Pith. sign in

REVIEW 3 major objections 6 minor 22 references

Density matrices and entropy operator for non-Hermitian quantum mechanics

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Density matrices, pure-state tests, and entropy operators can be extended consistently to non-Hermitian Hamiltonians in two ways: via a bounded invertible similarity and via a non-invertible intertwining map.

desk verdict The RDM half is correct and useful; the GDM half is underdetermined and Theorem 11 is false as stated — the paper needs major revision before the new claims can be trusted. read the letter →

arxiv 2501.11537 v1 pith:OTUTP4OH submitted 2025-01-20 math-ph math.MPquant-ph

Fabio Bagarello , Francesco Gargano , Lidia Saluto This is my paper · ORCID
classification math-phmath.MPquant-ph MSC 81Q1246N5047A10
keywords non-HermitianquantummechanicsdensitymatrixentropyoperatorRieszbasisintertwiningpurestatePT-symmetricHamiltonianSwanson
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks what a density matrix and its entropy should mean when the Hamiltonian is not self-adjoint. It proposes two natural extensions: an RDM, an operator similar to a standard density matrix $\rho_0$ via a bounded invertible similarity $\rho = R\rho_0 R^{-1}$, and a GDM, an operator intertwined with $\rho_0$ by a non-invertible $R$ satisfying $\rho R = R\rho_0$. The paper shows that in both settings the usual pure-state criteria—$\operatorname{tr}(\rho^2)=1$ and $S(\rho)=0$—remain equivalent, and that when $R^\dagger R$ is invertible the GDM has an explicit trace-one form. This matters because non-Hermitian Hamiltonians describe gain/loss and PT-symmetric systems, where the standard density-matrix language was previously not available.

What carries the argument

The machinery is the intertwining relation between a deformed operator and a standard density matrix. A normal DM $\rho_0$ has the spectral form $\rho_0 = \sum_j \lambda_j |e_j\rangle\langle e_j|$; the deformation operator $R$ moves the eigenbasis. When $R$ has a bounded inverse, the vectors $\varphi_j = Re_j$ form a Riesz basis with biorthogonal partners $\psi_j = (R^{-1})^\dagger e_j$, giving $\rho = \sum_j \lambda_j |\varphi_j\rangle\langle\psi_j|$ and $S(\rho) = R S(\rho_0) R^{-1}$. When $R$ is non-invertible, the same expansion is restricted to the closed span of the $\varphi_j$; if $R$ has the property that $R^\dagger R$ is invertible, the biorthogonal family is $\psi_j = R(R^\dagger R)^{-1}e_j$ and the trace-one formula $\rho = R\rho_0(R^\dagger R)^{-1}R^\dagger$ follows. In both branches the pure-state test $\operatorname{tr}(\rho^2)=1$ and the vanishing of the entropy operator are equivalent because both quantities are inherited from $\rho_0$.

What would settle it

Take a bounded non-invertible operator $R$ on an infinite-dimensional Hilbert space that does not satisfy the condition that $R^\dagger R$ is invertible, chosen so that $\{Re_j\}$ is complete and minimal but not a Schauder basis. If for such an $R$ a diagonal density matrix $\rho_0$ makes the biorthogonal expansion (2.17) fail for some $f$ in the closed span, then the GDM expansion (2.18), the generalized pure-state criterion, and the generalized entropy operator all lose their foundation; the existence of such complete minimal non-basis sequences is a standard fact.

Watch

Extended reading notes

Core claim

The central claim is that the density-matrix formalism extends to non-Hermitian Hamiltonians in two compatible ways. In the RDM case, $\rho = R\rho_0 R^{-1}$ with $R$ bounded and boundedly invertible preserves the spectrum of $\rho_0$, so it has trace one, and the equivalence $\operatorname{tr}(\rho^2)=1 \iff S(\rho)=0$ is inherited from the undeformed operator (Theorems 4 and 6). In the GDM case, where $R$ is non-invertible, the paper defines $\rho$ through the intertwining relation $\rho R = R\rho_0$ and proves that, whenever $R$ has the property that $R^\dagger R$ is invertible, $\rho$ can be written as $\rho = R\rho_0(R^\dagger R)^{-1}R^\dagger$, which again has trace one and satisfies the generalized pure-state criterion (Theorem 11). The paper also shows that without this property the trace of a GDM can fail to be one, as in the exceptional-point example of the finite-dimensional Swanson model.

