Evolution Operators for Quantum Chains

Discrete-time evolution operators in integrable quantum lattice models are sometimes more fundamental objects then Hamiltonians. In this paper we study an evolution operator for the one-dimensional integrable q-deformed Bose gas with XXZ-type impurities and find its spectrum. Evolution operators give a new interpretation of known integrable systems, for instance our system describes apparently a simplest laser with a clear resonance peak in the spectrum. Existence of a complete set of commuting operators is the basic principle of quantum integrability. However, often the commuting set does not provide a distinguished operator deserving the title of the Hamiltonian for a physical system. In particular, this is the common feature of quantum models obtained by quantization of classical equations of motion in wholly discrete space-time. Equations of motion define a discrete time translations for classical variables, a map A(τ, σ) → A(τ + 1, σ). Corresponding translation for quantum observables is produced by an evolution operator, (1) A(τ + 1, σ) = UA(τ, σ)U , UU = 1 . If the time unity interval coincides with the unity spacing, then there is no a small parameter expansion of U defining a lattice Hamiltonian. The evolution operator becomes the main object of the lattice quantum mechanics. Discrete-time evolution operators for quantum chains were considered in many papers. For instance, the study of the spectrum of evolution operator was in the focus of the quantum Liouville theory [1]. In this paper we discuss another example of quantum evolution system: the one-dimensional q-deformed Bose gas [2] with XXZ-type impurities. The simple test “what evolution operator is doing in the space of states” will give a new view on well known models. Quantum mechanics begins with the algebra of observables. In our case the local algebra of observables is the q-oscillator [3] with generators B,B,N: (2) BB = 1− q , BB = 1− q , 0 < q < 1 . 1991 Mathematics Subject Classification. 37K15.