Cyclic resonance spectrum

Several resonances from a finite coupling operator

For $N\ge3$ field amplitudes $a_j$ with $j$ taken modulo $N$, specify

$$i\dot a_j=\alpha a_j-\beta(a_{j-1}+a_{j+1}),\qquad \beta>0.$$

Here $\alpha,\beta$ have frequency units. A common sufficiently large $\alpha$ makes all frequencies positive if required by the application. The Fourier vectors $u_k(j)=e^{2\pi ikj/N}/\sqrt N$ diagonalize the operator, since each shifted component multiplies by $e^{\pm2\pi ik/N}$:

$$\omega_k=\alpha-2\beta\cos\frac{2\pi k}{N},\qquad k=0,\ldots,N-1.$$

Norm and the Hermitian quadratic energy are conserved. These resonances arise from the couplings themselves. They need not be direct harmonics of a common frequency. The amplitudes describe field components on the one physical manifold.

Degeneracy and a conditional filling rule

The modes $k$ and $N-k$ have equal frequency. The bottom mode $k=0$ is a singleton; for even $N$, the top mode $k=N/2$ is another singleton. With an independently specified twofold spin multiplicity and fermion occupation, filling the bottom singleton and $q$ complete pairs uses $2+4q$ particles. The rule applies before the top singleton is reached; filling that last singleton changes the count by two. Cyclic symmetry is $C_N$; the pair labels are not three-dimensional $SO(3)$ shells.

In units $\hbar=1$, examples at half filling are:

The four-ring has a partly occupied degenerate middle pair in the noninteracting description. These energies concern the specified model; chemical stability also depends on geometry, interactions, and occupation.

Alternating coupling in the four-ring

Set alternating edges to $t_1=\beta+\delta$, $t_2=\beta-\delta$, $|\delta|<\beta$. Separating the two sublattices gives off-diagonal block $-\left(\begin{smallmatrix}t_1&t_2\\t_2&t_1\end{smallmatrix}\right)$. Its singular values are $2\beta,2|\delta|$, hence the full relative spectrum is $\{-2\beta,-2|\delta|,2|\delta|,2\beta\}$. The four-particle relative energy is

$$E_{\rm fill}(\delta)=-4\beta-4|\delta|.$$

A distortion therefore lowers this electronic contribution. If the specified structural cost is $C\delta^2/2$, $C>0$, the total has minima at $|\delta|=4/C$ when this lies within $|\delta|<\beta$. A physical bond displacement needs its own map to $\delta$ and elastic coefficient.

• Counting mode tools — states the open frequency-selection question.

• Spectral survival recurrence — describes finite-mode return and recurrence.

• Resonance and profile tools — overview and verification.

resonance_profile_checks.py checks finite examples by quadrature, matrix evolution, and independent algebra.