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A unitary discretisation

Let L be a bounded selfadjoint operator on a separable Hilbert space H. After a rescaling tex2html_wrap_inline612 , which corresponds to a change of time in the evolution, we can assume that tex2html_wrap_inline614 . Assume tex2html_wrap_inline576 solves tex2html_wrap_inline618 . The unitary operators tex2html_wrap_inline620 are independent of the choice of tex2html_wrap_inline576 . Both tex2html_wrap_inline624 solve tex2html_wrap_inline626 and tex2html_wrap_inline628 has its spectrum in tex2html_wrap_inline630 . Here tex2html_wrap_inline632 is the n'th Chebychev polynomial of the first kind and tex2html_wrap_inline636 is the n'th Chebychev function of the second kind.

propo74

trivlist359

The discrete time evolution is obtained by iterating the map

  equation84

on tex2html_wrap_inline574 . The unitary nature of the evolution is also evident because

displaymath662

on tex2html_wrap_inline574 are conjugated by tex2html_wrap_inline666 using tex2html_wrap_inline668


Oliver Knill
Tue Aug 18, 1998