Generalized Eigenvectors for Resonances in the Friedrichs Model and Their Associated Gamov Vectors

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on R with finite-dimensional multiplicity space K, is constructed such that exactly the resonances (poles of the inverse of the Livsic-matrix) are (generalized) eigenvalues of H. The corresponding eigen-antilinearforms are calculated explicitly. Using the wave matrices for the wave (Moller) operators the corresponding eigen-antilinearforms on the Schwartz space S for the unperturbed Hamiltonian are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector, which is uniquely determined by restriction of S to the intersection of S with the Hardy space of the upper half plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup of the Toeplitz type for the positive half line on the Hardy space. That is: exactly those pre-Gamov vectors (eigenvectors of the decay semigroup) have an extension to a generalized eigenvector of H if the eigenvalue is a resonance and if the multiplicity parameter k is from that subspace of K which is uniquely determined by its corresponding Dirac type antilinearform.Comment: 16 page

Notions of Infinity in Quantum Physics

In this article we will review some notions of infiniteness that appear in Hilbert space operators and operator algebras. These include proper infiniteness, Murray von Neumann's classification into type I and type III factors and the class of F{/o} lner C*-algebras that capture some aspects of amenability. We will also mention how these notions reappear in the description of certain mathematical aspects of quantum mechanics, quantum field theory and the theory of superselection sectors. We also show that the algebra of the canonical anti-commutation relations (CAR-algebra) is in the class of F{/o} lner C*-algebras.Comment: 11 page

Twisted duality of the CAR-Algebra

We give a complete proof of the twisted duality property M(q)'= Z M(q^\perp) Z* of the (self-dual) CAR-Algebra in any Fock representation. The proof is based on the natural Halmos decomposition of the (reference) Hilbert space when two suitable closed subspaces have been distinguished. We use modular theory and techniques developed by Kato concerning pairs of projections in some essential steps of the proof. As a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator. This formula can be also applied to fermionic free nets, hence giving a formula of the modular operator for any double cone.Comment: 32 pages, Latex2e, to appear in Journal of Mathematical Physic

Dynamics of charged fluids and 1/L perturbation expansions

Some features of the calculation of fluid dynamo systems in magnetohydrodynamics are studied. In the coupled set of the ordinary linear differential equations for the spherically symmetric $\alpha^2-$dynamos, the problem represented by the presence of the mixed (Robin) boundary conditions is addressed and a new treatment for it is proposed. The perturbation formalism of large$-\ell$ expansions is shown applicable and its main technical steps are outlined.Comment: 16 p

Some Remarks on Group Bundles and C*-dynamical systems

We introduce the notion of fibred action of a group bundle on a C(X)-algebra. By using such a notion, a characterization in terms of induced C*-bundles is given for C*-dynamical systems such that the relative commutant of the fixed-point algebra is minimal (i.e., it is generated by the centre of the given C*-algebra and the centre of the fixed-point algebra). A class of examples in the setting of the Cuntz algebra is given, and connections with superselection structures with nontrivial centre are discussed.Comment: 22 pages; to appear on Comm. Math. Phy

Nonlinear theory of mirror instability near threshold

An asymptotic model based on a reductive perturbative expansion of the drift kinetic and the Maxwell equations is used to demonstrate that, near the instability threshold, the nonlinear dynamics of mirror modes in a magnetized plasma with anisotropic ion temperatures involves a subcritical bifurcation,leading to the formation of small-scale structures with amplitudes comparable with the ambient magnetic field

Protein phosphatase 1-dependent bidirectional synaptic plasticity controls ischemic recovery in the adult brain

Protein kinases and phosphatases can alter the impact of excitotoxicity resulting from ischemia by concurrently modulating apoptotic/survival pathways. Here, we show that protein phosphatase 1 (PP1), known to constrain neuronal signaling and synaptic strength (Mansuy et al., 1998; Morishita et al., 2001), critically regulates neuroprotective pathways in the adult brain. When PP1 is inhibited pharmacologically or genetically, recovery from oxygen/glucose deprivation (OGD) in vitro, or ischemia in vivo is impaired. Furthermore, in vitro, inducing LTP shortly before OGD similarly impairs recovery, an effect that correlates with strong PP1 inhibition. Conversely, inducing LTD before OGD elicits full recovery by preserving PP1 activity, an effect that is abolished by PP1 inhibition. The mechanisms of action of PP1 appear to be coupled with several components of apoptotic pathways, in particular ERK1/2 (extracellular signal-regulated kinase 1/2) whose activation is increased by PP1 inhibition both in vitro and in vivo. Together, these results reveal that the mechanisms of recovery in the adult brain critically involve PP1, and highlight a novel physiological function for long-term potentiation and long-term depression in the control of brain damage and repair