Emergent quantization from a dynamic vacuum

We show that adding quadratic temporal dispersion to a dynamic-vacuum acoustic model yields a fully analytic, exactly isospectral mapping to the hydrogenic Coulomb problem. In the regime <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>ω</a:mi> <a:mo>=</a:mo> <a:mi>D</a:mi> <a:msup> <a:mi>q</a:mi> <a:mn>2</a:mn> </a:msup> </a:mrow> </a:math> with <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>D</b:mi> <b:mo>=</b:mo> <b:mi>ℏ</b:mi> <b:mo>/</b:mo> <b:mo>(</b:mo> <b:mn>2</b:mn> <b:msub> <b:mi>m</b:mi> <b:mi>eff</b:mi> </b:msub> <b:mo>)</b:mo> </b:mrow> </b:math> , a proton-imprinted constitutive profile produces an inverse sound speed <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mn>1</c:mn> <c:mo>/</c:mo> <c:msubsup> <c:mi>c</c:mi> <c:mi>s</c:mi> <c:mn>2</c:mn> </c:msubsup> <c:mrow> <c:mo>(</c:mo> <c:mi>r</c:mi> <c:mo>)</c:mo> </c:mrow> <c:mo>=</c:mo> <c:mi>A</c:mi> <c:mrow> <c:mo>(</c:mo> <c:mi>ω</c:mi> <c:mo>)</c:mo> </c:mrow> <c:mo>+</c:mo> <c:mi>C</c:mi> <c:mrow> <c:mo>(</c:mo> <c:mi>ω</c:mi> <c:mo>)</c:mo> </c:mrow> <c:mo>/</c:mo> <c:mi>r</c:mi> </c:mrow> </c:math> and hence a time-harmonic operator <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mo>(</d:mo> <d:msup> <d:mi>∇</d:mi> <d:mn>2</d:mn> </d:msup> <d:mo>+</d:mo> <d:msubsup> <d:mi>k</d:mi> <d:mrow> <d:mi>eff</d:mi> </d:mrow> <d:mn>2</d:mn> </d:msubsup> <d:mo>)</d:mo> </d:mrow> </d:math> that is Coulombic at each bound eigenfrequency. Separation of variables yields the exact hydrogenic eigenfunctions <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:msub> <e:mi>R</e:mi> <e:mrow> <e:mi>n</e:mi> <e:mi>ℓ</e:mi> </e:mrow> </e:msub> <e:mrow> <e:mo>(</e:mo> <e:mi>r</e:mi> <e:mo>)</e:mo> </e:mrow> <e:msubsup> <e:mi>Y</e:mi> <e:mi>ℓ</e:mi> <e:mrow> <e:mspace width="0.16em"/> <e:mi>m</e:mi> </e:mrow> </e:msubsup> <e:mrow> <e:mo>(</e:mo> <e:mi>θ</e:mi> <e:mo>,</e:mo> <e:mi>ϕ</e:mi> <e:mo>)</e:mo> </e:mrow> </e:mrow> </e:math> ; the angular labels <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mo>(</g:mo> <g:mi>ℓ</g:mi> <g:mo>,</g:mo> <g:mi>m</g:mi> <g:mo>)</g:mo> </g:mrow> </g:math> emerge naturally from the Laplace-Beltrami spectrum on <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msup> <h:mi mathvariant="double-struck">S</h:mi> <h:mn>2</h:mn> </h:msup> </h:math> via rotational symmetry and boundary conditions (as in standard quantum mechanics), while localization follows from <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mi>A</j:mi> <j:mo>(</j:mo> <j:msub> <j:mi>ω</j:mi> <j:mi>n</j:mi> </j:msub> <j:mo>)</j:mo> <j:mo>&lt;</j:mo> <j:mn>0</j:mn> </j:mrow> </j:math> in a reactive stop band consistent with causal, passive dispersion. While angular-momentum quantization follows directly from rotational symmetry and boundary conditions in standard quantum mechanics (consistent with Noether's theorem), here it emerges within a classical-like dispersive acoustic framework without introducing additional wave-mechanical postulates beyond symmetry and self-adjointness. This highlights dispersion's role in bridging a hydrodynamic description to quantumlike spectral structure. Identifying <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:msub> <k:mi>q</k:mi> <k:mi>n</k:mi> </k:msub> <k:mo>≡</k:mo> <k:msub> <k:mi>κ</k:mi> <k:mi>n</k:mi> </k:msub> </k:mrow> </k:math> maps spatial scale to frequency, giving <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:msub> <l:mi>ω</l:mi> <l:mi>n</l:mi> </l:msub> <l:mo>=</l:mo> <l:mi>D</l:mi> <l:msubsup> <l:mi>κ</l:mi> <l:mi>n</l:mi> <l:mn>2</l:mn> </l:msubsup> <l:mo>∝</l:mo> <l:mn>1</l:mn> <l:mo>/</l:mo> <l:msup> <l:mi>n</l:mi> <l:mn>2</l:mn> </l:msup> </l:mrow> </l:math> and reproducing the Rydberg ladder. Calibration to the reduced-mass Rydberg frequency ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>ω</m:mi> <m:mo>*</m:mo> </m:msub> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mi>π</m:mi> <m:mi>c</m:mi> <m:msub> <m:mi>R</m:mi> <m:mi>H</m:mi> </m:msub> </m:mrow> </m:math> ) fixes <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mrow> <n:mi>D</n:mi> <n:mo>=</n:mo> <n:mi>ℏ</n:mi> <n:mo>/</n:mo> <n:mo>(</n:mo> <n:mn>2</n:mn> <n:mi>μ</n:mi> <n:mo>)</n:mo> </n:mrow> </n:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow> <o:msub> <o:mi>m</o:mi> <o:mi>eff</o:mi> </o:msub> <o:mo>=</o:mo> <o:mi>μ</o:mi> </o:mrow> </o:math> , with no free parameters. We determine the frequency dependence of <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:mi>A</p:mi> <p:mo>(<