8 Behavior of a particle in an almost periodic potential: a renormalization group technique applied to Schrodinger's and Harper's equations

If we consider the one-dimensional discretized Schrodinger equation

$$\Psi_{n+1} + \Psi_{n-1} = [E-\lambda V(n\omega )+ 2]\Psi_n$$ (8.1)

where $\omega$ is an irrational number and $V$ is a periodic function, it is possible to investigate the spectral and wave function properties of the different models using the scaling and multifractal analyses [18].

In order to scale the spectrum, the irrational number $\omega$ is replaced by a sequence of rational numbers which are obtained by truncating the continued fraction expansion of $\omega.$ For example, $1/\phi$ is approximated by a sequence of rational numbers

$$\frac{{F_{n - 1} }}{{F_n }} = \left\{ {\frac{1}{2}} \right... ...c{5}{8},\frac{8}{{13}},\ldots\left. {\frac{{}}{{}}} \right\},$$

where $F_n$ is a Fibonacci number defined recursively as $F_0=F_1=1, F_{n+1} = F_n + F_{n-1}.$

For the Fibonacci model, $V$ takes only two values: $1$ and $-1$ (the potential is constant, except for discontinuities) and the spectrum is purely singular continuous (critical wave functions).

Instead, when the potential $V$ is a smooth function, the spectrum is purely absolutely continuous (extended wave functions) for small $\lambda$ and a purely dense point (localized wave functions) for large $\lambda$. For an intermediate value of $\lambda,$ the spectrum is a mixture of absolutely continuous parts and dense point parts, which are separated by a finite number of mobility edges. There is no singular continuous part. The exception is the Harper model, where there is a singular continuous spectrum (critical states) at the critical point $\lambda = 2.$

Hence, a singular continuous spectrum rarely appears for a potential $V$ that is a smooth function of $x.$