Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Waves in open systems via a biorthogonal basis

P. T. Leung, Wai-Mo Suen, C. P. Sun, K. Young

Published May 1, 1998• 44 citations
Publisher

Abstract

Dissipative quantum systems are sometimes phenomenologically described in terms of a non-Hermitian Hamiltonian $H$, 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.

DOI
10.1103/physreve.57.6101

Related papers

Waves in Open Systems via Bi-orthogonal Basis

P. T. Leung, Wai-Mo Suen, C. P. Sun +1 more

Dissipative quantum systems are sometimes phenomenologically described in terms of a non-hermitian hamiltonian $H$, 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.

arXiv:math-ph/9903016•Mar 8, 1999•0 citations

Green Function and Perturbation Method for Dissipative Systems Based on Biorthogonal Basis

Zhang Li, Gao Yi-Bo, Cheng Hai Wang

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.

Communications in Theoretical Physics•Jun 1, 2009•5 citations

Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems

Alec Maassen van den Brink, K. Young

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.

arXiv:math-ph/9905019•Journal of Physics A Mathematical and General•Mar 19, 2001•20 citations

Observables in non-Hermitian systems: A methodological comparison

Karin Sim, Nicolò Defenu, Paolo Molignini +1 more

Despite acute interest in the dynamics of non-Hermitian systems, there is a lack of consensus in the mathematical formulation of non-Hermitian quantum mechanics in the community. Different methodologies are used in the literature to study non-Hermitian dynamics. This ranges from consistent frameworks like biorthogonal quantum mechanics and metric approach characterized by modified inner products, to normalization by time-dependent norms inspired by open quantum systems. In this work, we systematically explore the similarities and differences among these various methods. Utilizing illustrative models with exact solutions, we demonstrate that these methods produce not only quantitatively different results but also distinct physical interpretations. For dissipative systems where non-Hermiticity arises as an approximation, we find that simply dividing by the norm in the PT -broken regime closely aligns with the full master equation solutions. In contrast, for quantum systems where non-Hermiticity can be engineered exactly, incorporating metric dynamics is crucial for the probabilistic interpretation of quantum mechanics, necessitating the generalizations of similarity transformations and unitarity to non-Hermitian systems. This study lays the groundwork for further exploration of non-Hermitian Hamiltonians, potentially leveraging generalized transformations for novel physical phenomena.

arXiv:2406.05411•Physical Review Research•Mar 28, 2025•12 citations

Biorthogonal Renormalization

Elisabet Edvardsson, J Lukas K König, Marcus Stålhammar

The biorthogonal formalism extends conventional quantum mechanics to the non-Hermitian realm. It has, however, been pointed out that the biorthogonal inner product changes with the scaling of the eigenvectors, an ambiguity whose physical significance is still being debated. Here, we revisit this issue and argue when this choice of normalization is of physical importance. We illustrate in which settings quantities such as expectation values and transition probabilities depend on the scaling of eigenvectors, and in which settings the biorthogonal formalism remains unambiguous. To resolve the apparent scaling ambiguity, we introduce an inner product independent of the gauge choice of basis and show that its corresponding mathematical structure is consistent with quantum mechanics. Using this formalism, we identify a deeper problem relating to the physicality of Hilbert space representations, which we illustrate using the position basis. Apart from increasing the understanding of the mathematical foundations upon which many physical results rely, our findings also pave the way towards consistent comparisons between systems described by non-Hermitian Hamiltonians.

arXiv:2212.06004•Dec 12, 2022•1 citation

Open-quantum-system formulation of particle decay

Reinhold A. Bertlmann, Walter Grimus, Beatrix C. Hiesmayr

We consider an open quantum system which contains unstable states. The time evolution of the system can be described by an effective non-Hermitian Hamiltonian ${H}_{\mathrm{eff}}$, in accord with the Wigner-Weisskopf approximation, and an additional term of the Lindblad form, the so-called dissipator. We show that, after enlarging the original Hilbert space by states which represent the decay products of the unstable states, the non-Hermitian part of ${H}_{\mathrm{eff}}$---the ``particle decay''---can be incorporated into the dissipator of the enlarged space via a specific Lindblad operator. Thus the formulation of the time evolution on the enlarged space has a Hermitian Hamiltonian and is probability conserving. The equivalence of the formulation with the original one demonstrates that the time evolution which is governed by a non-Hermitian Hamiltonian and a dissipator of the Lindblad form is nevertheless completely positive, just as systems with Hermitian Hamiltonians.

arXiv:quant-ph/0602116•Physical Review A•May 1, 2006•95 citations
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.
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.
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.
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.
Despite acute interest in the dynamics of non-Hermitian systems, there is a lack of consensus in the mathematical formulation of non-Hermitian quantum mechanics in the community. Different methodologies are used in the literature to study non-Hermitian dynamics. This ranges from consistent frameworks like biorthogonal quantum mechanics and metric approach characterized by modified inner products, to normalization by time-dependent norms inspired by open quantum systems. In this work, we systematically explore the similarities and differences among these various methods. Utilizing illustrative models with exact solutions, we demonstrate that these methods produce not only quantitatively different results but also distinct physical interpretations. For dissipative systems where non-Hermiticity arises as an approximation, we find that simply dividing by the norm in the PT -broken regime closely aligns with the full master equation solutions. In contrast, for quantum systems where non-Hermiticity can be engineered exactly, incorporating metric dynamics is crucial for the probabilistic interpretation of quantum mechanics, necessitating the generalizations of similarity transformations and unitarity to non-Hermitian systems. This study lays the groundwork for further exploration of non-Hermitian Hamiltonians, potentially leveraging generalized transformations for novel physical phenomena.
The biorthogonal formalism extends conventional quantum mechanics to the non-Hermitian realm. It has, however, been pointed out that the biorthogonal inner product changes with the scaling of the eigenvectors, an ambiguity whose physical significance is still being debated. Here, we revisit this issue and argue when this choice of normalization is of physical importance. We illustrate in which settings quantities such as expectation values and transition probabilities depend on the scaling of eigenvectors, and in which settings the biorthogonal formalism remains unambiguous. To resolve the apparent scaling ambiguity, we introduce an inner product independent of the gauge choice of basis and show that its corresponding mathematical structure is consistent with quantum mechanics. Using this formalism, we identify a deeper problem relating to the physicality of Hilbert space representations, which we illustrate using the position basis. Apart from increasing the understanding of the mathematical foundations upon which many physical results rely, our findings also pave the way towards consistent comparisons between systems described by non-Hermitian Hamiltonians.
We consider an open quantum system which contains unstable states. The time evolution of the system can be described by an effective non-Hermitian Hamiltonian Heff{H}_{\mathrm{eff}}Heff​, in accord with the Wigner-Weisskopf approximation, and an additional term of the Lindblad form, the so-called dissipator. We show that, after enlarging the original Hilbert space by states which represent the decay products of the unstable states, the non-Hermitian part of Heff{H}_{\mathrm{eff}}Heff​---the ``particle decay''---can be incorporated into the dissipator of the enlarged space via a specific Lindblad operator. Thus the formulation of the time evolution on the enlarged space has a Hermitian Hamiltonian and is probability conserving. The equivalence of the formulation with the original one demonstrates that the time evolution which is governed by a non-Hermitian Hamiltonian and a dissipator of the Lindblad form is nevertheless completely positive, just as systems with Hermitian Hamiltonians.