Spectra of observables in the q-oscillator and q-analogue of the Fourier transform

Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa^+-qa^+a=1) are studied when q>1. These operators are symmetric but not self-adjoint. They have a one-parameter family of self-adjoint extensions. These extensions are derived explicitly. Their spectra and eigenfunctions are given. Spectra of different extensions do not intersect. The results show that the creation and annihilation operators a^+ and a of the q-oscillator for q>1 cannot determine a physical system without further more precise definition. In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Meromorphic extensions from small families of circles and holomorphic extensions from spheres

Let B be the open unit ball in C^2 and let a, b, c be three points in C^2 which do not lie in a complex line, such that the complex line through a and b meets B and such that is different from 1 if one of the points a, b is in B and the other in the complement of B and such that at least one of the numbers , is different from 1. We prove that if a continuous function f on the sphere bB extends holomorphically into B along each complex line which passes through one of the points a, b, c then f extends holomorphically through B. This generalizes recent work of L.Baracco who proved such a result in the case when the points a, b, c are contained in B. The proof is different from the one of Baracco and uses the following one variable result which we also prove in the paper and which in the real analytic case follows from the work of M.Agranovsky: Let D be the open unit disc in C. Given a in D let C(a) be the family of all circles in D obtained as the images of circles centered at the origin under an automorphism of D that maps the origin to a. Given distinct points a, b in D and a positive integer n, a continuous function f on the closed unit disc extends meromorphically from every circle T in either C(a) or C(b) through the disc bounded by T with the only pole at the center of T of degree not exceeding n if and only if f is of the form f(z) = g_0(z)+g_1(z)\bar z +...+ g_n(z)\bar z^n where the functions g_0, g_1, ..., g_n are holomorphic on D.Comment: This replaces the original version where an assumption was missing in Corollary 1.3. This assumption is now added and in the last section an example is added which shows that the added assumption is really necessar

Compactification-like extensions

Let $X$ be a space. A space $Y$ is called an extension of $X$ if $Y$ contains $X$ as a dense subspace. For an extension $Y$ of $X$ the subspace $Y\backslash X$ of $Y$ is called the remainder of $Y$. Two extensions of $X$ are said to be equivalent if there is a homeomorphism between them which fixes $X$ pointwise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous mapping of $Y'$ into $Y$ which fixes $X$ pointwise. Let $P$ be a topological property. An extension $Y$ of $X$ is called a $P$-extension of $X$ if it has $P$. If $P$ is compactness then $P$-extensions are called ompactifications. The aim of this article is to introduce and study classes of extensions (which we call compactification-like $P$-extensions, where $P$ is a topological property subject to some mild requirements) which resemble the classes of compactifications of locally compact spaces. We formally define compactification-like $P$-extensions and derive some of their basic properties, and in the case when the remainders are countable, we characterize spaces having such extensions. We will then consider the classes of compactification-like $P$-extensions as partially ordered sets. This consideration leads to some interesting results which characterize compactification-like $P$-extensions of a space among all its Tychonoff $P$-extensions with compact remainder. Furthermore, we study the relations between the order-structure of classes of compactification-like $P$-extensions of a Tychonoff space $X$ and the topology of a certain subspace of its outgrowth $\beta X\backslash X$. We conclude with some applications, including an answer to an old question of S. Mr\'{o}wka and J.H. Tsai: For what pairs of topological properties $P$ and $Q$ is it true that every locally-$P$ space with $Q$ has a one-point extension with both $P$ and $Q$?Comment: 86 page