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Journal of Physics A Mathematical and General

Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion

P. T. Leung, S. S. Tong, K. Young

Published Mar 21, 1997• 47 citations
Publisher

Abstract

The concept of eigenfunction expansions for the wave equation is generalized to open systems, in which waves escape to the outside. These non-conservative systems are non-Hermitian in the usual sense. It is shown that the natural framework is an eigenfunction expansion within a two-component formalism that treats the wavefunction and its conjugate momentum together. Provided the system approaches spatial infinity rapidly `without tails', and possesses spatial discontinuities, the expansion in terms of the eigenfunctions (which are now quasinormal modes) is shown to be valid.

DOI
10.1088/0305-4470/30/6/034

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The concept of eigenfunction expansions for the wave equation is generalized to open systems, in which waves escape to the outside. These non-conservative systems are non-Hermitian in the usual sense. It is shown that the natural framework is an eigenfunction expansion within a two-component formalism that treats the wavefunction and its conjugate momentum together. Provided the system approaches spatial infinity rapidly `without tails', and possesses spatial discontinuities, the expansion in terms of the eigenfunctions (which are now quasinormal modes) is shown to be valid.
Based on the approach of biorthogonal basis, we carry out the quasinormal modes (QNMs) expansions for a class of open systems described by the wave equation with outgoing wave boundary conditions. For such a non-Hermitian system, the eigenfunction perturbation expansions and Green function method, which are based on the orthogonal eigenvectors of the Hermitian Hamiltonian for the closed quantum system, can be generalized in terms of the biorthogonal basis, the two sets of eigenfunctions of H and its adjointness H † . The time-independent perturbation theory for the complex frequencies can be also developed.
The orthogonality requirement usually specified for the eigenfunctions φ of a quantum-mechanical operator P which has a continuous spectrum of eigenvalues λ, is ∫φ*(λ)φ(λ′) dλ=δ(λ−λ′). The derivation of this result is examined; it is seen to be merely a sufficient, but not a necessary condition for the consistent expansion of a bound-state wavefunction ψ in the form ∫ a(λ)φ(λ) dλ. The situation when P is a generalized momentum operator is investigated in detail. Such operators were discussed in the preceding paper. It is shown that any two of the corresponding momentum eigenfunctions are not usually orthogonal. Nevertheless, with the help of Fourier analysis, a consistent expansion ψ=∫ a(λ)φ(λ) dλ is established. The theory is illustrated with the familiar examples of a ``particle in a box'' and a ground-state hydrogen atom. When a space coordinate has a finite range, a quantum condition can be imposed on the generalized momentum eigenvalues so that the momentum eigenfunctions form a complete discrete orthogonal set. We attempt to justify the belief that such quantization is not essential unless it is necessary to ensure single-valued eigenfunctions.
Dissipative quantum systems are sometimes phenomenologically described in terms of a non-hermitian hamiltonian HHH, with different left and right eigenvectors forming a bi-orthogonal basis. It is shown that the dynamics of waves in open systems can be cast exactly into this form, thus providing a well-founded realization of the phenomenological description and at the same time placing these open systems into a well-known framework. The formalism leads to a generalization of norms and inner products for open systems, which in contrast to earlier works is finite without the need for regularization. The inner product allows transcription of much of the formalism for conservative systems, including perturbation theory and second-quantization.
Dissipative quantum systems are sometimes phenomenologically described in terms of a non-Hermitian Hamiltonian HHH, with different left and right eigenvectors forming a biorthogonal basis. It is shown that the dynamics of waves in open systems can be cast exactly into this form, thus providing a well-founded realization of the phenomenological description and at the same time placing these open systems into a well-known framework. The formalism leads to a generalization of norms and inner products for open systems, which in contrast to earlier works is finite without the need for regularization. The inner product allows transcription of much of the formalism for conservative systems, including perturbation theory and second quantization.
Dissipation can sometimes be described by a non-Hermitian Hamiltonian H , whose left and right eigenvectors { f j , f j } form a bi-orthogonal basis (BB). For waves in a class of open systems, this is known to lead to exact, complete BB expansions if ⟨ f j | f j ⟩≠0 for all j . If not, normalization seems impossible and many familiar formulae fail; examples are given. The problem is related to the merging of eigenmodes, so that H can only be diagonalized to Jordan blocks. The resolution involves a generalized BB containing extra vectors, whose dynamics are modified by polynomials in the time t . The splitting of merged modes under a perturbation is also treated. One thus obtains a non-trivial extension of the BB formalism for open systems.
Information on quantum systems can be obtained only when they are open (or opened) in relation to a certain environment. As a matter of fact, realistic open quantum systems appear in very different shape. We sketch the theoretical description of open quantum systems by means of a projection operator formalism elaborated many years ago, and applied by now to the description of different open quantum systems. The Hamiltonian describing the open quantum system is non-Hermitian. Most studied are the eigenvalues of the non-Hermitian Hamiltonian of many-particle systems embedded in one environment. We point to the unsolved problems of this method when applied to the description of realistic many-body systems. We then underline the role played by the eigenfunctions of the non-Hermitian Hamiltonian. Very interesting results originate from the fluctuations of the eigenfunctions in systems with gain and loss of excitons. They occur with an efficiency of nearly 100\%. An example is the photosynthesis.