# Vol. 15 (2009), no. 1

### Aleksandr Yakovlevich Povzner

MFAT **15** (2009), no. 1, 1-2

1-2

### Direct spectral problem for the generalized Jacobi Hermitian matrices

MFAT **15** (2009), no. 1, 3-14

3-14

In this article we will introduce and investigate some generalized Jacobi matrices. These matrices have three-diagonal block structure and they are Hermitian. We will give necessary and sufficient conditions for selfadjointness of the operator which is generated by the matrix of such a type, and consider its generalized eigenvector expansion.

### Origination of the singular continuous spectrum in the conflict dynamical systems

MFAT **15** (2009), no. 1, 15-30

15-30

We study the spectral properties of the limiting measures in the conflict dynamical systems modeling the alternative interaction between opponents. It has been established that typical trajectories of such systems converge to the invariant mutually singular measures. We show that "almost always" the limiting measures are purely singular continuous. Besides we find the conditions under which the limiting measures belong to one of the spectral type: pure singular continuous, pure point, or pure absolutely continuous.

### Spectral gaps of one-dimensional Schrödinger operators with singular periodic potentials

Vladimir Mikhailets, Volodymyr Molyboga

MFAT **15** (2009), no. 1, 31-40

31-40

The behavior of the lengths of spectral gaps $\{\gamma_{n}(q)\}_{n=1}^{\infty}$ of the Hill-Schrödinger operators $$ S(q)u=-u''+q(x)u,\quad u\in \mathrm{Dom}\left(S(q) \right), $$ with real-valued 1-periodic distributional potentials $q(x)\in H_{1\mbox{-}{\operatorname{per}}}^{-1}(\mathbb{R})$ is studied. We show that they exhibit the same behavior as the Fourier coefficients $\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,\varphi}$, $s>-1$, $\varphi\in \mathrm{SV}$. The case $q(x)\in L_{1\mbox{-}{\operatorname{per}}}^{2}(\mathbb{R})$, $s\in \mathbb{Z}_{+}$, $\varphi\equiv 1$, corresponds to the Marchenko-Ostrovskii Theorem.

### Inverse eigenvalue problems for nonlocal Sturm-Liouville operators

MFAT **15** (2009), no. 1, 41-47

41-47

We solve the inverse spectral problem for a class of Sturm-Liouville operators with singular nonlocal potentials and nonlocal boundary conditions.

### On $n$-tuples of subspaces in linear and unitary spaces

Yu. S. Samoilenko, D. Yu. Yakymenko

MFAT **15** (2009), no. 1, 48-60

48-60

We study a relation between brick $n$-tuples of subspaces of a finite dimensional linear space, and irreducible $n$-tuples of subspaces of a finite dimensional Hilbert (unitary) space such that a linear combination, with positive coefficients, of orthogonal projections onto these subspaces equals the identity operator. We prove that brick systems of one-dimensional subspaces and the systems obtained from them by applying the Coxeter functors (in particular, all brick triples and quadruples of subspaces) can be unitarized. For each brick triple and quadruple of subspaces, we describe sets of characters that admit a unitarization.

### Schrödinger operators with purely discrete spectrum

MFAT **15** (2009), no. 1, 61-66

61-66

We prove that $-\Delta +V$ has purely discrete spectrum if $V\geq 0$ and, for all $M$, $|{\{x\mid V(x) < M\}}|<\infty$ and various extensions.

### Necessary and sufficient condition for solvability of a partial integral equation

MFAT **15** (2009), no. 1, 67-73

67-73

Let $T_1: L_2(\Omega^2) \to L_2(\Omega^2)$ be a partial integral operator [4,7] with the kernel from $C(\Omega^3)$ where $\Omega=[a,b ]^ u$, $ u \in N$ is fixed. In this paper we investigate solvability of the partial integral equation $f-\varkappa T_1 f=g_0$ in the space $L_2(\Omega^2)$ in the case where $\varkappa$ is a cha ac eristic number. We prove a the theorem that gives a necessary and sufficient condition for solvability of the partial integral equation $f-\varkappa T_1 f=g_0.$

### Compact variation, compact subdifferetiability and indefinite Bochner integral

MFAT **15** (2009), no. 1, 74-90

74-90

The notions of compact convex variation and compact convex subdifferential for the mappings from a segment into a locally convex space (LCS) are studied. In the case of an arbitrary complete LCS, each indefinite Bochner integral has compact variation and each strongly absolutely continuous and compact subdifferentiable a.e. mapping is an indefinite Bochner integral.

### Integral representations for spectral functions of some nonself-adjoint Jacobi matrices

MFAT **15** (2009), no. 1, 91-100

91-100

We study a Jacobi matrix $J$ with complex numbers $a_n,\ n\in\mathbb Z_+,$ in the main diagonal such that $r_0 \leq {\rm Im}\, a_n \leq r_1,\ r_0,r_1\in\mathbb R$. We obtain an integral representation for the (generalized) spectral function of the matrix $J$. The method of our study is similar to Marchenko's method for nonself-adjoint differential operators.