We consider the time evolution of a particle bound by an attractive one-dimensional delta-function potential (at x = 0) when a uniform electrostatic field (F) is applied. We explore explicit expressions for the time-dependent wavefunction ψ_F(x,t) and the ionization probability {\mathcal{P}}(t), respectively, in the weak-field limit. In doing so, ψ_F(0,t) is a key element to their evaluation. We obtain a closed expression for ψ_F(0,t) which is an excellent approximation of the exact result being a numerical solution of the Lippmann-Schwinger integral equation. The resulting probability density |ψ_F(0,t)|^2, as a simple alternative to {\mathcal{P}}(t), is also in good agreement to its counterpart from the exact one. In doing this, we also find a new and useful integral identity of the Airy function.
We consider the time evolution of a particle subjected to both a uniform electrostatic field F and a one-dimensional delta-function potential well. We derive the propagator K_F(x,t|x',0) of this system, directly leading to the wavefunction psi_F(x,t), in which its essential ingredient K_F(0,t|0,0), accounting for the ionization-recombination in the bound-continuum transition, is exactly expressed in terms of the multiple hypergeometric functions F(z_1,z_2,...,z_n). And then we obtain the ingredient K_F(0,t|0,0) in an appropriate approximation scheme, expressed in terms of the generalized hypergeometric functions p_F_q(z) being much more transparent to physically interpret and much more accessible in their numerical evaluation than the functions F(z_1,z_2,...,z_n).
Nonperturbative dynamics of quantum fields out of equilibrium is often described by the time evolution of a hierarchy of correlation functions, using approximation methods such as Hartree, large
Nf, and
nPI-effective action techniques. These truncation schemes can be implemented equally well in a classical statistical system, where results can be tested by comparison with the complete nonlinear evolution obtained by numerical methods. For a
(1+1)-dimensional scalar field we find that the early-time behavior is reproduced qualitatively by the Hartree dynamics. The inclusion of direct scattering improves this to the quantitative level. We show that the emergence of nonthermal temperature profiles at intermediate times can be understood in terms of the fixed points of the evolution equations in the Hartree approximation. The form of the profile depends explicitly on the initial ensemble. While the truncated evolution equations do not seem to be able to get away from the fixed point, the full nonlinear evolution shows thermalization with a (surprisingly) slow relaxation.
The physical--observation dual-axis (PODA) structure treats physical and observation configurations as distinct arguments of joint dynamics. We study the analytic conditions under which this structure admits rigorous path integrals and finite complex measures for sequential quantum histories. On compact smooth joint geometry, a specified covariant Hamiltonian has a Feynman--Kac representation. A uniform complex-time product estimate permits simultaneous removal of Abel damping and time slicing at arbitrary relative speeds. For fixed finite bounded-readout protocols, class operators converge strongly and the trace-class branch-pair operators converge in trace norm. Finite instruments with complete Kraus families also yield trace-norm convergence of their record states. Energy-form regularity gives quantitative error bounds. A completed cylinder-test space carries a consistent, strongly positive history functional. The stronger finite-measure question is controlled by covariant transport: calibrated rank-one readouts on a finite-dimensional fibre satisfy the sharp estimate
∥μγ∥TV≤e2cγ, where
cγ is the off-diagonal covariant variation. In the specified common-action class, conserved energy and smooth, globally labelled rank-one readouts imply this regularity when the kinetic geometry is complete and the effective potential is bounded below, or when the chosen energy confines the trajectory to a compact set. In infinite dimension, finitely many readout sectors allow unbounded internal energies and bounded transitions. The construction extends to countable outcomes when the operator of block norms is bounded on
ℓ2; nonatomic readouts admit a double-Dyson construction when an absolute integral kernel is bounded on
L2. In these Dyson constructions, pure preparations with rank-one discrete readouts or scalar continuous fibres have exact variation
∥eTBv∥22. Whole-sector and spatial-domain cutoffs converge in variation, whereas nonatomic readout quantization converges weakly and remains separated in variation norm. Mixed curvature survives in the covariant constructions, and invariant quantum embeddings recover physical-sector histories. Unrestricted basis switching, a bounded-Hamiltonian countable example and free-particle position histories establish the limits of finite-measure existence; exact physical recovery transfers the last obstruction to the joint theory. Analytic two-axis models exhibit the resulting dynamics and verify the distinct convergence statements. A proposed two-mode trapped-ion realization links the magnetic oscillator's transport phase to independent spectral and driven-response tests. Exact finite-time Gaussian evolution shows that nearly complete wave-packet return can coexist with a substantial phase correction. The protocol therefore infers the coupling from finite-time interference and predicts the other observables without refitting, with calibration and uncertainty requirements explicit. These are model predictions, not measured data. Each construction is tied to its stated geometric, dynamical and readout hypotheses.
