We give a characterization, in one variable case, of those C∞ multipliers F such that the division problem is solvable in S′(R). For these functions F ∈ OM (R) we even prove that the multiplication operator MF (G) = FG has a continuous linear right inverse on S′(R), in contrast to what happens in the several variables case, as was shown by Langenbruch. 2010 Mathematics Subject Classification. P...

We study reducing subspaces for an analytic multiplication operator Mzn on the Bergman space L 2 a(Ar) of the annulus Ar, and we prove that Mzn has exactly 2 reducing subspaces. Furthermore, in contrast to what happens for the disk, the same is true for the Hardy space on the annulus. Finally, we extend the results to certain bilateral weighted shifts, and interpret the results in the context o...

The Lifshits–Krein spectral shift function is considered for the pair of operators H0 = (−4)l, l > 0 and H = H0 + V in L2(R), d ≥ 1; here V is a multiplication operator. The estimates for this spectral shift function ξ(λ;H,H0) are obtained in terms of the spectral parameter λ > 0 and the integral norms of V . These estimates are in a good agreement with the ones predicted by the classical phase...

Position uncertainty (delocalization) measures for a particle on the sphere are proposed and illustrated on several examples of states. The new measures are constructed using suitably the standard multiplication angle operator variances. They are shown to depend solely on the state of the particle and to obey uncertainty relations of the Schrödinger-Robertson type. A set of Hermitian operators ...

We generalize to several variables Kurbanov and Maksimov’s result that all linear polynomial operators can be expressed as a formal sum ∑∞ k=0 ak(X) D k in terms of the derivative D (or any degree reducing operator) and multiplication by x. In contrast, we characterize those linear operators that can be expressed as ∑∞ k=0 fk(D) X k and give several examples. Generalizations to several variable...

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of C2. The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. Several results and conjectures on the corresponding deformation of Jack symmetric functions are presented. A relationship between the quantum c...

Let M be a compact Riemannian manifold endowed with an isometric action of a compact, connected Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on M. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associa...

We introduce an algebra H consisting of difference-reflection operators and multiplication operators that can be considered as a q = 1 analogue of Sahi's double affine Hecke algebra related to the affine root system of type (C ∨ 1 , C1). We study eigenfunctions of a Dunkl-Cherednik-type operator in the algebra H, and the corresponding Fourier transforms. These eigenfunctions are non-symmetric v...

In this paper we present a definition of weighted Sobolev spaces on curves and find general conditions under which the spaces are complete. We also prove the density of the polynomials in these spaces for non-closed compact curves and, finally, we find conditions under which the multiplication operator is bounded on the completion of polynomials. These results have applications to the study of ...

Let V be a multiplication operator, whose negative part, V-(V-< 0) obeys-A + (1 + c)V->-c for some c, c > 0. Let W = Vx where x is the characteristic function of the exterior of a ball. Our main result asserts that the scattering for-A + V is complete if and only if that for-A + W is complete. Our technical estimates exploit Wiener integrals and the Feynman-Kac formula. We also make an applicat...