  Infinite quasiperiodic arrangements in space, such as quasicrystals, are
typically described as projections of higher-dimensional periodic lattices onto
the physical dimension. The concept of a reference higher-dimensional space,
called a superspace, has proved useful in relation to quasiperiodic systems.
Although some quantum-mechanical systems in quasiperiodic media have been shown
to admit quasiperiodic states, any sort of general Hamiltonian formalism in
superspace is lacking to this date. Here, we show how to extend generic
quantum-mechanical Hamiltonians to higher dimensions in such a way that
eigenstates of the original Hamiltonian are obtained as projections of the
Hamiltonian in superspace, which we call the super Hamiltonian. We apply the
super Hamiltonian formalism to a simple, yet realistic one-dimensional quantum
particle in a quasiperiodic potential without the tight-binding approximation,
and obtain continuously labelled eigenstates of the system corresponding to a
continuous spectrum. All states corresponding to the continuum are
quasiperiodic. We also obtain the Green's functions for continuum states in
closed form and, from them, the density of states and local density of states,
and scattering states off defects and impurities. The closed form of this
one-dimensional Green's function is equally valid for any continuum state in
any one-dimensional single-particle quantum system admitting continuous
spectrum. With the basis set we use, which is periodic in superspace, and
therefore quasiperiodic in physical space, we find that Anderson-localised
states are also quasiperiodic if distributional solutions are admitted, but
circumvent this difficulty by generalising the superspace method to open
boundary conditions (cont'd).