Load-bearing premise

The construction assumes that, even when the deformation operator $R$ is non-invertible, the deformed vectors $\varphi_j = Re_j$ form a genuine basis of their closed span with a unique biorthogonal family, so that the expansions (2.17)–(2.18) and the generalized entropy operator are legitimate; the paper posits this rather than proving it for all bounded non-invertible $R$.

Editorial extensions

If this is right

  • Any RDM shares the spectrum, trace, purity and entropy of its undeformed density matrix, so mixedness in a non-Hermitian system can be read off from the ordinary $\rho_0$ before deformation.
  • The pure-state characterization—$\operatorname{tr}(\rho^2)=1$ exactly when $S(\rho)=0$—continues to hold for RDMs and for GDMs with $R^\dagger R$ invertible, giving a practical test for purity in gain/loss systems.
  • Equation (2.19) gives a canonical trace-one GDM whenever $R$ is non-invertible but $R^\dagger R$ is invertible, so non-invertible deformations such as shifts on $\ell^2$ become usable.
  • In the finite-dimensional Swanson model, the exceptional point $\alpha_2 = -1/2$ drives purity to its minimum $1/3$ and entropy to its maximum $\log 3$, while away from it the RDM tends to a pure or half-mixed state depending on the regime; at the exceptional point a GDM can have trace different from one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The loss of unit trace for GDMs suggests that $\operatorname{tr}(\rho)$ itself could be used as a measure of how much information the non-invertible deformation discards; renormalizing by $\operatorname{tr}(\rho)$ would produce a normalized state whenever the trace is positive. This is an editorial inference, not a claim in the paper.
  • Because the trace-one formula (2.19) only needs $R$ to be injective with closed range, it should extend to a larger class of non-invertible maps; testing it on a concrete infinite-dimensional embedding is a natural next step the paper leaves open.
  • The maximal-mixing behaviour near the exceptional point hints that exceptional points may be used to prepare maximally mixed states in finite-dimensional non-Hermitian systems; the paper does not draw this conclusion.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes two extensions of the standard density-matrix formalism for non-Hermitian quantum mechanics. The first, a Riesz density matrix (RDM), is defined by similarity ρ = Rρ0R^{-1} with R invertible; the authors show that trace, purity, and the pure-state criterion tr(ρ²)=1 are preserved, and they define a corresponding entropy operator. The second, a generalized density matrix (GDM), is defined by the intertwining relation ρR = Rρ0 with R non-invertible; here the paper claims that a pure-state criterion and an entropy criterion remain available, and it gives a trace-one formula (2.19) under an additional property (PI: R†R invertible). These constructions are illustrated on a two-state PT-symmetric model and on a finite-dimensional truncated Swanson model, including the exceptional-point regime where R becomes non-invertible.

Significance. If correct, the GDM framework would give a systematic way to assign density matrices, purity, and entropy in non-Hermitian settings where a similarity transformation is unavailable. The RDM part is sound: Definitions 3 and 5, Theorem 4, and Theorem 6 are correct and easy to verify, and the worked examples for the two-state system and the finite Swanson model are useful illustrations. The GDM part, however, is not currently sound. Definition 7 is underdetermined, Theorem 11 is false as stated, and the basis expansion that underpins the non-invertible case is asserted rather than proved. These are load-bearing issues for the paper's central claim that the pure-state and entropy criteria extend to the intertwining (non-similarity) case. The paper deserves a major revision, not outright rejection, because the defects are local to the GDM definitions and could in principle be repaired by adding uniqueness or spectral-construction hypotheses.

