Localized wavefunctions in quantum systems with multiwell potentials

Abstract

One-dimensional systems with the multiwell potential V (x) = 2λ 1 cos x + 2λ 2 cos αx are considered. Localized wavefunctions, in the case when the energy is greater than the maximum value of the potential, are presented. Wigner and Weyl functions, corresponding to the wavefunctions of these systems, are also studied. It is shown that they are very sensitive to the value of the parameter α.

On the Nature of Electronic Wave Functions in One-Dimensional Self-Similar and Quasiperiodic Systems

“…In fact, one can implement numerically an empirical scaling analysis in which the QPS is approximated by a sequence of periodic systems with progressively larger unit cells of size defined by the optimal rational approximants to ; namely, = / . In this way, by checking that finer discretization produces almost the same results one can be confident enough of the reliability of the obtained results [44].…”

“…The existence of exponentially localized wave functions has been recently reported from numerical studies of (10) with the bichromatic potential ( ) = 2 1 cos + 2 2 cos with rational = / corresponding to certain specific energy values ( > 2 ( 1 + 2 )). This system then suggests the possible presence of a purepoint spectral component not restricted to low energy values in certain periodic systems [44].…”

“…Therefore, as the Vicsek fractal grows larger we can obtain a numerable infinite set of extended states in the thermodynamic limit. The energies corresponding to these states can be obtained from (44) by imposing the condition +1 = , ≥ 1, iteratively. The same condition can be arrived at by considering the local transfer matrices corresponding to the on-site hierarchical lattice model, which have the form:…”