Spectral counts in a responsive cavity

Summary: A gapped spatial band retains its cumulative count while its projector and coupling change; a reciprocal quasiperiodic phase gives a complete spatial response and nonlinear field evolution that test a low-frequency inertia reduction.

What the integrated count protects

For a discrete spatial operator $H$, define the cumulative count $\mathcal N(E)=\operatorname{Tr}\mathbf1_{(-\infty,E)}(H)$. In a finite cavity this is a count of modes. An infinite-volume integrated density of states instead takes an appropriate volume-normalized limit. The two normalizations must be stated before using an IDOS as a physical factor.

For $d$ independent oscillator directions, subtract the common zero-point energy and measure frequency in oscillator units. The cumulative band through degree $N$ has

$$\begin{aligned} D_N&=\sum_{k=0}^N\binom{k+d-1}{d-1}\\ &=\binom{N+d}{d}. \end{aligned} \tag{1}$$

Now add a bounded Hermitian spatial potential $V$. The variational eigenvalue bounds give $|\epsilon_j(H_0+V)-\epsilon_j(H_0)|\le\|V\|$. If $\|V\|<1/2$, the cutoff $N+1/2$ still encloses exactly $D_N$ states. Individual frequencies and spatial shapes may change inside that band.

For a smooth family with the cutoff in a gap, its spectral projector is

$$P=\frac1{2\pi i}\oint (z-H)^{-1}dz. \tag{2}$$

Differentiation, with $R(z)=(z-H)^{-1}$, gives

$$\dot P=\frac1{2\pi i}\oint R(z)\dot H R(z)dz. \tag{3}$$

In an eigenbasis, let $p_a$ be one inside the band and zero outside. Only the mixed blocks contribute:

$$(\dot P)_{ab} =\frac{p_a-p_b}{\epsilon_a-\epsilon_b}(\dot H)_{ab}. \tag{4}$$

Thus $\operatorname{Tr}\dot P=0$ and $P\dot P P=0$, while a physical probe $A$ may have $\operatorname{Tr}(A\dot P)\ne0$. A fixed count does not imply a fixed local contact or coupling.

The moving-cavity sum rule provides a particular kinetic law whose occupied translation inertia is proportional to that count. Under its stated assumptions, gap protection then protects that coefficient. Counting alone does not establish the mass functional needed to make the same inference for IDWT.

An actual quasiperiodically perturbed spatial band

Use a two-dimensional harmonic oscillator and a potential bounded by $\|V\|\le0.3$:

$$V(x)=0.1\sum_{j=1}^3\cos(k_j\cdot x+\phi_j), \tag{5}$$

with wavevectors

$$\begin{aligned} k_1&=(1,\sqrt2)/2,\\ k_2&=(\sqrt3,1)/2,\\ k_3&=(\sqrt5,\sqrt7)/2. \end{aligned} \tag{6}$$

The comparison has fifteen states through degree four for every choice of the three phases. The two phase choices $(0,0.7,1.3)$ and $(0.4,1.1,-0.2)$ illustrate the remaining freedom:

The same full projector derivative has norm $0.17415824$ and zero trace to numerical precision. An independent centered difference agrees with (4) to $4.61\times10^{-10}$ in matrix norm. The spatial cutoffs 24 and 32 agree in the reported energies and probe values to below $2\times10^{-7}$. These coordinates and wavevectors describe ordinary spatial waves, rather than auxiliary physical dimensions.

A lattice quasicrystal comparison

For a separate one-dimensional control, take an 89-site periodic rational approximant to the quasiperiodic operator

$$\begin{aligned} (H\psi)_j&=\psi_{j-1}+\psi_{j+1}+v_j\psi_j,\\ v_j&=1.4\cos(2\pi\alpha j+\phi). \end{aligned} \tag{7}$$

with $\alpha=55/89$. A fixed cutoff in the selected gap encloses 34 states, giving per-site IDOS $34/89$. Moving $\phi$ changes the local projector. The exact phase shift $\phi\mapsto\phi+2\pi\alpha$ translates the matrix by one site and leaves its spectrum unchanged, while the weight at a fixed site changes from $0.11477179$ to $0.68962240$.

