On the spectral properties of Hamiltonians without conservation of the particle number

Abstract

We consider quantum systems with variable but finite number of particles. For such systems we develop geometric and commutator techniques. We use these techniques to find the location of the spectrum, to prove absence of singular continuous spectrum and identify accumulation points of the discrete spectrum. The fact that the total number of particles is bounded allows us to give relatively elementary proofs of these basic results for an important class of many-body systems with non-conserved number of particles.

On the spectral analysis of many-body systems

“…Then H T has a structure analogous to a Fock space. Non-relativistic Hamiltonians affiliated to such C * -algebras C T S have been studied before in [33]. If we take T = {X} for an arbitrary X ∈ S then the associated subsystem has H X as state space and its Hamiltonian algebra is just C X = C X (S) · T X , the Hamiltonian algebra of the (generalized) N -body system determined by the semi-lattice S X .…”

On the Discrete Spectrum of a Model Operator in Fermionic Fock Space

“…obeys the equation ℎ(0) = min . (b) If (0) = 0, then the number = min is an eigenvalue of the operator ℎ(0) and the vector = ( 0 , 1 ), where 0 ∈ C 1 and 1 ∈ 2 (T 3 ) defined by (20), is the corresponding eigenvector.…”

“…In statistical physics [13,14], solid-state physics [15,16], and the theory of quantum fields [17][18][19], some important problems arise where the number of quasiparticles is bounded, but not fixed. The authers of [20] have developed geometric and commutator techniques to find the location of the spectrum and to prove absence of singular continuous spectrum for Hamiltonians without conservation of the particle number.…”

Analytic description of the essential spectrum of a family of 3×3 operator matrices

“…Recall that the study of systems describing N particles in interaction, without conservation of the number of particles is reduced to the investigation of the spectral properties of selfadjoint operators, acting in the cut subspace H (N ) of Fock space, consisting of n ≤ N particles [5,11,12,23].…”

“…In particular, in [21] it is described the essential spectrum of 4 × 4 operator matrix by the spectrum of the corresponding channel operators and proved the HWZ theorem. In [23] geometric and commutator techniques have been developed in order to find the location of the spectrum and to prove absence of singular continuous spectrum for Hamiltonians without conservation of the particle number.…”