Gibbs states defined by biorthogonal sequences

Abstract

Motivated by the growing interest on PT-quantum mechanics, in this paper we discuss some facts on generalized Gibbs states and on their related KMS-like conditions. To achieve this, we first consider some useful connections between similar (Hamiltonian) operators and we propose some extended version of the Heisenberg algebraic dynamics, deducing some of their properties, useful for our purposes.

A Note on States and Traces from Biorthogonal Sets

“…Sometimes ω 0 is written as ω 0 (X) = tr(ρX), where ρ := 1 Z 0 e −βH 0 . Hence, in view of (1), in [5] we have seen what happens if, in (2), we replace H 0 with H or with H * , and the e n 's with the ψ n 's or with the ϕ n 's. To do this, in [5] we assumed that H 0 , H and H † are closed and, at least, densely defined, and H 0 , H ∈ L † (D).…”

“…Hence, in view of (1), in [5] we have seen what happens if, in (2), we replace H 0 with H or with H * , and the e n 's with the ψ n 's or with the ϕ n 's. To do this, in [5] we assumed that H 0 , H and H † are closed and, at least, densely defined, and H 0 , H ∈ L † (D). A generalization of the above definition (2) could be useful therefore we assume H 0 , H ∈ L † (D).…”

“…and H † have only point spectra, and that all the eigenvalues coincide. In particular, in [5] several such triples of operators have been discussed and the following eigenvalues equations are deduced:…”

“…Here F e = {e n , n ≥ 0} is an orthonormal basis for the Hilbert space H, while F ϕ = {ϕ n , n ≥ 0} and F ψ = {ψ n , n ≥ 0} are two biorthogonal sets, ϕ n , ψ m = δ n,m , not necessarily bases for H. However, quite often, F ϕ and F ψ are complete in H and, see [5], they are also D-quasi bases, i.e., they produce a weak resolution of the identity in a dense subspace D of H:…”

Ordered Structures of Constructing Operators for Generalized Riesz Systems

“…Then these operators provide a link to quasi-Hermitian quantum mechanics, and its relatives. Many researchers have investigated such operators both from the mathematical point of view and for their physical applications [4][5][6][7][8][9]. Let { 푛 } be a generalized Riesz system with a constructing pair ( , ).…”