We analyze the spectrum and normal-mode representation of general quadratic bosonic forms $H$ not necessarily Hermitian. It is shown that, in the one-dimensional case, such forms exhibit either a harmonic regime where both $H$ and ${H}^{\ifmmode\dagger\else\textdagger\fi{}}$ have a discrete spectrum with biorthogonal eigenstates, and a coherent-like regime where either $H$ or ${H}^{\ifmmode\dagger\else\textdagger\fi{}}$ have a continuous complex twofold degenerate spectrum, while its adjoint has no convergent eigenstates. These regimes reflect the nature of the pertinent normal boson operators. Nondiagonalizable cases as well critical boundary sectors separating these regimes are also analyzed. The extension to $N$ -dimensional quadratic systems is also discussed.We analyze the spectrum and normal-mode representation of general quadratic bosonic forms $H$ not necessarily Hermitian. It is shown that, in the one-dimensional case, such forms exhibit either a harmonic regime where both $H$ and ${H}^{\ifmmode\dagger\else\textdagger\fi{}}$ have a discrete spectrum with biorthogonal eigenstates, and a coherent-like regime where either $H$ or ${H}^{\ifmmode\dagger\else\textdagger\fi{}}$ have a continuous complex twofold degenerate spectrum, while its adjoint has no convergent eigenstates. These regimes reflect the nature of the pertinent normal boson operators. Nondiagonalizable cases as well critical boundary sectors separating these regimes are also analyzed. The extension to $N$-dimensional quadratic systems is also discussed.

arXiv:1710.11266Physical Review ADec 26, 201712 citations

Projection of a quantum state on eigenstates of average Hamiltonian

Jae‐Seung Lee, Anatoly K. Khitrin

arXiv:0801.4098Journal of Magnetic ResonanceMar 6, 20095 citations

The bound state in the spectrum of the Lee–Friedrichs Hamiltonian

Roberto Passante, Franco Persico, Lucia Rizzuto

Physics Letters ASep 1, 20004 citations

Quantal phase factors accompanying adiabatic changes in the case of continuous spectra

Mustapha Maamache, Y. Saadi

By defining ``a virtual gap'' for the continuous spectrum through the notion of eigendifferential (Weyl's packet) and using the differential projector operator, we present a rigorous demonstration and discussion of the quantum adiabatic theorem and the validity condition of the adiabatic approximation for systems having a nondegenerate continuous spectrum. We show that a quantal system in an eigenstate, of operators with a continuous nondegenerate eigenvalue spectrum, slowly transported around a closed curve $C$ by varying parameters in its Hamiltonian, will acquire a generalized geometrical phase factor where an explicit formula is derived in terms of the eigenstates of the Hamiltonian. Examples are given for illustration.By defining ``a virtual gap'' for the continuous spectrum through the notion of eigendifferential (Weyl's packet) and using the differential projector operator, we present a rigorous demonstration and discussion of the quantum adiabatic theorem and the validity condition of the adiabatic approximation for systems having a nondegenerate continuous spectrum. We show that a quantal system in an eigenstate, of operators with a continuous nondegenerate eigenvalue spectrum, slowly transported around a closed curve $C$ by varying parameters in its Hamiltonian, will acquire a generalized geometrical phase factor where an explicit formula is derived in terms of the eigenstates of the Hamiltonian. Examples are given for illustration.

Physical Review ANov 12, 200821 citations

On ergodic limits of normal states in quantum statistical mechanics

Eduard Prugovečki, A. Tip

The asymptotic behavior for t→±∞ of S(t)=exp(−i H t)Sexp(+i H t) and its time average S̄ (t)=t−1∫0tdu S(u) is discussed. Here is S an element of the Banach space B1, constituted by the trace class of operators on the (separable or nonseparable) Hilbert space H, and H is the Hamiltonian, i.e., a bounded or unbounded self-adjoint operator on H. Necessary and sufficient conditions are given for the existence of the limits S̄(± ∞) and S(± ∞) with respect to the weak topology on B1, for the latter under the assumption that the continuous spectrum of H is absolutely continuous. In addition it is shown that if, for a normal state (density operator) ρ, ρ̄(t) has a weak limit, then the limit is again a normal state. This provides further insight in the nature of Emch's ``first ergodic paradox'' [G. G. Emch, J. Math. Phys. 7, 1413 (1966)].The asymptotic behavior for t→±∞ of S(t)=exp(−i H t)Sexp(+i H t) and its time average S̄ (t)=t−1∫0tdu S(u) is discussed. Here is S an element of the Banach space B1, constituted by the trace class of operators on the (separable or nonseparable) Hilbert space H, and H is the Hamiltonian, i.e., a bounded or unbounded self-adjoint operator on H. Necessary and sufficient conditions are given for the existence of the limits S̄(± ∞) and S(± ∞) with respect to the weak topology on B1, for the latter under the assumption that the continuous spectrum of H is absolutely continuous. In addition it is shown that if, for a normal state (density operator) ρ, ρ̄(t) has a weak limit, then the limit is again a normal state. This provides further insight in the nature of Emch's ``first ergodic paradox'' [G. G. Emch, J. Math. Phys. 7, 1413 (1966)].

Journal of Mathematical PhysicsMar 1, 19747 citations

Approach to Equilibrium for Quantum Systems with Continuous Spectrum

Roberto Laura

Nonlinear phenomena and complex systemsJan 1, 20041 citation

We analyze the spectrum and normal-mode representation of general quadratic bosonic forms

H

not necessarily Hermitian. It is shown that, in the one-dimensional case, such forms exhibit either a harmonic regime where both

H

and ${H}^{\ifmmode\dagger\else\textdagger\fi{}}$ have a discrete spectrum with biorthogonal eigenstates, and a coherent-like regime where either

H

or ${H}^{\ifmmode\dagger\else\textdagger\fi{}}$ have a continuous complex twofold degenerate spectrum, while its adjoint has no convergent eigenstates. These regimes reflect the nature of the pertinent normal boson operators. Nondiagonalizable cases as well critical boundary sectors separating these regimes are also analyzed. The extension to

N

-dimensional quadratic systems is also discussed.

C

by varying parameters in its Hamiltonian, will acquire a generalized geometrical phase factor where an explicit formula is derived in terms of the eigenstates of the Hamiltonian. Examples are given for illustration.