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Physical Review A

Eigenvalue spectrum of the independent-fermion kinetic-energy kernel

Daniel P. Joubert

Published Sep 1, 1996• 5 citations
Publisher

Abstract

The constrained minimization independent-fermion kinetic-energy kernel, ${\mathrm{\ensuremath{\delta}}}^{2}$${\mathit{T}}_{\mathit{s}}$[\ensuremath{\rho}]/\ensuremath{\delta}\ensuremath{\rho}(r)\ensuremath{\delta}\ensuremath{\rho}(${\mathbf{r}}^{\ensuremath{'}}$), has a zero mode for all \ensuremath{\rho}(r), while it is non-negative for \ensuremath{\rho}(r) noninteracting v representable. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.2479

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Spectral Decomposition in Non-Markovian Master Equations: No Golden Ratio or Base-60 Constants Found — E8 Intelligence Research

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FINDING: Spectral decomposition methods underpin non-Markovian master equations, with memory kernels encoding environment structure; no explicit golden ratio or base-60 constants appear in the listed sources. | MATH: Non-Markovian master equation: dρ(t)/dt = -i[H_S, ρ(t)] + ∫₀ᵗ K(t−s) ρ(s) ds, where K(t) is the memory kernel. Spectral decomposition: A = Σᵢ λᵢ Pᵢ (λᵢ eigenvalues, Pᵢ projectors). Feynman-Vernon influence functional yields exact time-dependent coefficients: γ(t) = ∫₀^∞ J(ω) coth(ℏω/2k_BT) sin(ωt)/ω dω (dissipation), Δ(t) = ∫₀^∞ J(ω) sin(ωt) dω (noise), with J(ω) = spectral density. | CONNECTION: None found in the provided sources. No 0.382, 0.618, 0.786, 1.618, 2.618, base-60, or crystallographic symmetries are mentioned. The spectral decomposition is linear-algebraic (eigenvalues/projectors), not harmonic-ratio-based. | DEPTH: 3 — The findings are standard mathematical physics (spectral theory applied to open quantum systems). The memory kernel is a convolution integral, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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The constrained minimization independent-fermion kinetic-energy kernel, {\mathrm{\ensuremath{\delta}}}^{2}$${\mathit{T}}_{\mathit{s}}[\ensuremath{\rho}]/\ensuremath{\delta}\ensuremath{\rho}(r)\ensuremath{\delta}\ensuremath{\rho}(r\ensuremath′{\mathbf{r}}^{\ensuremath{'}}r\ensuremath′), has a zero mode for all \ensuremath{\rho}(r), while it is non-negative for \ensuremath{\rho}(r) noninteracting v representable. \textcopyright{} 1996 The American Physical Society.
Abstract The quantal density functional theory (Q‐DFT) of excited states is the description of the physics of the mapping from any bound nondegenerate excited state of Schrödinger theory to that of the s‐system of noninteracting Fermions with equivalent density ρk(r), energy Ek, and ionization potential Ik. The s‐system may either be in an excited state with the same configuration as in Schrödinger theory or in a ground state with a consequently different configuration. The Q‐DFT description of the s‐system is in terms of a conservative field ℱk(r), whose electron‐interaction ℰee(r) and correlation‐kinetic 𝒵 (r) components are separately representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, and correlation‐kinetic effects, respectively. The sources of these fields are expectations of Hermitian operators taken with respect to the system wavefunction. The local electron‐interaction potential vee(r) of the s‐system, representative of all the many‐body correlations, is the work done to move an electron in the force of the field ℱk(r). The electron interaction Eee and correlation‐kinetic Tc components of the total energy Ek may be expressed in integral virial form in terms of their respective fields. The difference between the s‐system in its ground or excited state representation is due entirely to correlation‐kinetic effects. The highest occupied eigenvalue of the s‐system differential equation in either case is minus the ionization potential Ik. In this work we demonstrate the transformation of an excited state of Schrödinger theory, as represented by the first excited singlet state of the exactly solvable Hooke's atom, to that of noninteracting Fermions in their ground state with equivalent excited state density, energy, and ionization potential. To further prove the Fermions are in a ground state, we solve the corresponding singlet s‐system differential equation numerically for the vee(r) determined, and obtain the excited state density from the zero node orbitals generated. The resulting total energy is also the same. In addition, the single eigenvalue determined corresponds to minus the ionization potential of the excited state. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
FINDING: Spectral decomposition methods underpin non-Markovian master equations, with memory kernels encoding environment structure; no explicit golden ratio or base-60 constants appear in the listed sources. | MATH: Non-Markovian master equation: dρ(t)/dt = -i[H_S, ρ(t)] + ∫₀ᵗ K(t−s) ρ(s) ds, where K(t) is the memory kernel. Spectral decomposition: A = Σᵢ λᵢ Pᵢ (λᵢ eigenvalues, Pᵢ projectors). Feynman-Vernon influence functional yields exact time-dependent coefficients: γ(t) = ∫₀^∞ J(ω) coth(ℏω/2k_BT) sin(ωt)/ω dω (dissipation), Δ(t) = ∫₀^∞ J(ω) sin(ωt) dω (noise), with J(ω) = spectral density. | CONNECTION: None found in the provided sources. No 0.382, 0.618, 0.786, 1.618, 2.618, base-60, or crystallographic symmetries are mentioned. The spectral decomposition is linear-algebraic (eigenvalues/projectors), not harmonic-ratio-based. | DEPTH: 3 — The findings are standard mathematical physics (spectral theory applied to open quantum systems). The memory kernel is a convolution integral, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com