For the damped quantum harmonic oscillator at finite temperature, we study the ratio Δ_q⋆/g_F between a Lindblad-resolvent-induced quadratic form W_q⋆(A) = ⟨A · K_β⋆(A)⟩{ρ_β} (with K_β⋆ = (−ℒ⋆)^{−1} − I, ℒ⋆ the Heisenberg-picture Lindblad generator that admits the thermal state ρ_β as a stationary state) and the exponential-family Fisher variance g_F(A) = Var{ρ_β}(A), evaluated on polynomial perturbations A_p = (a + a†)^p. We establish five exact results. Theorem A (hypergeometric decomposition): the projector weight R_l onto the l-th Hermite eigenmode of −ℒ⋆ admits a closed hypergeometric form. Theorem B (closed-form factorial moments): all factorial moments admit closed product formulas. Theorem 1 (DC ratio): the resolvent-to-Fisher ratio equals −2 for every polynomial order p. Theorem 2 (off-diagonal closed form): gives the off-diagonal deviation matrix. Corollary 3: establishes negative semidefiniteness of the full deviation matrix. Proofs use only Hermite orthogonality, OU spectral decomposition, and the Bény–Osborne Wasserstein gradient flow formulation.
For the damped quantum harmonic oscillator at finite temperature, we study the ratio Δ_q⋆/g_F between a Lindblad-resolvent-induced quadratic form W_q⋆(A) = ⟨A · K_β⋆(A)⟩{ρ_β} (with K_β⋆ = (−ℒ⋆)^{−1} − I, ℒ⋆ the Heisenberg-picture Lindblad generator that admits the thermal state ρ_β as a stationary state) and the exponential-family Fisher variance g_F(A) = Var{ρ_β}(A), evaluated on polynomial perturbations A_p = (a + a†)^p. We establish five exact results. Theorem A (hypergeometric decomposition): the projector weight R_l onto the l-th Hermite eigenmode of −ℒ⋆ admits a closed hypergeometric form. Theorem B (closed-form factorial moments): all factorial moments admit closed product formulas. Theorem 1 (DC ratio): the resolvent-to-Fisher ratio equals −2 for every polynomial order p. Theorem 2 (off-diagonal closed form): gives the off-diagonal deviation matrix. Corollary 3: establishes negative semidefiniteness of the full deviation matrix. Proofs use only Hermite orthogonality, OU spectral decomposition, and the Bény–Osborne Wasserstein gradient flow formulation.
For the d-mode damped quantum harmonic oscillator at finite temperature, we ask whether the Lindblad-resolvent ratio Δ_q*/g_F derived for the 1D case (Dai 2026, DC paper / paper #52; with p → ∞ limit −2) generalises to a dimension-dependent limit −2d, as a naive "phase-space dimension" intuition might suggest. We establish that the limit is exactly −2, independent of d, for the collective polynomial perturbation A_p = (Σ_i (a_i + a_i†))^p. The argument rests on a collective-coordinate reduction theorem: under the symmetric d-mode thermal Lindblad (same n̄, γ, ω across modes), A_p depends only on the single bosonic mode b = (1/√d) Σ_i a_i, which is itself in a 1D thermal state at the same β under the original Lindblad. The d-mode problem reduces exactly to the 1D problem with rescaling A_p ↦ d^{p/2}(b+b†)^p. The dimensionless ratio Δ_q*/g_F is therefore strictly invariant under d. We further show: (i) For the asymmetric sum-of-powers observable B_p = Σ_i (a_i + a_i†)^p, the finite-p ratio has a closed-form d-correction DC(B_p) = DC_{1D}(p) − (d−1)⟨x^p⟩²_{1D}/g_{F,1D}(p), but the correction decays geometrically, so the limit is still −2. (ii) For anisotropic temperatures β_i ≠ β_j, the symmetry is broken at every p, but numerical evidence (d=2, p up to 12) is consistent with the limit again being −2, suggesting the −2 ceiling reflects a single-mode property of the thermal Lindblad eigenstructure, not phase-space geometry. We discuss why the "phase-space dim = 2d" intuition fails, and propose this as a precise mathematical instantiation of emergence: the d-mode system, in the collective-observable sector, behaves identically to a 1-mode system at the same β.
For the d-mode damped quantum harmonic oscillator at finite temperature, we ask whether the Lindblad-resolvent ratio derived for the 1D case (Dai 2026) generalises to a dimension-dependent limit, as a naive "phase-space dimension" intuition might suggest. We establish that the limit is exactly -2, independent of d, for the collective polynomial perturbation. The argument rests on a collective-coordinate reduction theorem: under the symmetric d-mode thermal Lindblad (same γ, n̄, β across modes), the ratio depends only on the single bosonic mode, which is itself in a 1D thermal state at the same β under the original Lindblad. The d-mode problem reduces exactly to the 1D problem with rescaling. The dimensionless ratio is therefore strictly invariant under d. We further show: (i) For the asymmetric sum-of-powers observable, the finite-p ratio has a closed-form d-correction, but the correction decays geometrically (factor ~1/4 per +2 in p), so the limit is still -2. (ii) For anisotropic temperatures, the symmetry is broken at every p, but our numerical evidence (d=2, p up to 12) is consistent with, but far from establishing, the same -2 limit. We discuss why the "phase-space dim = 2d" intuition fails, and give this as a mathematically explicit example of dimensional reduction associated with the collective-observable sector.
