The spectral transition conjecture for the extended Harper model

Let $H^{\mathrm{EHM}}_{\lambda,\alpha,\theta}$ be the extended Harper model, with parameter $clambda=(\lambda_1,\lambda_2,\lambda_3)$ and hopping function

$$c_{\lambda}(\theta)=\lambda_1 e^{-2\pi i(\theta+\frac{\alpha}{2})}+\lambda_2+\lambda_3 e^{2\pi i(\theta+\frac{\alpha}{2})}.$$

Let $L(\lambda)$ be its Lyapunov exponent on the spectrum, let

$$\gamma(\alpha,\theta):=\limsup -\frac{\ln \|2\theta+n\alpha\|}{|n|},$$

and define

$$\mathcal{R}_1:=\{\lambda:0<\max(\lambda_1+\lambda_3,\lambda_2)<1\},$$ $$\mathcal{R}_s:=\{\lambda:\lambda_1=\lambda_3\geq\frac{\lambda_2}{2},\text{ or }\lambda_1+\lambda_3=\lambda_2\}.$$

For the continued-fraction denominators $q_n$ of $calpha$, set

$$\tilde{\delta}(\alpha,\theta)=\limsup_{n\to\infty}\frac{\ln q_{n+1}+\sum_{\theta':c_{\lambda}(\theta')=0}\ln\|q_n(\theta-\theta')\|}{q_n},$$

where zeros are counted with multiplicities. The function $c_{\lambda}$ has one or two, possibly coinciding, zeros in this regime.

Spectral transition conjecture. For $\lambda\in\mathcal{R}_1\cap\mathcal{R}_s$, the operator $H^{\mathrm{EHM}}_{\lambda,\alpha,\theta}$ has purely singular continuous spectrum if

$$L(\lambda)<\tilde{\delta}(\alpha,\theta)+\gamma(\alpha,\theta),$$

and pure point spectrum if

$$L(\lambda)>\tilde{\delta}(\alpha,\theta)+\gamma(\alpha,\theta).$$

This conjecture proposes a sharp spectral transition controlled by the Lyapunov exponent, frequency-resonance strength, phase-resonance strength, and anti-resonance contribution. The equality case is not covered by the statement, and the source gives no resolution status.