Abstract: 

Dirac operators with point interactions in one dimension and metric graphs are considered. Unlike the Schrödinger operator, the Dirac operator is not semi-bounded. In the case of compact graphs, the spectrum of the Dirac operator is discrete and tends to be both positive as well as negative infinity. For non-compact graphs, the same operator has a continuous spectrum with a gap and covers(-∞ , -c²/2)∪( c²/2 , +∞). 

The spectrum is discrete inside the gap. Our project focuses on the study of the discrete spectrum of the Dirac operator and the relation between the Dirac operator and its discreet counterpart, Jacobi matrix. We investigate the problem of finding the number of eigenvalues within the gap. We also study the boundary triplets and corresponding M-function of the Dirac operator. By using a suitable boundary triplet, we construct a Jacobi matrix for the Dirac operator and see how the spectral properties i.e. selfadjointness and discreteness of the Dirac operator correlate with the spectral properties of the Jacobi matrix. 

Evaluation Committee  

• Dr.  Muhammad Usman(Advisor)
• Dr. Ali Ashher Zaidi (Thesis committee member/evaluator)