Spectral Theory

Spectral theory can be seen as a generalization of the ideas and concepts of eigenvalues and eigenvectors of a square matrix to a broader class of operators acting in different spaces. Dr. Usman’s research interest lies in the spectral theory of Quantum graphs. A quantum graph is a metric graph with one-dimensional Schrödinger operators acting on the edges and equipped with some appropriate boundary conditions at vertices. The main focus is on quantities related to the discrete spectrum of these operators. Discrete spectrum or the set of discrete eigenvalues of a Schrödinger operator corresponds to energies of bound states of a quantum mechanical system. His work is focused on finding certain estimates on the discrete spectrum of quantum graphs and how the spectrum is related to the topology of graph.