Quantum theory of non-integrable systems

Petrosky, Tomio; Prigogine, Ilya; Tasaki, Seiji

doi:10.1016/0378-4371(91)90257-d

Cited by 202 publications

(241 citation statements)

References 13 publications

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“…When the discrete states are in resonance with the continuum, they generally decay into the continuum. In order to explain the exponential decay process in terms of the microscopic dynamics, we consider the complex eigenvalue problem in the extended Hilbert space [10] …”

Section: Model and Effective Hamiltonianmentioning

confidence: 99%

“…Prigogine and one of the authors (T.P.) et al have clarified that the spectrum of the effective Hamiltonian coincides with that of the total Hamiltonian, so that the Hermitian Hamiltonian of the total system can have a complex spectrum due to the resonance if we extend the eigenvector space from the ordinary Hilbert space into a dual vector space, called the extended Hilbert space, where the Hilbert norm of the eigenvector vanishes [8,10,33].…”

Section: Introductionmentioning

confidence: 99%

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Higher-order time-symmetry-breaking phase transition due to meeting of an exceptional point and a Fano resonance

et al. 2016

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We have theoretically investigated the time-symmetry breaking phase transition process for two discrete states coupled with a one-dimensional continuum by solving the nonlinear eigenvalue problem for the effective Hamiltonian associated with the discrete spectrum. We obtain the effective Hamiltonian with use of the Feshbach-Brillouin-Wigner projection method. Strong energy dependence of the self-energy appearing in the effective Hamiltonian plays a key role in the time-symmetry breaking phase transition: as a result of competition in the decay process between the Van Hove singularity and the Fano resonance, the phase transition becomes a higher-order transition when both the two discrete states are located near the continuum threshold.

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Section: Model and Effective Hamiltonianmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Higher-order time-symmetry-breaking phase transition due to meeting of an exceptional point and a Fano resonance

et al. 2016

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“…In the early days of quantum mechanics, it was noticed that we can find the resonance pole as an eigenstate of the timeindependent Schrödinger equation under the Siegert boundary condition [5]. Later on, Feshbach [27,28] as well as other researchers [49] established the Feshbach formalism, which algebraically produces resonance poles. In the present paper we will review the two methods respectively in Secs.…”

Section: Modelmentioning

confidence: 98%

Equivalence of the effective Hamiltonian approach and the Siegert boundary condition for resonant states

Hatano

2012

Fortschritte der Physik

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“…Note that Σ + (z) is defined by the Cauchy integral with the branch cut from -1 to 1 in the energy plane; we define Σ + (z) by taking the analytic continuation from the upper half energy plane as denoted by the + superscript [21]. This is the same self-energy that brings about the boundstate-in-continuum (BIC) as studied in Ref.…”

Section: Complex Eigenvalue Problemmentioning

confidence: 99%

“…In this section we briefly summarize the complex eigenvalue problem with use of the BWF projection method. One could refer to the literatures for details [13,21,23].…”

mentioning

confidence: 99%

Fano absorption spectrum with the complex spectral analysis

et al. 2017

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A new aspect of understanding a Fano absorption spectrum is presented in terms of the complex spectral analysis. The absorption spectrum of an impurity embedded in semi-infinite superlattice is investigated. The boundary condition on the continuum causes a large energy dependence of the self-energy, enhances the nonlinearity of the eigenvalue problem of the effective Hamiltonian, yielding several nonanalytic resonance states. The overall spectral features is perfectly reproduced by the direct transitions to these discrete resonance states. Even with a single optical transition path the spectrum exhibits an asymmetric Fano profile, which is enhanced for the transition to the nonanalytic resonance states. Since this is the genuine eigenstates of the total Hamiltonian, there is no ambiguity in the interpretation of the absorption spectrum, avoiding the arbitrary interpretation based on the quantum interference. The spectral change around the exceptional point is well understood when we extract the resonant state component.

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Quantum theory of non-integrable systems

Cited by 202 publications

References 13 publications

Higher-order time-symmetry-breaking phase transition due to meeting of an exceptional point and a Fano resonance

Higher-order time-symmetry-breaking phase transition due to meeting of an exceptional point and a Fano resonance

Equivalence of the effective Hamiltonian approach and the Siegert boundary condition for resonant states

Fano absorption spectrum with the complex spectral analysis

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