With the identity matrix on the stand, the bare trace reports
252 and the originally normalized prepared state reports
1: the first evaluates the trace over the full matter carrier, the second the action within the fixed preparation. For the fixed
S01/dx1 external source with the co-directional readout operator, zero output momentum, double Laplace parameters
z=w=6c(1−i), the same preparation, and an incoming/observing two-ball window of radius
B, we prove that in the complete five-term response both the uncentred (raw) and the centred (connected) quadratic coefficient tensors have two negative and one positive eigenvalue. Here "curvature" means the quadratic coefficient after the non-analytic linear term of the sharp window is removed, not the Hessian of the complete response at the origin. Two branches start from the same action containing an independent dual matter field; that action, its background, matter carrier, and constants are generated from common source data with no fitted parameter. Under the
P6 grading, the closed trace of any finite admissible ordered gauge-propagation word equals the closed trace of its block-diagonal part; the prescribed one-directional scalar-insertion closed trace vanishes, and so does every open chain with an arbitrary admissible propagation word between two co-directional insertions. The prepared state, the boundary weight
K and the complete second-quantized word give different evaluations; the actual return operator produces a nonzero reading, while an intermediate preparation projection erases it. The field-equation branch passes from
289 fields through auxiliary, Ward and symmetry reductions to a
103-field system, counts matter feedback once, and constructs a rational
55-dimensional bosonic kernel together with field/source recovery maps. The matter block at the origin has rank
42; keeping six light matter directions yields a
61-dimensional second-order system whose constant term has rank
56, and its five-dimensional light kernel gives both linear and quadratic momentum time scales and is read out as propagation through the dynamical
g00 source. The two quantum source legs, the readout contact term and the two field propagation legs form the complete five-term response. The non-analytic linear term of the true ball intersection, the full complex Gram matrix, implicit root derivatives, full-direction integrals over paired frames and outward-rounded rational remainders together determine the quadratic tensor. The raw three-axis coefficients lie in
[−1.355200,−1.309214],
[2.140271,2.186257] and
[−1.742600,−1.696614] times
10−20 in the units of the original action; a uniform bound on all mixed terms makes the inertia statement apply to the complete tensor. The mean term is kept separately; its second-axis interval contains zero. The actual Fourier Hessian of the same original action is also identified with the complete 289-field matrix. Five origin directions and a 98-dimensional coordinate complement generate an exact Schur kernel for all four complex momenta, a source-generated invertibility radius and a strict cubic remainder. The actual ordered current reconstructs the denominator-cleared whole field, retaining contact and all nine null directions. For another original configuration preparation, the complete ordered five-factor word, its actual Noether contact and four Fourier keys generate a real-field half-axis response operator. A source-generated created-unit/background difference then observes the same history: within its positive source-generated nonlinear time interval, it produces the full
289×289 quantum tensor
ΠT, and the finite integral of the mother-Euler amplitude derivative is exactly
[WTH289−ΠT]a. Coincident clocks give
WT=T; the complete linearized half-axis retains its own growth condition. The same original Green returns this preparation's response to all fields, with conditions stated separately from those of the specified two-ball window.
The exact multi-resolvent hierarchy of dense many-body spectra encodes higher-order spectral-weight correlations, but closing it predictively for interacting Hamiltonians remains a challenge. We introduce the multi-resolvent self-consistency (MRSC) equations, which separate an exact algebraic hierarchy from an approximate finite-parameter closure. The exact Feshbach geometric identity organizes r-resolvent cumulants into a ladder over irreducible vertices Gamma^(r), and the projected hierarchy expresses those vertices as functionals of the cumulants. The closure works in finite parameters: a connectivity basis organizes a general Hamiltonian's seed into class amplitudes, and a causal r-fold Stieltjes representation with a moment-field identity reduce moment matching to a finite-dimensional problem closed at a predictive fixed point. On the free-fermion benchmark the tested exact identities are reproduced and the Faddeeva closure is recovered at r=1. In a blind test on the interacting Ising chain, with the exact two-resolvent data withheld, the Hamiltonian-derived closure predicts the tested probes to 0.35%-0.97%. At the observable level the exact two-resolvent sector supplies a sign-correct, energy-resolved correction to the single-resolvent baseline for the off-diagonal eigenstate-thermalization-hypothesis (ETH) variance, of order one relative to it. Closure errors organize hierarchically: a level-by-level deficit decomposition, anchor measurements of the isolatable terms, and an explicit propagation bound. The same moments further yield a diagnostic chain for the window conditions of local rotational invariance -- an exact drift-kernel identity, a proven single-state identifiability boundary, and a transferability census over couplings and realizations -- a finite-order framework, not a proof of universal emergence.