major comments (3)
  1. [Definition 7 and Theorem 11 (p1'')] The intertwining relation (2.15) does not determine ρ on the orthogonal complement of Ran R: if ρ satisfies ρR = Rρ0, then for every bounded T with T(Ran R) = {0}, ρ+T also satisfies the relation. This is not an infinite-dimensional artifact. In the paper's own finite-dimensional example of Section III.2.4, take α1 = 1, R as in (3.21), ρ0 = |e1⟩⟨e1|, and ρ = |e2⟩⟨e2| so that ρR = Rρ0. Let u = (1,0,-√2) and P_u = |u⟩⟨u|/3. Since P_u R = 0, the one-parameter family ρ_γ = ρ + γP_u satisfies (2.15) for every γ, and by Definition 10 each ρ_γ is a GPS because ρ0 is pure. However, tr(ρ_γ²) = 1 + γ², which equals 2 for γ = 1. Thus p1'' of Theorem 11 is false for a GDM as defined. The theorem must be restricted to a uniquely specified spectral construction, for example the form (2.18), or Definition 7 must be tightened with a condition on the complement of Ran R (e.g., ρ vanishes on (Ran R)⊥). Without such a restriction, trace, purity, and entropy of a GDM are not well-defined by (2.15).
  2. [Section II.3, Eqs. (2.16)-(2.18)] The statement that Fφ = {Re_j} is a basis for Hφ with a unique biorthogonal basis Fψ, and the expansions (2.17)-(2.18), are asserted without proof. For a general bounded non-invertible R, linear independence and totality of Fφ in Hφ do not imply that Fφ is a Schauder basis of its closed span; the footnote about degenerate eigenvalues does not supply the missing argument. Since the spectral decomposition (2.18) is what later justifies the GDM entropy and purity computations, this is a load-bearing gap. The authors should either prove the basis property under explicit hypotheses or restrict the GDM construction to cases where it holds, such as the property-I condition introduced only later, or to finite-rank situations.
  3. [Definition 10 and Theorem 11 (p2'')] The generalized entropy operator is defined only by the intertwining relation S(ρ)R = RS(ρ0). Like (2.15), this leaves S(ρ) completely unspecified on vectors outside Ran R, so the statement 'S(ρ) = 0 on Hφ' is not a well-defined criterion unless S(ρ) is additionally required to be constructed uniquely from ρ0 and R (for instance, as the spectral operator analogous to (2.18)). The proof of Theorem 11 is omitted with 'similar to the one of Theorem 6', but the similarity argument used there is unavailable for a non-invertible R, and in any case the failure of p1'' already invalidates the theorem as stated. The entropy part needs to be restated and proved under a corrected GDM definition.
minor comments (6)
  1. [Abstract and general text] There are several typographical errors that should be corrected, including 'waht' in the abstract, 'becames' and 'analize' in Section I, and 'continous' in Section II.3.
  2. [Section II.1 and II.2] The set of all DMs is denoted G in Section II.1, while Section II.2 uses G(R, ρ0) for the set of (R,ρ0)-RDMs; these notations are too close and should be distinguished.
  3. [Section II.3] The identity operator is written as '1 1' in several places (e.g., after (2.4) and in Definition 9); a single symbol such as I or 1l would be clearer.
  4. [Section III.1.1] The sentence 'the families Fφ and Fψ~ = {ψ~±} are also bi-orthogonal Riesz-basis' has subject-verb disagreement and nonstandard notation; it should be rewritten.
  5. [Section III.2.3] The phrase 'immersion in a heath bath' should read 'heat bath', and the caption of Figure 2, 'α > −1/2', should be 'α1α2 > −1/2' to match the text.
  6. [Section III.2.4] The sentence 'the case λ1 → 1 is to be considered singular' is vague; the limiting density matrix ρ0 = |e1⟩⟨e1| is not singular in an operator-theoretic sense, and the boundary values of purity and entropy at λ1 = 1 should be stated explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the RDM/GDM criteria are consequences of the defining similarity/intertwining relations, with only non-load-bearing self-citations.

