# Radial Coulomb foundation: operator, exact spectrum, and limits

Fixed Grid String Lab 0.2.0 / Theory of Reality 1.77.

This module supplies a concrete bound-state Hamiltonian for a deliberately
limited question: does the implemented radial finite-difference Coulomb model
reproduce its exact spectrum and its continuum limit? It supplies an internal
mathematical and numerical gate. It does not establish a physical spatial
lattice, a fundamental time tick, a hidden dimension, or a new hydrogen law.

## Frozen conventions

Use nonrelativistic atomic units, with unit mass, unit Coulomb coupling, and
unit reduced Planck constant. The continuum radial S-state operator is

\[
h=-\frac12\frac{d^2}{dr^2}-\frac1r,\qquad r>0,\qquad u(0)=0.
\]

Here \(u=rR\) is the reduced radial amplitude; its continuum norm is
\(\int_0^\infty |u(r)|^2dr=1\). The S-state angular factor is
\(Y_{00}=1/\sqrt{4\pi}\). Radial coordinates alone do not provide a cubic
three-dimensional grid, angular dynamics, or a particle trajectory.

The frozen numerical choices are:

| Quantity | Choice |
|---|---|
| Radial spacing \(a\) | \(0.4,0.2,0.1,0.05\) |
| Primary spacing | \(a=0.2\) |
| Outer Dirichlet boundary \(R\) | \(80,160\) |
| Retained energy levels | \(n=1,2,3,4\) |
| Tick \(\tau\) | \(0.1\) atomic time units |
| Primary dimensionless diagnostic | \((E_3-E_1)/(E_2-E_1)\) |
| Spectrum gate at \(R=160\) | Absolute finite-matrix/exact difference \(\le10^{-8}\) atomic energy units |
| Empirical fitted parameters | None |

These are benchmark choices, not estimates of natural constants. The smaller
box is a domain-error diagnostic and is not silently substituted for the
larger box if an error is inconvenient. This document defines the mathematics;
the numerical report supplies the measured computational errors and decisions.

For an eventual finite-proton-mass interpretation, define the reduced mass
\(\mu\), length \(b_\mu=4\pi\epsilon_0\hbar^2/(\mu e^2)\), energy
\(\mathcal E_\mu=\hbar^2/(\mu b_\mu^2)\), and time
\(t_\mu=\hbar/\mathcal E_\mu\). Then dimensional spacings would be
\(\Delta r=ab_\mu\) and \(\Delta t=\tau t_\mu\). No numerical SI conversion
or spectroscopy comparison is needed for the present dimensionless gate.

## 1. Declared discrete Hamiltonian

On \(\ell^2(\mathbb N)\), with \(u_0=0\), declare

\[
(H_a u)_j=-\frac{u_{j+1}-2u_j+u_{j-1}}{2a^2}
             -\frac{u_j}{ja},\qquad j\ge1.
\tag{1}
\]

Thus the diagonal is \(1/a^2-1/(ja)\), and each nearest-neighbor
off-diagonal is \(-1/(2a^2)\). The sequence norm is
\(\sum_{j\ge1}|u_j|^2=1\). Under continuum comparison,
\(u_j\simeq\sqrt a\,u(ja)\); omitting this square-root factor would mix two
different normalization conventions.

At fixed positive \(a\), the kinetic operator \(K_a\) obeys

\[
\langle u,K_a u\rangle
=\frac1{2a^2}\sum_{j=0}^{\infty}|u_{j+1}-u_j|^2,
\qquad 0\le K_a\le\frac2{a^2}I.
\tag{2}
\]

The real diagonal potential lies between \(-1/a\) and zero. Consequently
\(H_a\) is bounded and self-adjoint on all of \(\ell^2\), with
\(-I/a\le H_a\le2I/a^2\). There is no undefined potential evaluation at
the origin: the origin is a Dirichlet boundary, not a dynamical site.

The Coulomb coupling is imported from ordinary nonrelativistic
electrostatics. It is not derived from the older four-component string rule.
No transverse electromagnetic field, photon sector, or general local gauge
coupling is implemented here. Thus Appendix AR's complete electromagnetic
physical gate remains open even if this smaller bound-state gate passes.

