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Results on the number of zero modes of the Weyl-Dirac operator

Research output: Thesis › Doctoral Thesis

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Standard

Results on the number of zero modes of the Weyl-Dirac operator. / Ta, Tri.
Lancaster University, 2009. 106 p.

Research output: Thesis › Doctoral Thesis

Harvard

Ta, T 2009, 'Results on the number of zero modes of the Weyl-Dirac operator', PhD, Lancaster University.

APA

Ta, T. (2009). Results on the number of zero modes of the Weyl-Dirac operator. [Doctoral Thesis, Lancaster University]. Lancaster University.

Vancouver

Ta T. Results on the number of zero modes of the Weyl-Dirac operator. Lancaster University, 2009. 106 p.

Author

Ta, Tri. / Results on the number of zero modes of the Weyl-Dirac operator. Lancaster University, 2009. 106 p.

Bibtex

@phdthesis{c72a0cd66052462eafd2ca2832b21ad5,
title = "Results on the number of zero modes of the Weyl-Dirac operator",
abstract = "For a given magnetic potential A one can define the Weyl-Dirac operator σ·(−i∇−A) on R 3 . An L 2 eigenfunction of σ · (−i∇ − A) corresponding to 0 is called a zero mode. In this thesis we will be concerned with the zero mode problem for the WeylDirac operator and some related problems. The main results are (i) upper bounds for the number of zero modes of the Weyl-Dirac operator in three dimensions when scaling a given magnetic field. A similar version for the Dirac operator in two dimensions is also obtained. There are also related results to estimate the number of zero modes of the massless Dirac operator, and the dimension of the eigenspaces at threshold energies for the Dirac operator with positive mass. (ii) construction of Dirac operators on the unit ball S 2 of R 3 as well as the determination of their spectrum in case of “constant” magnetic fields. We also show another proof for the Aharonov-Casher theorem for S 2 based on results about spectral properties of Dirac operators that we have obtained. (iii) a formula giving the number of zero modes of the Weyl-Dirac operator for a special magnetic field, which is the result of pullbacks from the “constant” volume form of S 2 . We also obtain a lower bound for the number of zero modes for the Weyl-Dirac operator corresponding to certain scaled magnetic fields; the magnetic fields are parallel to fibres of the Hopf fibration (pulled-back to R 3 using inverse stereographic projection).",
author = "Tri Ta",
year = "2009",
month = dec,
day = "4",
language = "English",
publisher = "Lancaster University",
school = "Lancaster University",

}

RIS

TY - BOOK

T1 - Results on the number of zero modes of the Weyl-Dirac operator

AU - Ta, Tri

PY - 2009/12/4

Y1 - 2009/12/4

N2 - For a given magnetic potential A one can define the Weyl-Dirac operator σ·(−i∇−A) on R 3 . An L 2 eigenfunction of σ · (−i∇ − A) corresponding to 0 is called a zero mode. In this thesis we will be concerned with the zero mode problem for the WeylDirac operator and some related problems. The main results are (i) upper bounds for the number of zero modes of the Weyl-Dirac operator in three dimensions when scaling a given magnetic field. A similar version for the Dirac operator in two dimensions is also obtained. There are also related results to estimate the number of zero modes of the massless Dirac operator, and the dimension of the eigenspaces at threshold energies for the Dirac operator with positive mass. (ii) construction of Dirac operators on the unit ball S 2 of R 3 as well as the determination of their spectrum in case of “constant” magnetic fields. We also show another proof for the Aharonov-Casher theorem for S 2 based on results about spectral properties of Dirac operators that we have obtained. (iii) a formula giving the number of zero modes of the Weyl-Dirac operator for a special magnetic field, which is the result of pullbacks from the “constant” volume form of S 2 . We also obtain a lower bound for the number of zero modes for the Weyl-Dirac operator corresponding to certain scaled magnetic fields; the magnetic fields are parallel to fibres of the Hopf fibration (pulled-back to R 3 using inverse stereographic projection).

AB - For a given magnetic potential A one can define the Weyl-Dirac operator σ·(−i∇−A) on R 3 . An L 2 eigenfunction of σ · (−i∇ − A) corresponding to 0 is called a zero mode. In this thesis we will be concerned with the zero mode problem for the WeylDirac operator and some related problems. The main results are (i) upper bounds for the number of zero modes of the Weyl-Dirac operator in three dimensions when scaling a given magnetic field. A similar version for the Dirac operator in two dimensions is also obtained. There are also related results to estimate the number of zero modes of the massless Dirac operator, and the dimension of the eigenspaces at threshold energies for the Dirac operator with positive mass. (ii) construction of Dirac operators on the unit ball S 2 of R 3 as well as the determination of their spectrum in case of “constant” magnetic fields. We also show another proof for the Aharonov-Casher theorem for S 2 based on results about spectral properties of Dirac operators that we have obtained. (iii) a formula giving the number of zero modes of the Weyl-Dirac operator for a special magnetic field, which is the result of pullbacks from the “constant” volume form of S 2 . We also obtain a lower bound for the number of zero modes for the Weyl-Dirac operator corresponding to certain scaled magnetic fields; the magnetic fields are parallel to fibres of the Hopf fibration (pulled-back to R 3 using inverse stereographic projection).

M3 - Doctoral Thesis

PB - Lancaster University

ER -