full rationale

The derivation chain is definitional and internal rather than circular. Definition 3 defines an RDM as rho = R rho0 R^{-1}, so Theorem 6's criteria p1' (tr(rho^2)=1) and p2' (S(rho)=0) follow from cyclicity of the trace and similarity invariance of the entropy construction; they are immediate consequences of the defining similarity, not independently fitted or predicted quantities. For GDMs, Definition 7 uses only the intertwining relation rho R = R rho0, and formulas (2.16), (2.18), and (2.19) are derived from rho0's spectral decomposition plus the explicit choice psi_j = R(R^dagger R)^{-1}e_j under property I; the trace-one statement tr(rho)=tr(rho0)=1 follows from tr(AB)=tr(BA). No parameter is fitted to data, and the examples in Section III choose R and the weights lambda_j by hand and then compute purity/entropy curves, so those curves are illustrations of the definitions rather than predictions that reduce to their own inputs. The only notable self-citation is Definition 9's attribution of property I to [19], by one of the present authors, but the needed construction is re-derived in the text and is not load-bearing; other self-citations are background references for non-Hermitian frameworks. The paper also explicitly flags limitations: Section II.3 states that a GDM need not have trace one, and Section III.2.4 provides a case where R lacks property I and tr(rho) != 1. A separate correctness concern is that (2.15) leaves rho unconstrained on the complement of Ran R, so Theorem 11's omitted proof ('The proof is similar to the one of Theorem 6 and will not be repeated') would need additional hypotheses; that is a support gap, not a circular reduction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 3 invented entities

The central framework rests on standard operator theory plus several unproved or under-justified assumptions in the GDM part: entropy convergence in infinite dimensions, basis properties of images of ONBs under non-invertible R, and the physical status of non-positive deformed DMs. The application sections add manually chosen occupation weights that drive the reported purity and entropy curves.

free parameters (3)
  • Initial occupation weights λ_j (RDM I) = λ_j = |µ_j|² / Σ_k |µ_k|²
    Chosen by hand so ρ0(0) is positive with unit trace for all α1,α2; the asymptotic purity 1/2 and entropy log 2 are direct consequences of λ2,λ3→1/2.
  • Initial thermal weights λ_j (RDM II) = λ_j = e^{-βµ_j} / Σ_k e^{-βµ_k}
    Standard Boltzmann weights imported into the deformed setting; β is an external temperature parameter and the purity and entropy curves are functions of β and α1α2.
  • Degeneracy parameter λ1 (GDM example) = free in [0,1], with λ2=λ3=(1-λ1)/2
    Imposed to make the non-invertible R intertwining solvable; the resulting ρ has trace (1+λ1)/2, so the reported trace loss is an artifact of this choice.
assumptions (5)
  • standard math Bounded positive operators are self-adjoint and admit a spectral decomposition with eigenvalues in [0,1] for trace-one positive ρ0.
    Used in Section II.1 to justify dropping the Hermiticity requirement in Definition 1; based on the Hilbert-Schmidt theorem.
  • domain assumption The entropy series S(ρ0)=Σ_j -λ_j log λ_j P^o_j converges in the operator norm for the trace-one positive ρ0 considered.
    Section II.1, Remark (1): the paper explicitly says a general convergence proof is 'not so easy' and not provided; the entropy operator and its similarity transform are central to the framework.
  • ad hoc to paper For non-invertible R, the sequence Fφ={Re_j} is a basis for Hφ and has a unique biorthogonal basis Fψ with expansions (2.17)-(2.18).
    Section II.3, immediately after Eq. (2.16): linear independence and totality are asserted but do not imply the needed basis expansion without a closed-range or frame condition.
  • domain assumption Operators ρ that are not positive and not self-adjoint can be treated as density matrices as long as they are similar to or intertwined with a standard DM.
    Section II.2, after Eq. (2.8) and Eq. (2.13): the paper acknowledges Φρ is not a positive state and no longer gives mean values, yet still calls ρ a DM; physical interpretation is assumed rather than derived.
  • standard math The cyclic property tr(AB)=tr(BA) applies to the products Rρ0R^{-1}, Rρ0(R†R)^{-1}R†, and related trace-class products.
    Used in Sections II.2-II.3 to show trace preservation for RDMs and, under property I, for GDMs.
invented entities (3)
  • Riesz Density Matrix (RDM)
    purpose: Density-matrix-like operator obtained as ρ=Rρ0R^{-1} with R bounded invertible; extends DM to non-Hermitian settings via similarity.
    Formal mathematical object; no empirical handle is provided. It generalizes biorthogonal DMs already in the literature.
  • Generalized Density Matrix (GDM)
    purpose: Operator ρ satisfying ρR=Rρ0 with non-invertible R; intended to handle exceptional points.
    New mathematical construct; the paper itself defers its concrete physical relevance to future work (Section IV).
  • Generalized Entropy Operator (GEO)
    purpose: Entropy operator S(ρ) satisfying the intertwining relation S(ρ)R=RS(ρ0).
    Introduced in Section II.3, Definition following Eq. (2.22); no independent verification beyond the paper's examples.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Density matrices and entropy operator for non-Hermitian quantum mechanics." pith.science (2026). https://pith.science/paper/OTUTP4OH

