Spectral and Scattering Problems for Differential Operators on Quantum Star Graphs
Spectral theory is a generalization of ideas of eigenvalues and eigenvectors of square matrices to broader classes of linear operators defined in a variety of infinite dimensional vector spaces. Spectrum of non-compact quantum star graphs, star shaped metric graph equipped with differential operators and set of vertex conditions, is studied. The selection of vertex or boundary conditions play a vital role in making the operators self-adjoint as well as ensuring that edges are connected or coupled with each other. Generally, a closed and densely defined self-adjoint linear differential operator defined on a separable Hilbert space has real spectrum which consists of a set of negative eigenvalues with finite multiplicity and a continuous part covering the positive semi-axis. The number of negative eigenvalues are finite if the potential function(s) are rapidly decaying at the boundary or a countable infinite set if the potential decays slowly with zero as the only accumulation point. In case of complex-valued potential the corresponding non-selfadjoint graphs has complex eigenvalues. For Schrödinger operators we develop trace formulas for resolvent difference of perturbed and unperturbed operators. Which lead to explicit expressions for perturbation determinant, Levinson's formula and spectral shift function for star shaped quantum graphs. These mathematical objects describe continuous as well as discrete part of the spectrum in terms of quantities which are directly to the operator (potential functions) and boundary parameters. Moreover, among many spectral problems associated with differential operators, one is, to find the bounds on the series of moments of negative eigenvalues in terms of potentials and boundary parameters of the operator. We derive sharp upper bounds for a fourth order differential operator with two real-valued potentials on semi-axis with the "Generic" three parameters self-adjoint boundary conditions at zero. Such bounds are commonly known as Lieb-Thirring inequalities. We also obtain bounds for eigenvalues of star graphs with complex valued potentials having n edges with δ and δ’ vertex conditions. For two edge graphs sharp and explicit bounds on the complex eigenvalues are obtained in terms of potential and boundary parameters.
- M. D. Zia and M. Usman, Sharp Spectral Inequalities for Fourth Order Differential Operators on Semi-Axis, Math. Phys. Anal. Geom. 22(24), (2019). https://link.springer.com/article/10.1007/s11040-019-9321-9
- M. D. Zia and M. Usman, A Trace Formula, Perturbation Determinant and Levinson’s Theorem for a class of star graphs. (Submitted).
- M. D. Zia and M. Usman, Perturbation Determinant and Levinson’s Formula for Schrödinger Operators with Generalized Point Interaction. (Submitted).
- M. D. Zia and M. Usman, Eigenvalue Bounds for Non-selfadjoint Star Graphs, in preparation.