For a phase-insensitive thermal Lindblad oscillator, the Heisenberg generator restrictedto polynomials in one rotating-frame quadrature is exactly an Ornstein–Uhlenbeck generator. We use this restriction to determine the integrated autocorrelation time of Ap = Xpfor every degree p. The result is a parity-unified finite rational sum, independent of temperature. We prove the controlled large-degree expansion γτp = 4/p + 2/p2 + 8/p3 + 36/p4 +O(p−5) and show that replacing τp by the inverse spectral gap overestimates it by a factorp/2 − 1/4 − 7/(8p) + O(p−2). For even powers, the same Hermite spectral average equalsthe ratio of the Wasserstein-2 and Fisher pullback metrics along Gaussian Gibbs deformations: γτ2n = 2gWnn/gFnn. The normalized shortfall from the gap bound is therefore an exactFisher–Wasserstein defect. We express its normalization-cleared sequence as a terminatinghypergeometric sum, prove factorial leading growth, and derive an order-three recurrencewith polynomial coefficients and a creative-telescoping certificate. The result provides anexactly solvable hierarchy in which a fixed generator has observable-dependent relaxationtimes parametrically smaller than its inverse gap, while the geometric and dynamical descriptions remain identical term by term.
Abstract The three canonical families of the hypergeometric orthogonal polynomials in a continuous real variable (Hermite, Laguerre, and Jacobi) control the physical wavefunctions of the bound stationary states of a great number of quantum systems [Correction added after first online publication on 21 December, 2022. The sentence has been modified.]. The algebraic ‐norms of these polynomials describe many chemical, physical, and information theoretical properties of these systems, such as, for example, the kinetic and Weizsäcker energies, the position and momentum expectation values, the Rényi and Shannon entropies and the Cramér‐Rao, the Fisher‐Shannon and LMC measures of complexity. In this work, we examine review and solve theq‐asymptotics and the parameter asymptotics (i.e., when the weight function's parameter tends towards infinity) of the unweighted and weighted ‐norms for these orthogonal polynomials. This study has been motivated by the application of these algebraic norms to the energetic, entropic, and complexity‐like properties of the highly excited Rydberg and high‐dimensional pseudo‐classical states of harmonic (oscillator‐like) and Coulomb (hydrogenic) systems, and other quantum systems subject to central potentials of anharmonic type (such as, e.g., some molecular systems) [Correction added after first online publication on 21 December, 2022. Oscillatorlike has been changed to oscillator‐like.].
Abstract After replacing the reduced density matrix of a quantum harmonic oscillator linearly coupled to a thermal bath by its ‘characteristic function’ (a non-negative definite function defined on the Heisenberg-Weyl group), the master equation of the reduced density matrix can be transformed into a first-order linear partial differential equation of the characteristic function. We derive stationary solutions for the Caldeira–Leggett (CL) master equation as well as for the late-time Hu–Paz–Zhang master equation, and examine the positivity problem for these solutions. In order to illustrate to what extent classical properties of quantum states of the damping oscillator emerge dynamically due to coupling to the environment, we discuss the problem of how and under what conditions a steady state predicted by the CL master equation could be expressed as a statistical mixture of Gaussian packets having a uniform position variance. Similar studies are carried out for the solutions of a Lindblad-form generalization of the CL master equation, where the positivity problem is certainly absent. Our results show that in the low-temperature regime the steady states predicted by the Lindblad-type master equation may depend on the damping strength of the bath and there are cases which may not show Gaussian-packet decomposition.
We study the spectral excitation functional D(C ∥χ) := −R ∞0 C(ω)log Fβ(ω) dω,where Fβ(ω) = (1 − e−βω)/(βω) is the Bogoliubov–Kubo–Mori (BKM) thermalweight and C is a positive-frequency spectral density. Three new results, allproved rigorously and verified numerically, are:(i) Exact closed-form evolution. Under KMS Lindblad dynamics, the fulltime trajectory of D is D(Ct ∥ χt) = Deq +Pk(nk(0) − n¯k) w(ωk) e−Γkt,where w(ω) = − log Fβ(ω) and Γk = γk(2n¯k + 1). This is not merely amonotonicity statement; it gives the trajectory in closed form.(ii) Bernstein characterization. The function t 7→ D(Ct ∥ χt) − Deq iscompletely monotone if and only if every mode is initially over-occupiedrelative to thermal equilibrium. This links the conditional monotonicityof D to the Bernstein–Widder theorem for Laplace transforms of positivemeasures.(iii) Spectral H-theorem. The weighted L1Pimbalance functional F(t) :=k|nk(t) − n¯k| w(ωk) is universally non-increasing under KMS Lindbladdynamics, decays exponentially with rate Γmin, and satisfies |D(Ct ∥χt) −Deq| ≤ F(t) for all t ≥ 0. This resolves the open question of whethera universally monotone spectral functional exists and characterises theprecise sense in which D can violate monotonicity.We also establish self-containedly: the Todd identification F−1β = td(βω), theBose–Einstein decomposition of D, the Laplace harmonic series, the completePetz information-loss chain with energy-independent bounds, the conditionalmonotonicity theorem in sharp form with an explicit two-mode counterexample,and the large-N limiting density (the folded semicircle on [0, 2√2] with meanenergy 8√2/(3π)). All results are verified numerically to machine precision.