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We consider the partial non-stationary perturbation determinant $$ \Delta _{H/A}^{(1)} (t): = \det \left( {e^{itA} P_1 e^{ - itH} |_{\mathcal{H}_1 } } \right),t \in \mathbb{R}.$$ Here H is a self-adjoint operator in some Krein space \( \mathcal{K}\) and A is a self-adjoint operator in the Hilbert space \( \mathcal{H}_1\), which is the positive component of a fundamental decomposition of \( \mathcal{H}_1\) with corresponding orthogonal projection P1. The asymptotic behavior of δ H/A (1) (t) for t → ∞ and the spectral shift function for H and its diagonal part are studied. Analogous results for the case if the underlying space is a Hilbert space were obtained in [1].

The description of all solutions of the truncated Stieltjes matrix moment problem consisting in finding all s × s matrix measures dσ (t) on [0,∞) with given first 2n+1 power s×s matrix moments (Cj) j=0 n is obtained in a general case, when the block Hankel matrix Γn ≔ (Cj+k) j,k=0 n may be non-invertible. Special attention is paid to the description of canonical solutions for which dσ (t) is a sum of at most sn + s point matrix “masses” with the minimal sum of their ranks.

A partial non-stationary perturbation determinant Δ1 (t) is defined as follows: $$\begin{array}{*{20}{c}} {{\Delta _1}(t): = \det \left( {{e^{itA}}{P_1}{e^{ - itH}}{|_{{\mathcal{H}_1}}}} \right),}&{t > 0;} \end{array}$$ here A is a self-adjoint operator in some Hilbert space \({\mathcal{H}_1}\), H is a self-adjoint operator in a larger Hilbert space \(\mathcal{H} \supset {\mathcal{H}_1}\), P 1 i the orthogonal projection in \(\mathcal{H}\) onto \({\mathcal{H}_1}\) and \({P_1}(H - A){|_{{\mathcal{H}_1}}}\) is a trace class operator in \({\mathcal{H}_1}\). If the operator P 1 H(I- P 1) is finite-dimensional, Δ1(t) is expressed by the resolvent kernel of a system of Fredholm integral equations on (0, t) of second kind. Moreover, in a particular situation the asymtotic behavior of Δ1(t) for t → ∞ is studied.

The conditions of solvability and description of all solu- tions of the truncated Stieltjes moment problem are obtained using as the starting point earlier results on the Hamburger truncated moment problem. An algebraic algorithm for the explicit solution of both prob- lems is proposed.

This paper deals with quadratic operator pencils $$ L(\lambda ) = \lambda ^2 A + \lambda B + C, $$ where A, B are non-invertible non-negative operators and C is a selfadjoint operator, which is bounded from below and has a compact resolvent. Such pencils naturally arise in stability problems of mechanics and resistive magnetohydrodynamics. Under certain assumptions on A, B and C the description of the spectrum of the pencil L is given.

The papers [AL], [AdLMSau] were devoted to the characterization of certain spectral subspaces of selfadjoint operators L, which are generated by symmetric 2 x 2 operator matrices of the form \( {L_0} = \left( {\begin{array}{*{20}{c}} AB the entries A, B and D are not necessarily bounded operators in or between these spaces. One of the main assumptions in those investigations was the weak separation of the spectra of the selfadjoint operators A and D,i.e., max σ (D) ≤ min σ (A).

The truncated Hamburger and Stieltjes moment problems [1] were solved elsewhere (see [3] and references therein) using the methods of the extension theory of Hermitian operators and, in particular, the results on extensions of nonnegative operators and matrices. Moreover, applying methods of the extension theory we found a purely algebraic algorithm for the solution of both problems. Here we solve the corresponding moment problems with point constraints.

A mathematical model for a simple microscheme is constructed on the basis of the scattering theory for a pair of different self-adjoint extensions of the same symmetric ordinary differential operator on a one-dimensional manifold, which consist of a finite number of semiinfinite straight outer lines attached to a “black box” in a form of a flat connected graph. An explicit expression for the scattering is given under a continuity condition at the graph vertices.

Explicit expressions for the wave functions and dispersion equation for the band p - electrons in single-wall carbon nanotubes are obtained within the method of zero-range potentials. They are then used to investigate the absorption spectrum of polarized light caused by direct interband transitions in isolated nanotubes. It is shown that, at least, under the above approximations, the circular dichroism is absent in chiral nanotubes for the light wave propagating along the tube axis. The results obtained are compared with those calculated in a similar way for a graphite plane., Comment: 16 pages, 8 figures, 1 table