Vol. 15 (2009), no. 4

The characteristic properties of scattering data for the Schrodinger operator on the axis with a triangular $2\times 2$ matrix potential are obtained under the simple or multiple virtual levels being possibly present. Under a multiple virtual level, a pole for the reflection coefficient at $k=0$ is possible. For this case, the modified Parseval equality is constructed.

A subset $A$ of $\mathbb R^n$ is said to be max-min convex if, for any $x,y\in A$ and any $t\in \mathbb R$, we have $x\oplus t\otimes y\in A$ (here $\oplus$ stands for the coordinatewise maximum of two elements in $\mathbb R^n$ and $t\otimes (y_1,\dots,y_n)=(\min\{t,y_1\},\dots, \min\{t,y_n\})$). It is proved that the hyperspace of compact max-min convex sets in the Euclidean space $\mathbb R^n$, $n\ge2$, is homeomorphic to the punctured Hilbert cube. This is a counterpart of the result by Nadler, Quinn and Stavrokas proved for the hyperspace of compact convex sets. We also investigate the maps of the hyperspaces of compact max-min convex sets induced by the projection maps of Euclidean spaces. It is proved that this map is a Hilbert cube manifold bundle.

In this article we will investigate an inverse spectral problem for three-diagonal block Jacobi type Hermitian real-valued matrices with "almost" semidiagonal matrices on the side diagonals.

Let $X$ be a Baire space, $Y$ be a compact Hausdorff space and $f:X \times Y \to \mathbb{R}$ be a separately continuous mapping. For each $y \in Y$, we define a game $G(Y, \{ y \})$ between players $O$ and $P$, to show that if in this game either $O$ player has a winning strategy or $X$ is $\alpha$-favorable and $P$ player does not have a winning strategy, then for each countable subset $E$ of $Y$, there exists a dense $G_\delta$ subset $D$ of $X$ such that $f$ is jointly continuous on $D \times E$.

Walter Roth studied one form of the uniform boundedness theorem in \cite{Rot98}. We investigate some other versions of the uniform boundedness theorem for barreled and upper-barreled locally convex cones. Finally, we show some applications of this theorem.

A model operator $H$ associated to a system describing four particles in interaction, without conservation of the number of particles, is considered. We describe the essential spectrum of $H$ by the spectrum of the channel operators and prove the Hunziker-van Winter-Zhislin (HWZ) theorem for the operator $H.$ We also give some variational principles for boundaries of the essential spectrum and interior eigenvalues.

A classification of irreducible *-representations of a certain deformation of twisted canonical commutation relations is given.

We introduce the notion of an $ls$-Ponomarev-system $(f, M, X, \{\mathcal{P}_{\lambda,n}\})$, and give necessary and sufficient conditions such that the mapping $f$ is a compact (compact-covering, sequence-covering, pseudo-sequence-covering, sequentially-quotient) mapping from a locally separable metric space $M$ onto a space $X$. As applications of these results, we systematically get characterizations of certain compact images of locally separable metric spaces.