## 2. Exact negative-energy quantization

For a negative eigenvalue \(E\), introduce \(\kappa>0\) by

\[
\cosh\kappa=1-a^2E.
\]

Equation (1) becomes

\[
u_{j+1}+u_{j-1}
=\left(2\cosh\kappa-\frac{2a}{j}\right)u_j.
\tag{3}
\]

A nonzero solution has \(u_1\ne0\), because \(u_0=u_1=0\) would force
every subsequent component to vanish. Let

\[
G(z)=\sum_{j=1}^{\infty}u_jz^j=zF(z),\qquad F(0)=u_1.
\]

Multiply (3) by \(jz^j\) and sum. Writing \(c=\cosh\kappa\), the result is

\[
(1-2cz+z^2)F'(z)+2(z+a-c)F(z)=0.
\tag{4}
\]

The solution analytic near zero is

\[
G(z)=u_1z
(1-e^\kappa z)^{\nu-1}
(1-e^{-\kappa}z)^{-\nu-1},
\qquad \nu=\frac a{\sinh\kappa}>0.
\tag{5}
\]

For an \(\ell^2\) sequence, Cauchy--Schwarz ensures that \(G\) is analytic
throughout \(|z|<1\). The point \(z=e^{-\kappa}\) lies inside that disk.
The second factor in (5) is nonzero and analytic there, so the singularity
of the first factor is removable exactly when \(\nu-1\) is a nonnegative
integer. Therefore

\[
\nu=n\in\{1,2,3,\ldots\},\qquad
\sinh\kappa_n=\frac an.
\tag{6}
\]

Conversely, for each such integer the numerator is a polynomial, and the
remaining singularity is at \(z=e^{\kappa_n}>1\). The coefficients then
decay exponentially times a polynomial and are square summable. These
conditions are both necessary and sufficient for negative-energy bound
states of (1).

The exact negative eigenvalues are consequently

\[
\boxed{E_n(a)=\frac{1-\sqrt{1+(a/n)^2}}{a^2}
=-\frac1{n^2\left(1+\sqrt{1+(a/n)^2}\right)}}.
\tag{7}
\]

The rationalized expression avoids cancellation as \(a\) becomes small.
These are energies of the infinite half-line operator, not exact eigenvalues
of a finite box. They are ordered \(E_1<E_2<\cdots<0\), approaching zero.

For explicit excited-state construction, put
\(q_n=e^{-\kappa_n}=\sqrt{1+(a/n)^2}-a/n\). Equation (5) reads

\[
G_n(z)=u_1z\frac{(1-z/q_n)^{n-1}}{(1-q_nz)^{n+1}}.
\tag{8}
\]

Its coefficient of \(z^j\) is a finite binomial sum:

\[
\frac{u_j}{u_1}=q_n^{j-1}
\sum_{k=0}^{\min(n-1,j-1)}(-1)^k
\binom{n-1}{k}\binom{n+j-k-1}{n}q_n^{-2k}.
\tag{9}
\]

This gives an eigenvector up to normalization. It is also a standard
Meixner generating function: set \(x=n-1\), \(\beta=2\), \(c=q_n^2\),
and replace the generating variable by \(q_nz\) in
[NIST DLMF, equation 18.23.4](https://dlmf.nist.gov/18.23.E4).
This identification supplies mathematical context; equation (3) and the
analyticity argument above establish the result in this model's conventions.

## 3. Exactly normalized ground state

For \(n=1\), set

\[
\kappa=\operatorname{arsinh}a,\qquad q=e^{-\kappa}.
\]

Equation (8) gives \(u_j=Cjq^{j-1}\). The identity

\[
\sum_{j=1}^{\infty}j^2q^{2(j-1)}
=\frac{1+q^2}{(1-q^2)^3}
\]

therefore yields

\[
\boxed{u_j^{(1)}=
\sqrt{\frac{(1-q^2)^3}{1+q^2}}\,j q^{j-1}
=\frac{2a^{3/2}}{(1+a^2)^{1/4}}j e^{-j\operatorname{arsinh}a}}.
\tag{10}
\]

It is strictly positive, has \(u_0=0\), and has exactly unit infinite-sequence
norm. Direct substitution verifies (3). As \(a\to0\), with \(r=ja\) fixed,
\(u_j^{(1)}/\sqrt a\to2re^{-r}\), the normalized continuum reduced radial
ground state. Restricting (10) to a finite box removes a nonzero tail; its
restricted vector must be renormalized before a finite-matrix overlap test.

## 4. Separating box error from spacing error

For the finite problem, require \(R/a=M\) to be an integer, retain sites
\(j=1,\ldots,M-1\), and impose \(u_0=u_M=0\). The matrix size is therefore
\(M-1\), and the outer wall is exactly at \(R\). A matrix of size \(M\)
with the same stencil would instead place the wall at \(R+a\).

