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arXiv:2408.04980 · Open Systems & Information Dynamics

On the Liouville–von Neumann Equation for Unbounded Hamiltonians

Davide Lonigro, Alexander Hahn, Daniel Klaus Burgarth

Published Dec 1, 2024• 5 citations

Abstract

The evolution of mixed states of a closed quantum system is described by a group of evolution superoperators whose infinitesimal generator (the quantum Liouville superoperator, or Liouvillian) determines the mixed-state counterpart of the Schrödinger equation: the Liouville–von Neumann equation. When the state space of the system is infinite-dimensional, the Liouville superoperator is unbounded whenever the corresponding Hamiltonian is. In this paper, we provide a rigorous, pedagogically-oriented, and self-contained introduction to the quantum Liouville formalism in the presence of unbounded operators. We present and discuss a characterization of the domain of the Liouville superoperator originally due to M. Courbage; starting from that, we develop some simpler characterizations of the domain of the Liouvillian and its square. We also provide, with explicit proofs, some domains of essential selfadjointness (cores) of the Liouvillian.

DOI: 10.1142/s1230161224500185

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The evolution of mixed states of a closed quantum system is described by a group of evolution superoperators whose infinitesimal generator (the quantum Liouville superoperator, or Liouvillian) determines the mixed-state counterpart of the Schrödinger equation: the Liouville–von Neumann equation. When the state space of the system is infinite-dimensional, the Liouville superoperator is unbounded whenever the corresponding Hamiltonian is. In this paper, we provide a rigorous, pedagogically-oriented, and self-contained introduction to the quantum Liouville formalism in the presence of unbounded operators. We present and discuss a characterization of the domain of the Liouville superoperator originally due to M. Courbage; starting from that, we develop some simpler characterizations of the domain of the Liouvillian and its square. We also provide, with explicit proofs, some domains of essential selfadjointness (cores) of the Liouvillian.

We present a Schrödinger (Liouville) type of equation for a quantum open system. It has a correlated part, and various master equations may be its special cases. It also has significant applications for constructing decoherence-free subspace for quantum computation. It is related to the original Schrödinger (Liouville) equation for the total system through a non-unitary similarity transformation. It is unnecessary for its correlated part to be self-adjoint, so there is a complex spectrum for the corresponding Hamiltonian (Liouvillian), which enables the time evolution of states to be asymmetric. This shows just the correlation to produce evolution of world.

In the quantum theory of matter, the wavefunctions Ψ form a linear space {Ψ} which serves as a carrier space for the operators F corresponding to physical observables as well as for the system operatorsΓ associated with physical ensembles. The mappings of the operators T in turn are called superoperators, and, of particular importance are the Liouvillian and the superevolution operator used in describing the time dependence of the system operators Γ(t). The theory of such superoperators is reviewed in some detail. By studying the time dependence of expectation values (F) and time-correlation functions (F(t)G(t')), the connection between the superoperator approach and the conventional propagator method is explored. The eigenvalue problem of the superoperators is treated, and the principle importance of the superresolvents and the practical usefulness of the partitioning technique are employed. The “inner projections” of superoperators with respect to both orthogonal projectors and skew projectors are studied, and it is shown that they render “rational approximations” of the quantities involved which are always convergent. It is finally shown that the partitioning technique may be used also in solving the time-dependent Liouville equation.

Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function

S^{(K)}(q,p,t)

whose physical meaning is unknown. We show that a different S(q, p, t), with well-defined physical meaning, may be introduced without destroying the attractive “quantum-like” mathematical features of the KvN theory. This new S(q, p, t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrödinger’s equation if functions on phase space are reduced to functions on configuration space. This new kind of “quantization” does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing

\frac{\partial }{\partial p}

by 0 and p by

\frac{\hbar }{\imath } \frac{\partial }{\partial q}

, thus providing an explanation for the common quantization rules.

The problem of construction a quantum mechanical evolution for the Schrödinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C *-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C *-algebras are investigated and it is shown that pure states could evolve into mixed states.