Search

In the present paper we study the spectrum of a diagonlizable 3×3 block operator matrix A2 arising in the spectral analysis of the energy operator of a system of three non conserved number of particles on a lattice. An analytic description of the essential spectrum of A2 is established and its new branches are singled out. Further, a formula for the point spectrum of A2 is derived. The relation for the discrete spectrum of A2 is obtained. [ABSTRACT FROM AUTHOR]

We consider 2 × 2 operator matrix Aμ, μ > 0 acting in the direct sum of the one- and two-particle subspaces of the bosonic Fock space. We describe the essential spectrum of spectrum of Aμ via the spectrum of a family of generalized Friedrichs models and define its new branches. For the special cases the location of these branches are investigated. [ABSTRACT FROM AUTHOR]

We consider the family of 3x3 operator matrices H(K), K ∈ T³ := (-π; π]³ associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set Λ ⊂ T³ to prove the existence of infinitely many eigenvalues of H(K) for all K ∈ ⊂ when the associated Friedrichs model has a zero energy resonance. It is found that for every K ∈ ⊂, the number N(K, z) of eigenvalues of H(K) lying on the left of z, z < 0, satisfies the asymptotic relation ... with 0 < U0 < ∞, independently on the cardinality of ... . Moreover, we prove that for any K ∈ ⊂ the operator H(K) has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model. [ABSTRACT FROM AUTHOR]

We consider a $2\times2$ operator matrix $A$ (so-called generalized Friedrichs model) associated with a system of at most two quantum particles on ${\rm d}$-dimensional lattice. This operator matrix acts in the direct sum of zero- and one-particle subspaces of a Fock space. We investigate the structure of the closure of the numerical range $W(A)$ of this operator in detail by terms of its matrix entries for all dimensions of the torus ${\bf T}^{\rm d}$. Moreover, we study the cases when the set $W(A)$ is closed and give necessary and sufficient conditions under which the spectrum of $A$ coincides with its numerical range.

This is an English translation of A. Lichnerowicz’ 1964 paper “Propagateurs, commutateurs et anticommutateurs en relativité générale”. It was based on lectures at a Les Houches school, and gave a classic summary of the author’s work on quantum propagators in curved space-times, introducing tensorial and spinorial propagators which generalise the flat space propagators of Jordan and Pauli. It gives the quantum commutators for the Maxwell and Dirac fields and for perturbations of the gravitational field, and numerous intermediate formulae. The results are basic to quantum field theory in curved space-times. The paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note written by G. Esposito and by a brief biography written by M. A. H. MacCallum. [ABSTRACT FROM AUTHOR]