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Том 50, № 3 (2016)
- Год: 2016
- Статей: 10
- URL: https://journal-vniispk.ru/0016-2663/issue/view/14562
Article
Krein’s trace formula for unitary operators and operator Lipschitz functions
Аннотация
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.
167-175
Automorphisms of the solution spaces of special double-confluent Heun equations
Аннотация
Two new linear operators determining automorphisms of the solution space of a special double-confluent Heun equation in the general case are obtained. This equation has two singular points, both of which are irregular. The obtained result is applied to solve the nonlinear equation of the resistively shunted junction model for an overdamped Josephson junction in superconductors. The new operators are explicitly expressed in terms of structural polynomials, for which recursive computational algorithms are constructed. Two functional equations for the solutions of the special double-confluent Heun equation are found.
176-192
Hyperquasipolynomials and their applications
Аннотация
For a given nonzero entire function g: C → C, we study the linear space F(g) of all entire functions f such that
193-203
On the convergence of bloch eigenfunctions in homogenization problems
Аннотация
We study the convergence of continuous spectrum eigenfunctions for differential operators of divergence type with ε-periodic coefficients, where ε is a small parameter. Two cases are considered, the case of classical homogenization, where the coefficient matrix satisfies the ellipticity condition uniformly with respect to ε, and the case of two-scale homogenization, where the coefficient matrix has two phases and is highly contrast with hard-to-soft-phase contrast ratio 1: ε2.
204-218
219-224
Brief Communications
225-227
Projections of orbital measures for classical Lie groups
Аннотация
In this paper we compute the radial parts of the projections of orbital measures for the compact Lie groups of B, C, and D type, extending previous results obtained for the case of the unitary group by Olshanski and Faraut. Applying the method of Faraut, we show that the radial part of the projection of an orbital measure is expressed in terms of a B-spline with knots located symmetrically with respect to zero.
228-232
Stark–Wannier ladders and cubic exponential sums
Аннотация
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.
233-236
Diffusion processes on the Thoma cone
Аннотация
The Thoma cone is a certain infinite-dimensional space that arises in the representation theory of the infinite symmetric group. The present note is a continuation of a paper by A. M. Borodin and the author (Electr. J. Probab. 18 (2013), no. 75), where a 2-parameter family of continuous-time Markov processes on the Thoma cone was constructed. The purpose of the note is to show that these processes are diffusions.
237-240
Homogenization of Schrödinger-type equations
Аннотация
We consider a self-adjoint elliptic operator Aε, ε> 0, on L2(Rd; 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 Hs(Rd;Cn) (with suitable s) to L2(Rd;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∂τuε(x, τ) = Aεuε(x, τ).
241-246