What the one-dimensional potentials page establishes, assumes and defers

What is established. The page fixes a declared effective one-dimensional model of a spinless massive particle (The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions) and proves, as conditional consequences of P1, P2, P4, P5 and the cited mathematics, the following statements. The mathematical lemmas The one-dimensional free Schr"odinger operator as the weak second-derivative operator and Positive scalar multiples of self-adjoint operators: domains, resolvents, spectra and eigenbases identify the free operator with the weak second-derivative operator and transfer self-adjointness, resolvents, spectra and eigenfamilies along positive scaling; The one-dimensional continuity identity and the constant flux of stationary solutions proves the one-dimensional continuity identity and the constancy of the current of a stationary solution. The line model's energy observable, flow and stationary states establishes that the declared energy operator of the line model with a real bounded measurable potential is an assigned sharp observable, generates the unitary flow of the model, and that its stationary preparations are exactly the eigenrays with point-mass energy law. Interval models: discrete spectra, complete eigenbases, min-max bounds and nodal counts establishes, for the declared Dirichlet and separated Robin interval models with continuous real potentials, self-adjointness with compact resolvent, complete orthonormal eigenbases, discreteness and simplicity of the spectra, the min-max comparison of the Dirichlet energies with the free box energies and the nodal counts, together with the Robin eigenvalue condition. Finite square well: bound-state conditions, simplicity and count gives the exact root conditions, simplicity, parity and count of the bound states of the finite square well, and The zero-range (delta) model: self-adjointness and the unique attractive bound-state preparation gives the self-adjointness of the declared zero-range interface operator and its unique bound state for attractive coupling with the exact energy $-m{g}^2\mathrm{/}(2{\mathrm{\hslash }}^2)$. Rectangular barrier: scattering solution, flux balance and transmission gives the unique scattering solution of the rectangular barrier, its conserved current, the exact coefficients $T$ and $R$, the flux balance $T+R=1$, the reading of $T$ and $R$ as fractions of the reference current, and the thick-barrier error bound.

What is assumed, not derived. The page's model data are declared, not derived: the reduction of the three-dimensional massive-particle model to one spatial dimension, the position and momentum assignments of the line model, the Dirichlet and Robin boundary conditions with their boundary parameters, the zero-range interface condition with its coupling, and the scattering reading of the current ratios. Each is an assumption of the pair in the sense of the preparation and apparatus discipline P7; an identification of a concrete material system with any of them is a tested identification with uncertainty and is not produced by these items. The operators rest on the mathematical supplier Real multiplication operators and free plus bounded-potential Schrödinger operators and on the interval and contact mathematics of One-dimensional Sobolev spaces on intervals and the items of the intervals pair, whose Axiom of Choice and Countable Choice hypotheses are inherited by every consumer above.

What is not addressed. No wave-packet or time-dependent scattering limit is computed; no arrival-time, detector-response, efficiency or background model is declared; no count, rate, cross-section or measured spectrum appears. The scattering solution of the barrier item is not an element of the model state space, so no Born probability is read from it. Potentials with unbounded or singular growth, the harmonic oscillator, the Coulomb problem, higher-dimensional scattering, resonant and above-barrier behaviour, and the Airy/WKB connection problem for smooth barriers belong to other pages of the category and are not consumed or anticipated here. The example page of the pair carries exact finite checks of the barrier and well formulas; those are model computations, not measurements.

Evidence status. The page is a conditional model development. It contains no experiment, no reported observation, no uncertainty statement and no statistical inference; agreement of any of its formulas with data would be evidence about the combined model, apparatus and analysis assumptions and would not prove the model.