@misc{pith2026250111537,
  author       = {Pith},
  title        = {Pith review of: Density matrices and entropy operator for non-Hermitian quantum mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OTUTP4OH}},
  note         = {Machine review of arXiv:2501.11537}
}
abstract

In this paper we consider density matrices operator related to non-Hermitian Hamiltonians. In particular, we analyse two natural extensions of what is usually called a density matrix operator (DM), of pure states and of the entropy operator: we first consider those {\em operators} which are simply similar to a standard DM, and then we discuss those which are intertwined with a DM by a third, non invertible, operator, giving rise to waht we call Riesz Density Matrix operator (RDM). After introducing the mathematical framework, we apply the framework to a couple of applications. The first application is related to a non-Hermitian Hamiltonian describing gain and loss phenomena, widely considered in the context of $PT$-quantum mechanics. The second application is related to a finite-dimensional version of the Swanson Hamiltonian, never considered before, and addresses the problem of deriving a milder version of the RDM when exceptional points form in the system.

Figures

Figures reproduced from arXiv: 2501.11537 by the authors.

Figure 1
Figure 1. (a) Behavior of purity tr(ρ 2 (t)) and entropy tr(S(ρ(t))) in the vicinity of the ex￾ceptional point α2 = −1/2, with α1 = 1 for the RDM I example. At the exceptional point, R is not invertible, and the purity tends to its minimal value of 1/3, while the entropy to its maximal value of log(3). (b) Behavior of purity and entropy for decreasing values of α2 with α1 = 1. As α2 moves away from the exceptional point, the … view at source ↗
Figure 2
Figure 2. (a) Behavior of the purity tr(ρ 2 ) for various value of β and with α > −1/2, α1 = 1 for the RDM II example. As α2 → −1/2, ρ tends to a Riesz pure state, whereas for α2 → ∞, ρ is a fully mixed state. (b)) Same as (a) but for the entropy tr(S(ρ)). which are different from those considered so far, one can check that (2.15) is satisfied by taking ρ =   (1 − λ1)/2 0 0 0 λ1 0 (1 − λ1)/2 √ 2α1 0 0   . (3.22) We obse… view at source ↗
Figure 3
Figure 3. Behavior of purity tr(ρ 2 ) and entropy tr(S(ρ)) as function of the parameter λ1 for the GDM example . Hence our R has not PI, as we have already observed after Definition 9. For this reason, it is not possible to use (2.19) in the present context. IV Conclusions In this paper we have proposed some natural extensions of the notions of density matrix, pure state and entropy operators. Our main aim was to use our prop… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 19 canonical work pages

  1. [1]

    Bender, A

    C. Bender, A. Fring, U. G¨ uenther, H. Jones Eds, Special issue on quantum physics with non-Hermitian operators, J. Phys. A: Math. and Ther., 45 (2012)

    work page 2012

  2. [2]

    C. M. Bender, P TSymmetry In Quantum and Classical Physics , World Scientific Pub- lishing Europe Ltd., London (2019)

    work page 2019

  3. [3]

    Bagarello, J

    F. Bagarello, J. P. Gazeau, F. H. Szafraniec e M. Znojil Eds., Non-selfadjoint operators in quantum physics: Mathematical aspects , John Wiley and Sons (2015)

    work page 2015

  4. [4]

    Bagarello, R

    F. Bagarello, R. Passante, C. Trapani, Non-Hermitian Hamiltonians in Quantum Physics; Selected Contributions from the 15th International Conference on Non-Hermitian Hamil- tonians in Quantum Physics , Palermo, Italy, 18-23 May 2015, Springer (2016)

    work page 2016

  5. [5]

