卷 50, 编号 3 (2016)

The main result of this paper is a description of the space of functions on the unit circle, for which Krein’s trace formula holds for arbitrary pairs of unitary operators with trace class difference. This space coincides with the space of operator Lipschitz functions on the unit circle.

For a given nonzero entire function g: C → C, we study the linear space F(g) of all entire functions f such that

where φ₁, ψ₁,..., φ_n, ψ_n: C → C. In the case of g ≡ 1, the expansion characterizes quasipolynomials, that is, linear combinations of products of polynomials by exponential functions. (This is a theorem due to Levi-Civita.) As an application, all solutions of a functional equation in the theory of trilinear functional equations are obtained.

We present an explicit formula for the series S_2,p(t) of Virasoro singular vectors.

A two-sided estimate of local multiplicities of Maxwell sets of isolated singularities of smooth functions is proved. This estimate is sharp for semi-homogeneous functions.

Given a one-dimensional Stark–Wannier operator, we study the reflection coefficient and its poles in the lower half of the complex plane far from the real axis. In particular, the reflection coefficient is described asymptotically in terms of regularized infinite cubic exponential sums.

We consider a self-adjoint elliptic operator A_ε, ε> 0, on L₂(R^d; Cⁿ) given by the differential expression b(D)*g(x/ε)b(D). Here \(b(D) = \sum\nolimits_{j = 1}^d {b_j D_j }\) is a first-order matrix differential operator such that the symbol b(ξ) has maximal rank. The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice. We study the operator exponential \({e^{ - i\tau {A_\varepsilon }}}\), where τ ∈ R. It is shown that, as ε → 0, the operator \({e^{ - i\tau {A_\varepsilon }}}\) converges to \({e^{ - i\tau {A^0}}}\) in the norm of operators acting from the Sobolev space H^s(R^d;Cⁿ) (with suitable s) to L₂(R^d;Cⁿ). Here A⁰ is the effective operator with constant coefficients. Order-sharp error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation i∂_τu_ε(x, τ) = A_εu_ε(x, τ).