The integers 89, 55 and 34 specify this numerical comparison; they are not particle mode assignments. It is a finite approximant, rather than a proof about every infinite quasicrystal. The relation between Harper and Fibonacci quasiperiodic models has a useful mathematical treatment in Kraus and Zilberberg. The phase labels a spatial potential; it is not an extra spatial coordinate.

Let one spatial phase respond to the wave

A further inverse control makes the first phase in (5) reciprocal. Restore the oscillator zero-point energy and use $\hbar=1$ and unit oscillator frequency in this control. Write

$$\begin{aligned} H(\theta)={}&H_{\rm rest}\\ &+\kappa(C\cos\theta-S\sin\theta). \end{aligned} \tag{8}$$

where $C=\cos(k_1\cdot x)$, $S=\sin(k_1\cdot x)$, $\kappa=0.1$, and $H_{\rm rest}$ contains the harmonic confinement and the two remaining fixed phase terms. Here $\theta$ is a parameter of an actual spatial profile. It has no independent particle interpretation.

Let $\Phi$ be a frame of fifteen orthogonal complex occupied modes, with $\Phi^\dagger\Phi=I$, and put $\rho=\Phi\Phi^\dagger$. This occupation is a stated preparation. For the first-order canonical action $i\operatorname{Tr}(\Phi^\dagger\dot\Phi)-E$, take

$$\begin{aligned} E(\rho,\theta)&=\operatorname{Tr}(\rho H_{\rm rest})\\ &\quad+\kappa(c\cos\theta-s\sin\theta),\\ c&=\operatorname{Tr}(\rho C),\\ s&=\operatorname{Tr}(\rho S). \end{aligned} \tag{9}$$

Both the field and phase vary in this one energy. With no bare phase kinetic term, the phase equation is algebraic. Its minimizing branch, for $r=\sqrt{c^2+s^2}>0$, has

$$\cos\theta=-\frac c r,\qquad \sin\theta=\frac s r. \tag{10}$$

Eliminating it gives the exact energy and field derivative

$$\begin{aligned} E_{\rm red}(\rho)&=\operatorname{Tr}(\rho H_{\rm rest})-\kappa r,\\ H_{\rm red}(\rho)&=H_{\rm rest}-\kappa\frac{cC+sS}{r},\\ i\dot\Phi&=H_{\rm red}(\rho)\Phi. \end{aligned} \tag{11}$$

This derives the phase adjustment from the declared action. No rule steers the phase toward compatibility independently of the field variation. The resulting global moment dependence is instantaneous. A causal reduction from local master dynamics has not been supplied, and the harmonic confinement remains prescribed. Thus (11) is an inverse comparison, not a completed same-field particle model.

The complete response and its low-frequency limit

At a stationary lowest-band projector $P$, define $V=\partial_\theta H$, $W=\partial_\theta^2H$ and $\Delta_{ba}=\epsilon_b-\epsilon_a>0$, for $a$ occupied and $b$ complementary. The direct curvature and relaxed curvature are

$$\begin{aligned} K_{\rm dir}&=\operatorname{Tr}(PW),\\ K_0&=K_{\rm dir} -2\sum_{a,b}\frac{|V_{ba}|^2}{\Delta_{ba}}. \end{aligned} \tag{12}$$

The second expression follows either by differentiating the occupied energy or by varying the complete field and phase together. Since all complementary gaps are positive, $K_0>0$ makes the quadratic energy positive on this occupied-subspace orbit, after its irrelevant frame rotations are removed. This establishes linear stability of this specified stationary preparation.

