A Criterion for the Uniform Eventual Positivity of Operator Semigroups

Daners, Daniel; Glück, Jochen

doi:10.1007/s00020-018-2478-y

Cited by 32 publications

(57 citation statements)

References 22 publications

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“…Then, the transcendental equation for finding eigenenergies of the Hamiltonian reads:

\begin{matrix} true \\ left (() \frac{1}{Λ_{-}} + \frac{1}{Λ_{+}}) π E^{1 / 2} prefixcos π E^{1 / 2} + (() \frac{1}{{normalΛ}_{-} {normalΛ}_{+}} - π^{2} E) prefixsin π E^{1 / 2} \\ left = 0 \end{matrix}

It immediately shows that for the opposite signs of the equal magnitudes of the extrapolation lengths (as mentioned above, we will call such a geometry an antisymmetric one and denote the corresponding quantities by the subscript ‘ A ’) the energies are:

\begin{matrix} true \\ right E_{A_{0}} & = & left - \frac{1}{π^{2} Λ^{2}}, E_{A_{n}} = n^{2}, n = 1, 2, \dots, \\ right {normalΛ}_{-} & = & left - Λ_{+} \equiv normalΛ \end{matrix}

So, interaction of the two interfaces yields the Dirichlet spectrum supplemented by the BC split‐off state whose negative energy is equal to its counterpart of the single attractive wall . The emergence of the level with

E < 0

is caused by the negative extrapolation length when for

Λ_{+}

approaching zero from the left the waveform, as is shown below, becomes more and more localized at the corresponding surface and the energy in the same limit falls down as

- {normalΛ}^{- 2}

what is true for any l ‐dimensional domain . Normalized to unity,

0true \int_{- 1 / 2}^{1 / 2} Ψ^{2} (Λ; x) d x = 1

…”

Section: Formulation Energy Spectrum and Position Waveformsmentioning

confidence: 99%

Quantum Information Measures of the One‐Dimensional Robin Quantum Well

Olendski

2018

Annalen der Physik

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Shannon quantum information entropies S x ,k , Fisher informations I x ,k , Onicescu energies O x ,k , and statistical complexities e S x ,k O x ,k are calculated both in the position (subscript x ) and momentum (k) representations for the Robin quantum well characterized by the extrapolation lengths − and + at the two confining surfaces. The analysis concentrates on finding and explaining the most characteristic features of these quantum information measures in the whole range of variation of the Robin distance for the symmetric, − = + = , and antisymmetric, − = − + = , geometries. Analytic results obtained in the limiting cases of the extremely large and very small magnitudes of the extrapolation parameter are corroborated by the exact numerical computations that are extended to the arbitrary length . It is confirmed, in particular, that the entropic uncertainty relation S x n + S k n ≥ 1 + ln π and general inequality e S O ≥ 1, which is valid both in the position and momentum spaces, hold true at any Robin distance and for every quantum state n. For either configuration, there is a range of the extrapolation lengths where the rule S x n+1 ( ) + S k n+1 ( ) ≥ S x n ( ) + S k n ( ) that is correct for the Neumann ( = ∞) or Dirichlet ( = 0) boundary conditions, is violated. Other analytic and numerical results for all measures are discussed too and their physical meaning is highlighted. S(2)

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“…Then, the transcendental equation for finding eigenenergies of the Hamiltonian reads:

\begin{matrix} true \\ left (() \frac{1}{Λ_{-}} + \frac{1}{Λ_{+}}) π E^{1 / 2} prefixcos π E^{1 / 2} + (() \frac{1}{{normalΛ}_{-} {normalΛ}_{+}} - π^{2} E) prefixsin π E^{1 / 2} \\ left = 0 \end{matrix}

\begin{matrix} true \\ right E_{A_{0}} & = & left - \frac{1}{π^{2} Λ^{2}}, E_{A_{n}} = n^{2}, n = 1, 2, \dots, \\ right {normalΛ}_{-} & = & left - Λ_{+} \equiv normalΛ \end{matrix}

E < 0

is caused by the negative extrapolation length when for

Λ_{+}

approaching zero from the left the waveform, as is shown below, becomes more and more localized at the corresponding surface and the energy in the same limit falls down as

- {normalΛ}^{- 2}

what is true for any l ‐dimensional domain . Normalized to unity,

0true \int_{- 1 / 2}^{1 / 2} Ψ^{2} (Λ; x) d x = 1

…”

