GENERALIZED WEAK WEYL RELATION AND DECAY OF QUANTUM DYNAMICS

Abstract

Let H be a self-adjoint operator on a Hilbert space [Formula: see text], T be a symmetric operator on [Formula: see text] and K(t)(t ∈ ℝ) be a bounded self-adjoint operator on [Formula: see text]. We say that (T, H, K) obeys the generalized weak Weyl relation (GWWR) if e-itHD(T) ⊂ D(T) for all t ∈ ℝ and Te-itHψ = e-itH(T+K(t))ψ, ∀ψ ∈ D(T) (D(T) denotes the domain of T). In the context of quantum mechanics where H is the Hamiltonian of a quantum system, we call T a generalized time operator of H. We first investigate, in an abstract framework, mathematical structures and properties of triples (T, H, K) obeying the GWWR. These include the absolute continuity of the spectrum of H restricted to a closed subspace of [Formula: see text], an uncertainty relation between H and T (a "time-energy uncertainty relation"), the decay property of transition probabilities |〈ψ,e-itHϕ〉|2 as |t| → ∞ for all vectors ψ and ϕ in a subspace of [Formula: see text], where 〈·,·〉 denotes the inner product of [Formula: see text]. We describe methods to construct various examples of triples (T, H, K) obeying the GWWR. In particular, we show that there exist generalized time operators of second quantization operators on Fock spaces (full Fock spaces, boson Fock spaces and fermion Fock spaces) which may have applications to quantum field models with interactions.