    Breuer, F

    H. Breuer, F. Petruccione, The theory of open quantum systems , Oxford University Press (2002)

    work page 2002

  6. [6]

    Bagarello, Pseudo-Bosons and Their Coherent States , Springer, Mathematical Physics Studies, 2022

    F. Bagarello, Pseudo-Bosons and Their Coherent States , Springer, Mathematical Physics Studies, 2022

    2022

  7. [7]

    Heil, A basis theory primer: expanded edition , Springer, New York, (2010)

    C. Heil, A basis theory primer: expanded edition , Springer, New York, (2010)

    2010

  8. [8]

    Christensen, An Introduction to Frames and Riesz Bases , Birkh¨ auser, Boston, (2003) 24

    O. Christensen, An Introduction to Frames and Riesz Bases , Birkh¨ auser, Boston, (2003) 24

    work page 2003

Show all 22 references
  1. [9]

    D. C. Brody, Biorthogonal Quantum Mechanics , J. Phys. A: Math. Theor. 47 035305 (2013)

    2013

  2. [10]

    Mostafazadeh, Pseudo-hermitian quantum mechanics , Int

    A. Mostafazadeh, Pseudo-hermitian quantum mechanics , Int. J. Geom. Methods Mod. Phys., 7, 1191-1306 (2010)

    2010

  3. [11]

    Sinha, A

    A. Sinha, A. Ghosh, B. Bagchi, Exceptional points and ground-state entanglement spectrum for a fermionic extension of the Swanson oscillator , arXiv:2401.17189 (2024)

    2024 arXiv

  4. [12]

    Herviou, N

    L. Herviou, N. Regnault, J. H. Bardarson, Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models , SciPost Phys. 7, 069 (2019)

    2019

  5. [13]

    Sergi, K

    A. Sergi, K. G. Zloshchastiev, Non-Hermitian quantum dynamics of a two-level system and models of dissipative environments , Int. J. Mod. Phys. B, 27, 1350163 (2013)

    2013

  6. [14]

    Kuru S., Tegmen A., Vercin A., Intertwined isospectral potentials in an arbitrary dimen- sion, J. Math. Phys, 42, No. 8, 3344-3360, (2001); Kuru S., Demircioglu B., Onder M., Vercin A., Two families of superintegrable and isospectral potentials in two dimensions , J. Math. Phy...

    2001

  7. [15]

    Merzbacher, Quantum mechanics , Second Edition, John Wiley and Sons, New York (1970)

    E. Merzbacher, Quantum mechanics , Second Edition, John Wiley and Sons, New York (1970)

    1970

  8. [16]

    Roman, Advanced quantum theory, Addison-Wesley, Reading (1965)

    P. Roman, Advanced quantum theory, Addison-Wesley, Reading (1965)

    1965

  9. [17]

    S. Reed, B. Simon, Methods of modern mathematical physics , Vol I: Functional analysis, Academic Press-to, New York (1972)

    1972

  10. [18]

    Bratteli and D.W

    O. Bratteli and D.W. Robinson, Operator algebras and Quantum statistical mechanics 1 , Springer-Verlag, Berlin, (1987) 25

    1987

  11. [19]

    Bagarello, Intertwining operators for non self-adjoint Hamiltonians and bicoherent states, J

    F. Bagarello, Intertwining operators for non self-adjoint Hamiltonians and bicoherent states, J. Math. Phys., 57, 103501 (2016)

    2016

  12. [20]

    M. S. Swanson, Transition elements for a non-Hermitian quadratic hamiltonian , J. Math. Phys., 45, 585 (2004)

    2004

  13. [21]

    Fernandez, R

    V. Fernandez, R. Ramirez, M. Reboiro, Swanson Hamiltonian: non-PT-symmetry phase , J. Phys. A, 55, 15303 (2022)

    2022

  14. [22]

    Felski, A

    A. Felski, A. Beygi, C. Karapoulitidis, S.P. Klevansky Three perspectives on entropy dy- namics in a non-Hermitian two-state system , preprint arXiv:2404.03492 (2024). 26

    2024 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.