Section: Formulation Energy Spectrum and Position Waveformsmentioning

confidence: 99%

Quantum Information Measures of the One‐Dimensional Robin Quantum Well

Olendski

2018

Annalen der Physik

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“…First, consider heat capacity behavior at the cold temperatures, β → ∞, and not very small extrapolation lengths, 1. Then, Equation (13) reduces to (14) and the corresponding specific heat reads:…”

Section: Canonical Ensemblementioning

confidence: 99%

“…The mathematical and physical reasons for this strong binding in general n-dimensional domain were explained and analyzed before [9][10][11][12][13][14][15][16][17][18] and repeated for our geometry in a preceding paper [8] where also quantum-information measures of the structure were computed. The mathematical and physical reasons for this strong binding in general n-dimensional domain were explained and analyzed before [9][10][11][12][13][14][15][16][17][18] and repeated for our geometry in a preceding paper [8] where also quantum-information measures of the structure were computed.…”

Section: Introductionmentioning

confidence: 99%

“…So, it is the Dirichlet spectrum supplemented by the BC split-off state whose negative energy inversely depends on the square of the Robin distance. The mathematical and physical reasons for this strong binding in general n-dimensional domain were explained and analyzed before [9][10][11][12][13][14][15][16][17][18] and repeated for our geometry in a preceding paper [8] where also quantum-information measures of the structure were computed. Even though it is not a main subject of the present research, let us mention that the geometry from Equation (10) does satisfy the requirement of the Kenneth-Klich theorem [19] about the Casimir interaction of the two bodies related by reflection and, accordingly, the Casimir force between the plates will be attractive whereas the theorem does not apply for the surfaces with the same extrapolation lengths considered here too and the corresponding interaction can be repulsive.…”

Section: Introductionmentioning

confidence: 99%

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Thermodynamic Properties of the 1D Robin Quantum Well

Olendski

2018

Annalen der Physik

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The thermodynamic properties of the Robin quantum well with extrapolation length are analyzed theoretically both for the canonical and two grand canonical ensembles, with special attention being paid to the situation when the energies of one or two lowest-lying states are split off from the rest of the spectrum by the large gap that is controlled by the varying . For the single split-off level, which exists for the geometry with the equal magnitudes but opposite signs of the Robin distances on the confining interfaces, the heat capacity c V of the canonical averaging is a nonmonotonic function of the temperature T with its salient maximum growing to infinity as ln 2 for the decreasing to zero extrapolation length and its position being proportional to 1/( 2 ln ). The specific heat per particle c N of the Fermi-Dirac ensemble depends nonmonotonically on the temperature too, with its pronounced extremum being foregone on the T axis by the plateau, whose value at the dying is (N − 1)/(2N)k B , with N being the number of fermions. The maximum of c N , similar to the canonical averaging, unrestrictedly increases as goes to zero and is largest for one particle. The most essential property of the Bose-Einstein ensemble is the formation, for a growing number of bosons, of the sharp asymmetric shape on the c N −T characteristics, which is more protrusive at the smaller Robin distances. This cusp-like structure is a manifestation of the phase transition to the condensate state. For two split-off orbitals, one additional maximum emerges whose position is shifted to colder temperatures with the increase of the energy gap between these two states and their higher-lying counterparts and whose magnitude approaches a -independent value. All these physical phenomena are qualitatively and quantitatively explained by the variation of the energy spectrum by the Robin distance. Parallels with other structures are drawn and similarities and differences between them are highlighted. Generalization to higher dimensions is also provided.

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“…Since the publication of Ref. 1, problem () received attention in a series of papers, 5–11 mainly due to its novelty and the interesting properties hidden behind its apparent simplicity.…”

Section: Introductionmentioning

confidence: 99%

On Wiener's violent oscillations, Popov's curves, and Hopf's supercritical bifurcation for a scalar heat equation

Guidotti

Merino

2021

Stud Appl Math

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A parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that admits an interpretation as a thermostat device or in the context of an agent‐based price formation model for a market. The existence and the stability of periodic self‐oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation are proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. They follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only, it can be extended naturally to arbitrary Euclidean dimension and to manifolds.

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A Criterion for the Uniform Eventual Positivity of Operator Semigroups

Cited by 32 publications

References 22 publications

Quantum Information Measures of the One‐Dimensional Robin Quantum Well

Quantum Information Measures of the One‐Dimensional Robin Quantum Well

Thermodynamic Properties of the 1D Robin Quantum Well

On Wiener's violent oscillations, Popov's curves, and Hopf's supercritical bifurcation for a scalar heat equation

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