This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The proposed project is on approximation of operator functions, whose prototype in the case of functions of a scalar argument is the Taylor polynomial approximation. Values of operator functions do not commute in general, which makes the analysis of such functions and, in particular, their approximations much subtler than in the classical case. Under certain assumptions, traces of the remainders of the first and second order approximations can be represented via spectral shift functions, which originate from Lifshits' work on the quantum theory of crystals in 1952. While higher order Taylor-type approximations are also of interest in applications (for instance, in perturbation theory for Schrodinger operators with long-range potentials), very little is known about the structure of their error terms. The project will concentrate on the study of the higher order Taylor-type approximations, in particular, on testing Koplienko's conjecture of 1984 on existence of higher order spectral shift measures.
Perturbation theory has originated as mathematical modeling of some problems of quantum mechanics, where physical quantities are described by self-adjoint operators acting on a separable Hilbert space. The change of a value of an operator function under a perturbation of its argument is reflected in the spectral shift functions. A comprehensive theory with various applications, including those to perturbation theory for Schrodinger operators, scattering theory, and spectral flow, has been constructed for these functions. Finding higher order analogs of the spectral shift functions is one of the goals of the project. Many operators can be naturally affiliated with von Neumann algebras (for instance, the integrated density of states for some operators can be expressed in terms of the corresponding von Neumann algebras). We will work in both the original and the von Neumann algebra setting of the perturbation theory.
