Concordia University

Thesis defences

PhD Oral Exam - Hassan Harb, Mathematics and Statistics

Spectral Comparison Theorems for Klein Gordon Equation in d≥1 dimensions

Date and time
Date & time

January 30, 2020
10 a.m. – 1 p.m.


Room LB 921.1
J.W. McConnell Building
1400 De Maisonneuve Blvd. W.
Sir George Williams Campus


This event is free

Wheelchair accessible
Wheelchair accessible



School of Graduate Studies


Jennifer Sachs

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.


We first study bound-state solutions of the Klein--Gordon equation φ"(x)=[m2 - (E-V(x))2] φ(x), for vector potentials which in one spatial dimension have the form V(x) = vf(x), where f(x)≤0 is the shape of a finite symmetric central potential that is monotone non-decreasing on [0, ∞) and vanishes as x⟶∞, and v>0 is the coupling parameter.

We characterize the graph of spectral functions of the form v= G(E) which represent solutions of the eigen-problem in the coupling parameter v for a given E: they are concave, and at most uni-modal with a maximum near the lower limit E = -m of the energy E ∈ (-m, m). This formulation of the spectral problem immediately extends to central potentials in d > 1 spatial dimensions. Secondly, for each of the dimension cases, d=1 and d≥ 2, a comparison theorem is proven, to the effect that if two potential shapes are ordered f1(r) ≤ f2(r), then so are the corresponding pairs of spectral functions G1(E) ≤ G2(E) for each of the existing eigenvalues. These results remove the restriction to positive energies necessitated by earlier comparison theorems for the Klein--Gordon equation by Hall and Aliyu. Corresponding results are obtained when scalar potentials S(x) are also included.

We then weaken the condition for the ground states by proving that if ∫x0 [f2(t) - f1(t)]φ(t) dt≥0, the corresponding coupling parameters remain ordered, where φi= 1, or φi is the bound state solution of the Klein--Gordon equation with potential Vi, i = 1, 2. These results are valid for any energy E ∈ (-m, m)., but they are restricted to the ground states.

We finally present a complete recipe for finding upper and lower spectral bounds for both bounded and unbounded potentials, and we exhibit specific result for the applications for the Woods-Saxon, Gaussian, sech-squared, and Yukawa potentials in dimensions d = 1 and d = 3.

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