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Next: Relations with other numerical Up: A remark on quantum Previous: Quantum dynamics versus random

Numerical experiments

We illustrate the discrete time evolution (2.1) in two numerical experiments.

(i) The first numerical experiment deals with an electron in the plane under a constant magnetic field B reduced by a Landau gauge to a one-dimensional situation, where one considers the one-dimensional Schrödinger operator L on tex2html_wrap_inline1026 of the form tex2html_wrap_inline1028 and take an initial condition which is a localized wave tex2html_wrap_inline1030 at the origin k=0. We use Wiener's theorem to get numerically information about the discrete part of the spectral measure. For this illustration, we take the almost Mathieu (or Harper) operator tex2html_wrap_inline1034 , where much about the spectrum is known (see [24, 20, 11] for reviews). Note that most of the known results hold only for almost all or generic tex2html_wrap_inline1036 and under some assumptions on the magnetic flux tex2html_wrap_inline606 .

We did a numerical determination of tex2html_wrap_inline1040 using (2.1), up to n=40'000 as a function of tex2html_wrap_inline1044 in the almost Mathieu operator with tex2html_wrap_inline1046 . tex2html_wrap_inline770 is the spectral measure on the circle belonging to the vector tex2html_wrap_inline1050 localized at the origin in tex2html_wrap_inline1052 . The value of tex2html_wrap_inline1054 was computed using evolution (2.1) with initial condition tex2html_wrap_inline1056 on the grid [-n/2,n/2] so that the boundary effects the value tex2html_wrap_inline768 only after n steps: tex2html_wrap_inline1064 has support in [-n,n] and the boundary begins to affect tex2html_wrap_inline678 after n/2 time steps and so to influence tex2html_wrap_inline768 after n steps. The numerical experiment is in agreement with the now established fact that there is no point spectrum for tex2html_wrap_inline1076 for almost all tex2html_wrap_inline606 (for tex2html_wrap_inline1080 see [8]) and some point spectrum for tex2html_wrap_inline1082 [12]. Longer runs, ( tex2html_wrap_inline1084 ) indicated that indeed tex2html_wrap_inline1086 for tex2html_wrap_inline1088 .

(ii) In a second numerical experiment, we take a two dimensional operator L which is the Hamiltonian for an electron in the discrete plane, where the magnetic field B is randomly taking values in U(1) (see [15] for some theoretical results or [3, 2] for other numerical experiments on this model). If the distribution of tex2html_wrap_inline1096 is a Haar measure of U(1), then this field can be generated with a vector potential tex2html_wrap_inline1100 with independent random variables tex2html_wrap_inline1102 having the uniform distribution in tex2html_wrap_inline1104 . There is no free parameter. The ergodic operator, which we consider, is

displaymath1106

and the open question is what is the spectral type of L.

While one knows the moments of the density of states of L (the n-th moment of the density of states is the number of closed paths in tex2html_wrap_inline1114 of length n starting at 0 which give zero winding number to all plaquettes [16]), nothing about the spectral type of L seems to be known. For the two dimensional experiment, we experimented on a tex2html_wrap_inline604 lattice, were we can compute the first 1000 Fourier coefficients of the spectral measure exactly. Our experiments indicate no eigenvalues. If eigenvalues exist, they would have to be extremely uniformly distributed because tex2html_wrap_inline1124 must be small. The measurements done on a usual workstation indicate that tex2html_wrap_inline1126 goes to zero monotonically: tex2html_wrap_inline1128 . Longer runs with better computers on larger lattices are needed to confirm this picture.

Remark. We made also numerical experiments with a Aharonov-Bohm problem on the lattice. This is the situation when the magnetic field B is different from 1 only at one plaquette n=(0,0). The vector potential A in this situation can not be chosen differently from 1 in a compact set. However, in a suitable gauge, the operator L is a compact perturbation of the free operator by a result of Mandelstham-Jitomirskaja [22] see [16] for an other proof of this fact). As expected, there was no indication of some discrete spectrum. The numerical experiments suggest that tex2html_wrap_inline1142 is bounded which would mean that the spectral measures are in tex2html_wrap_inline756 .


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Next: Relations with other numerical Up: A remark on quantum Previous: Quantum dynamics versus random

Oliver Knill, Tue Aug 18, 1998