Logarithmic Threshold Return

A trap driven toward its binding threshold can lose almost all bound-state return as its cycle becomes arbitrarily slow.

One particle starts bound to a single attractive site on a two-dimensional square lattice. The attraction is weakened and restored along the same quadratic schedule. The question is how much returns to the original bound orbital after this coherent, closed cycle.

Read next	Purpose
Reading guide · Single-book route · Tutorial bridge	Learn the model and mechanism from one external book
Proof and quantifiers · Claim map	Follow the return law and its proof dependencies
Limits and accuracy · Assumptions · Prior-work comparison	Check the asymptotic domain, physical premises and attribution
Evidence and reproduction · Scope and evidence	Inspect the checks and their interpretation
LLM guide	Identify relevant questions and authoritative files

The fixed question

Measure energy in $J$ and time in $\hbar/J$. The Hamiltonian and return probability are

$$\begin{aligned} H(t)&=H_0-\left[u+(4-u)(t/T)^2\right]\lvert0\rangle\langle0\rvert, \qquad -T\le t\le T,\\\ P_\infty(T,u)&=\left|\langle b_4\rvert\mathcal U(T,-T)\lvert b_4\rangle\right|^2. \end{aligned}$$

Here $H_0$ is the nearest-neighbor square-lattice Laplacian with band $[0,8]$, $\lvert b_4\rangle$ is the normalized endpoint bound orbital, and $T$ is the half-cycle duration. The endpoint attraction is 4 for every $0\le u<4$. The measurement projects onto that orbital. Site occupation is a different observable. The probability is unconditional; population that escapes remains in continuum states.

The logarithmic return law

The leading slow-cycle law at exact touching is

$$\boxed{P_\infty(T,0)\sim\frac{\pi^2}{4\ln^2T}\longrightarrow0.}$$

With units restored, the logarithm is $\ln(T_{\rm physical}J/\hbar)$. The proof map gives the exact quantifiers, and limits and accuracy explains the remainder and finite-duration interpretation.

A trap with a shrinking positive minimum also retains the effect. Define

$$\rho_0=\frac1{4\pi},\qquad L+\tfrac12\ln L=\ln\frac{32T}{\sqrt{(4-u)\rho_0}},\qquad b=\rho_0uL.$$

For $0\le b(T)\le1-\delta$ with fixed $0<\delta\le1$, the uniform law is

$$P_\infty(T,u(T))= \frac{\pi^2}{4L^2[1-b(T)]^2}[1+o(1)].$$

A strictly positive minimum $u=o(1/\ln T)$ keeps the zero-minimum coefficient. Along the sufficient periodic-square family $n(T)=2\lceil4.5T\rceil+1$, in the same residual-depth domain, the law also holds using each lattice's own endpoint bound state. The size is sufficient, not optimal, and the expansion does not apply at $b=1$.

Limit	Return behavior
Infinite lattice, exact touching, then slow cycle	Vanishes as an inverse square logarithm
Shrinking minimum within the fixed-margin domain	Vanishes with the displayed minimum-dependent coefficient
Sufficient growing periodic squares in that same domain	Retains the logarithmic law
Fixed positive minimum or fixed finite square	Ultimately approaches adiabatic return

These are different orders of limits. Limits and accuracy explains why the joint family and fixed-gap recovery are compatible.

Why a logarithm changes return

The lower-edge local Green function and weak binding obey

$$G(\eta)=\rho_0\ln(32/\eta)+O\!\left(\eta\ln(1/\eta)\right),\qquad \eta(U)\sim32e^{-4\pi/U}.$$

The gap becomes exponentially small at weak attraction. The lower-edge density of states is finite and nonzero; the logarithm belongs to the local Green function. The middle-band van Hove singularity is a separate feature. See spectral scope.

Fourier transformation turns the quadratic rank-one drive into a scalar energy equation. Its bound-channel reflection becomes weak in the logarithmic threshold limit. Retarded reconstruction, equal channel normalization and controlled outer matching connect that reflection to the physical return probability. The local bridge explains the steps; the documentation map locates their proofs.

One tutorial, then this result

The selected learning foundation is:

Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and Their Applications in Atomic Physics, Plenum Press (1988).

Publisher record · DOI

The single-book route focuses on Chapters 1–2 and 8–10, with precise section locations, checkpoints and an access ledger covering the publisher contents and inspected previews. The tutorial bridge supplies the lattice threshold, normalization, matching and joint-limit steps locally.

Boundaries and prior work

The claim concerns one particle, one local square-lattice trap, the specified quadratic cycle and unconditional orbital return. A bath, free waiting interval or postselection changes the task. The assumption map and readout audit distinguish the ideal Hamiltonian and observable from an implemented device. Fixed positive anisotropy has a supporting scope check; its zero-hopping limit is not uniform.

Threshold-touching loss, static essential binding and the energy-domain method have predecessors. A local power-law tangent predicts the leading prefactor; the contribution is the controlled logarithmic physical limit and its joint domain. Scientific context, the Devdariani comparison and the 2014 construction comparison give the attributions. Scope and evidence connects the result to its supporting records.

Evidence and reproduction

python -m pip install -r requirements.txt
python verify.py --integrity-only
python verify.py --output-dir verification-artifacts

The five preserved scientific suites cover 28 groups and 272 finite controls. The 60 supplementary checks are counted separately. The runner writes to a fresh directory, retains observed outputs and complete differences, and preserves saved references. Verification policy distinguishes assertions, numerical agreement and exact bytes.

Purpose and contact

This repository serves as a record of the work and a guide for the author’s self-directed learning. For discussion or potential collaboration, please contact Ruge Lin at gogoko699@gmail.com.

The LLM guide gives relevant questions, search terms and an authoritative reading order. Code is available under the MIT license.