At fixed \(a\), the finite matrices are Dirichlet restrictions of (1).
The min--max principle gives, for each fixed ordered level,

\[
E_n^{(R)}(a)\ge E_n(a),\qquad
E_n^{(R)}(a)\downarrow E_n(a)\quad(R\to\infty).
\tag{11}
\]

The first comparison is the box error
\(E_n^{(R)}(a)-E_n(a)\). It must not be counted as a spatial-lattice
correction. Once this error is controlled, expansion of (7) gives the exact
spacing dependence

\[
E_n(a)=-\frac1{2n^2}+\frac{a^2}{8n^4}
        -\frac{a^4}{16n^6}+O(a^6).
\tag{12}
\]

Thus the leading continuum energy error is positive and quadratic. Halving
the spacing gives an asymptotic factor of four reduction, rather than an
identity at finite spacing. The larger excited states need a larger box;
their generating functions decay with
\(\kappa_n=\operatorname{arsinh}(a/n)\), multiplied by a polynomial.

The frozen spectral diagnostic is

\[
D(a)=\frac{E_3(a)-E_1(a)}{E_2(a)-E_1(a)},\qquad
D(0)=\frac{32}{27},
\]

\[
D(a)=\frac{32}{27}\left[1+\frac{5a^2}{144}+O(a^4)\right].
\tag{13}
\]

Its scale independence makes it a useful internal spectrum check. A gap is
not by itself an emission-line prediction: an interaction and its matrix
elements determine which transitions are observable. In particular, all
states retained here are S states; ordinary one-photon electric-dipole
S-to-S matrix elements vanish by parity. This module does not calculate
two-photon rates or a measured optical spectrum.

## 5. Exact tick evolution and its non-identification result

For the self-adjoint Hamiltonian, or its finite-matrix restriction, define

\[
U_\tau=e^{-i\tau H_a},\qquad u^{(m)}=U_\tau^m u^{(0)}.
\tag{14}
\]

The same generator at every tick gives

\[
U_\tau^\dagger U_\tau=I,\qquad
U_\tau^{-1}=U_{-\tau},\qquad
\boxed{U_\tau^m=e^{-im\tau H_a}}.
\tag{15}
\]

Norm and energy expectation are conserved, and exact reversal is obtained
by reversing the tick sign. These statements are exact mathematical
identities; their numerical residuals measure implementation error.

Equation (15) is also an exact counterexample to time-tick identification.
The continuous-time solution \(u(t)=e^{-itH_a}u(0)\), sampled at
\(t=m\tau\), gives the identical state at every recorded tick. All
observables of those states agree. Consequently this update cannot identify
whether time itself is discrete, regardless of how accurately it is run.
The exponential of the nearest-neighbor Hamiltonian is generally dense; it
does not impose a one-site-per-tick propagation rule.

There is a second non-uniqueness even when \(\tau\) is fixed. In a finite
eigenbasis write \(H=\sum_s E_sP_s\). For arbitrary integers \(k_s\),

\[
H'=\sum_s\left(E_s+\frac{2\pi k_s}{\tau}\right)P_s
\quad\Longrightarrow\quad
e^{-i\tau H'}=e^{-i\tau H}.
\tag{16}
\]

Choosing unequal \(k_s\) changes relative energies and intermediate-time
evolution but leaves every integer-tick state identical. An energy-window
assumption or additional dynamical information can restrict these branches.
The alternative \(H'\) need not retain the Coulomb potential or
nearest-neighbor form; the conclusion concerns what tick observations alone
identify, not equivalence inside an already fixed Hamiltonian class.

## 6. Attribution and physical boundary

Finite-difference hydrogen models predate this work. A primary example is
Frederick T. Wall, [*Discrete wave mechanics: The hydrogen atom*, PNAS
83(16), 5753--5755 (1986)](https://www.pnas.org/doi/10.1073/pnas.83.16.5753),
DOI 10.1073/pnas.83.16.5753. Its abstract explicitly discusses a
finite-difference treatment restricted to spherically symmetric states.
That is prior art, not experimental support for a fundamental lattice.
The self-contained derivation above fixes this implementation's Hamiltonian
and energy convention; it does not adopt the article's additional
relativistic interpretation or assert priority for the spectrum.

The operator imports a nonrelativistic S-state reduction, an electrostatic
Coulomb potential, complex amplitudes, and the usual quantum observation
rule. It is distinct from the earlier integer-valued string's state space
and update. Its relationship to that string has not been derived.

Relativistic structure, radiative/Lamb corrections, higher recoil terms,
finite nuclear size, hyperfine structure, and preparation-dependent line
effects are absent. No difference between (7) and measured hydrogen is
interpreted as a new interaction. The \(a\)-dependence in this release is a
controlled discretization effect. Treating \(a\) as a physical parameter
would require a separate declared theory, ordinary precision corrections
with uncertainties, and a frozen external comparison.

Successful numerical gates establish that the declared operator was solved
correctly and approaches the stated continuum model. They do not supply
evidence that nature uses this radial grid or this time sampling.