Retaining both conjugate frequency branches gives the complete scalar phase-response kernel

$$K(\Omega)=K_{\rm dir} -2\sum_{a,b}\frac{\Delta_{ba}|V_{ba}|^2} {\Delta_{ba}^2-\Omega^2}. \tag{13}$$

The coupled frequencies are its zeros, together with any modes that decouple from the phase. At low frequency,

$$\begin{aligned} K(\Omega)&=K_0-\mathcal I_\theta\Omega^2+O(\Omega^4),\\ \mathcal I_\theta&=2\sum_{a,b}\frac{|V_{ba}|^2}{\Delta_{ba}^3}. \end{aligned} \tag{14}$$

This is a derived phase-inertia coefficient in the comparison. The expansion is controlled below the first coupled transition gap. It cannot be used to predict a frequency beyond that range merely by setting $\Omega^2=K_0/\mathcal I_\theta$.

The spatially refined stationary phase is $3.1109292764$, with $K_0=0.6768828640$ and $\mathcal I_\theta=0.02718833687$. The low-frequency estimate is $4.98959664$, but the first transition gap is only $0.92383914$. The full kernel has a first coupled frequency $0.92383905$. Among the numerically resolved modes below 2.5, the mode with the largest phase amplitude has frequency $0.9878377084$. That explicitly stated excitation is used for the time-domain control; it is not a proposed particle selection rule.

Complete nonlinear spatial evolution

For a coupled frequency $\Omega$, set the positive-frequency phase amplitude initially to one. The corresponding complete density variation has mixed eigenbasis entries

$$\begin{aligned} (\delta\rho_+)_{ba}&=\frac{V_{ba}}{\Omega-\Delta_{ba}},\\ (\delta\rho_+)_{ab}&=-\frac{V_{ab}}{\Omega+\Delta_{ba}}. \end{aligned} \tag{15}$$

It satisfies both the complete linearized field equation and the phase constraint. The program normalizes this whole variation and constructs the initial occupied frame by a unitary exponential, rather than dropping components or fitting a field amplitude to an eigenvalue.

Direct integration of (11) retains all complex spatial coefficients. For perturbation sizes $0.002$ and $0.001$, the errors in the linear phase prediction are $8.29\times10^{-9}$ and $2.02\times10^{-9}$. Their approximately fourfold decrease resolves the first nonlinear correction. The full energies are conserved to about $10^{-13}$ and the canonical Gram matrices to about $10^{-15}$ over two periods.

Time refinement changes the complete fields by $5.40\times10^{-12}$. Increasing the spatial cutoff from 32 to 40 changes the full trajectories by $2.32\times10^{-8}$, after one fixed initial unitary alignment of the occupied spectator frame. The response equation evaluated in a larger spatial complement has residual $2.53\times10^{-8}$ at cutoff 40. These checks retain the changing phase and the complete spatial fields.

The bounded potential keeps the instantaneous spectral count at fifteen. Unitarity separately preserves the rank of the occupied density matrix. During motion that density need not equal the instantaneous spectral projector. Equating them would discard the response being calculated.

Connection to cavity mode selection

Laser mathematics supplies examples of nonlinear, self-consistent mode selection in spatial cavities. Ge, Chong and Stone develop such equations, including gain saturation and changes in which modes lase. Their gain and threshold conditions are part of the stated laser system. Using that mathematics for IDWT would require the relevant energy transport and confinement to arise from its same field; laser mode selection does not itself give the particle mass law.

The useful connection here is precise. The cumulative count can survive spatial changes, while the actual projector, local compatibility and reciprocal dynamics change. The frequency-resolved cavity response calculates the corresponding spatial and transition weights. The common-size balance states what additional energy and kinetic scaling is needed before this count becomes the proposed mass spectrum.

spectral_count_checks.py verifies the projector, bounded perturbation and lattice controls, and records the fixed benchmark frequency units. quasiperiodic_phase_checks.py verifies the reciprocal phase action, complete spectral response and refined nonlinear complex spatial trajectories.

Harmonic Phase Persistence and Reciprocal Transfer

The cavity phase-lock construction distinguishes an invariant spatial-band count from temporal comb occupation. Positive candidate modal energies admit locked sawtooth or adjacent-band branches and reciprocal difference-frequency transfer. A quartic endpoint control shows why a spatially retained band matters; a separate eight-wave action preserves support but cannot populate absent harmonics. Neither temporal action derives the simplex index or the cavity confinement.