A New Class of Point Interactions in One Dimension

AbstractWe present a class of self-adjoint extensions of the symmetric operator −Δ|C∞0(R1\{0}) which correspond formally to perturbations of the Laplacian by pseudopotentials involving δ2. These operators, which provide new examples of generalized point interactions in the sense of Šeba, are defined by the boundary conditions ƒ(0+) = e−zƒ(0−), rƒ(0+) + ƒ′(0+) = ez[rƒ(0−) + ƒ′(0−)], for z ∈ C, r ∈ R. We calculate their spectra, resolvents, and scattering matrices, and show that they can be realized as limits of Schrödinger operators with local short-range potentials

Torus fibrations and localization of index II

We give a framework of localization for the index of a Dirac-type operator on an open manifold. Suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets, each of which has a structure of the total space of a torus bundle. Under an acyclic condition we define the index of the Dirac-type operator by using the Witten-type deformation, and show that the index has several properties, such as excision property and a product formula. In particular, we show that the index is localized on the compact set.Comment: 47 pages, 2 figures. To appear in Communications in Mathematical Physic

Topological features of massive bosons on two dimensional Einstein space-time

In this paper we tackle the problem of constructing explicit examples of topological cocycles of Roberts' net cohomology, as defined abstractly by Brunetti and Ruzzi. We consider the simple case of massive bosonic quantum field theory on the two dimensional Einstein cylinder. After deriving some crucial results of the algebraic framework of quantization, we address the problem of the construction of the topological cocycles. All constructed cocycles lead to unitarily equivalent representations of the fundamental group of the circle (seen as a diffeomorphic image of all possible Cauchy surfaces). The construction is carried out using only Cauchy data and related net of local algebras on the circle.Comment: 41 pages, title changed, minor changes, typos corrected, references added. Accepted for publication in Ann. Henri Poincare

Around the Van Daele–Schmüdgen Theorem

For a {bounded} non-negative self-adjoint operator acting in a complex, infinite-dimensional, separable Hilbert space H and possessing a dense range R we propose a new approach to characterisation of phenomenon concerning the existence of subspaces M\subset H such that M\capR=M^\perp\capR=\{0\}. We show how the existence of such subspaces leads to various {pathological} properties of {unbounded} self-adjoint operators related to von Neumann theorems \cite{Neumann}--\cite{Neumann2}. We revise the von Neumann-Van Daele-Schm\"udgen assertions \cite{Neumann}, \cite{Daele}, \cite{schmud} to refine them. We also develop {a new systematic approach, which allows to construct for any {unbounded} densely defined symmetric/self-adjoint operator T infinitely many pairs of its closed densely defined restrictions T_k\subset T such that \dom(T^* T_{k})=\{0\} (\Rightarrow \dom T_{k}^2=\{0\}$) k=1,2 and \dom T_1\cap\dom T_2=\{0\}, \dom T_1\dot+\dom T_2=\dom T

BMS field theory and holography in asymptotically flat space-times

We explore the holographic principle in the context of asymptotically flat space-times by means of the asymptotic symmetry group of this class of space-times, the so called Bondi-Metzner-Sachs (BMS) group. In particular we construct a (free) field theory living at future (or past) null infinity invariant under the action of the BMS group. Eventually we analyse the quantum aspects of this theory and we explore how to relate the correlation functions in the boundary and in the bulk.Comment: 36 pages, updated introduction, conclusions and references; added a discussion on Schwartzschild background in section