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Thesis defences

PhD Oral Exam - Petr Zorin, Mathematics

Spectral comparison theorems in relativistic quantum mechanics


Date & time
Monday, August 1, 2016
2 p.m. – 5 p.m.
Cost

This event is free

Organization

School of Graduate Studies

Contact

Sharon Carey
514-848-3802, ext. 3802

Where

J.W. McConnell Building
1400 De Maisonneuve W.
Room LB 921-4

Wheel chair accessible

Yes

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

The classic comparison theorem of quantum mechanics states that if the comparison potentials are ordered then the corresponding energy eigenvalues are ordered as well, that is to say if Va  Vb, then Ea  Eb. The nonrelativistic Schr¨odinger Hamiltonian is bounded below and the discrete spectrum may be characterized variationally. Thus the above theorem is the direct consequence of the min–max characterization of the discrete spectrum [1, 2]. The classic comparison theorem does not allow the graphs of the comparison potentials to cross over each other. The refined comparison theorem for the Schr¨odinger equation [3] overcomes this restriction by establishing conditions under which graphs of the comparison potentials can intersect and still preserve the ordering of eigenvalues. The relativistic Hamiltonian is not bounded below and it is not easy to define the eigenvalues variationally. Therefore comparison theorems must be established by other means than variational arguments. Attempts to prove the nonrelativistic refined comparison theorem without using the min–max spectral characterization suggested the idea of establishing relativistic comparison theorems for the ground states of the Dirac and Klein–Gordon equations [4, 5]. Later relativistic comparison theorems were proved for all excited states by the use of monotonicity properties [6]. In the present work, refined comparison theorems have now been established for the Dirac x4.2.1 and x4.2.2 [7] and Klein–Gordon x4.1.1 and x4.1.2 [8] equations. In the simplest one–dimensional case, the condition Va  Vb is replaced by Ua  Ub, where Ui =R x 0 Vidt, x 2 [0; 1), and i = a or b.

Special refined comparison theorems for spin–symmetric and pseudo–spin–symmetric relativistic problems [9], which also allow very strong potentials such as the harmonic oscillator x4.1.2, x4.2.1, and x4.2.2 [8, 10], are proved.


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