M.Sc Student | Simkhovich Boris |
---|---|

Subject | Factorization Properties of Finite Spaces |

Department | Department of Physics |

Supervisors | Professor Emeritus Michael Revzen |

Professor Ady Mann | |

Professor Emeritus Joshua Zak | |

Full Thesis text |

A finite phase space of dimension M,
where coordinate and momentum have M possible values, is a frequent component
of various physical and mathematical problems. In this work we consider
problems of Fast Fourier Transform, Schwinger factorization of unitary
operators and generation of Zak bases which all relate to a finite phase space.
In particular, we consider the possibility of performing operations in M=M_{1}M_{2}
- dimensional phase space as acting in factorized dimensions M_{1} and
M_{2}. For coprime numbers M_{1} and M_{2} all the
three listed problems have solutions. The FFT problem was solved by Good and
achieves fast operation by factorization of the M - dimensional Discrete
Fourier Transform (DFT) into smaller M_{1} and M_{2} -
dimensional DFTs. The problem of factorization of unitary operators was solved
by Schwinger through the use of the Fermat-Euler theorem. The two mutually
unbiased Zak bases in the coprime case can be generated using the two complete
sets of commuting operators. In our work we show that all these solutions are
interconnected by the use of the Chinese Remainder Theorem from number theory.
For non-coprime numbers M_{1} and M_{2}, only in the case of
the Fast Fourier Transform there exists the solution proposed by Cooley and Tukey.
The solution is based on the use of “conjugate” Division Algorithm
Representations for coordinate and momentum variables. Following this solution,
we propose a permutation operator, which transforms a single coordinate
variable between “conjugate” Division Algorithm Representations of it. This
special choice of the permutation operator allows us to provide the solution
for the Schwinger factorization of unitary operators for all possible
factorizations of M. We derive complementary pairs of unitary operators, each
pair describing one of the factorized dimensions M_{1} or M_{2}.
Then we apply them to the Harper-like Hamiltonian model. We generalize the idea
of previous work for energy spectrum design. For each possible factorization of
the dimension M=M_{1}M_{2} we propose Harper-like Hamiltonians
which have M_{1} energy levels, each M_{2} degenerate. Such
energy spectra control may be of interest for practical realization in solid
state physics for electrons in magnetic field. We also apply the new
factorization to the problem of Zak bases and generate a pair of mutually
unbiased bases, one of which is the original Zak basis and the other is a
Zak-like basis. The new Zak-like basis has particular quasi-periodicity
properties in comparison to the original Zak bases.