured plane which are square integrable with respect to a weight. We determine when these spaces are finite-dimensional and which are composition operators acting on them, studying their cyclicity and hypercyclicity properties.     

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.     

Limit-point / limit-circle classification of second-order differential operators arising in PT quantum mechanics

We consider a second-order differential equation −y^″ + q(x)y(x) = λy(x) with complex-valued potential q and eigenvalue parameter λ ∈ ℂ. In PT quantum mechanics the potential q is given by q(x) = −(ix)^N+2 on a contour Γ ⊂ ℂ. Via a parametrization we obtain two differential equations on [0, ∞) and (−∞, 0]. We give a limit-point/limit-circle classification of this problem via WKB-analysis. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Covariant differential operators

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

Rings of differential operators on curves

Let k be an algebraically closed field of characteristic 0 and let A be a finitely generated k-algebra that is a domain whose Gelfand-Kirillov dimension is in [2, 3). We show that if A has a nonzero locally nilpotent derivation then A has quadratic growth. In addition to this, we show that A either satisfies a polynomial identity or A is isomorphic to a subalgebra of D(X), the ring of differential operators on an irreducible smooth affine curve X, and A is birationally isomorphic to D(X).     

Multilinear differential operators on modular forms

We construct multilinear differential operators on modular forms and prove that they are essentially unique. We also discuss certain homogeneous polynomials associated to such differential operators as well as some related multilinear differential operators that do not produce modular forms.

     

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.     

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.     

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.