Trace Formulas Of Several Kinds Of Differential Operators On Lasso Graphs

The theory of ordinary differential operator is a branch of mathematics based on Fourier analysis,Sturm-Liouville theory and unbounded operator theory in Hilbert space.It has important applications in quantum mechanics,mathematical physics and other fields.The problems studied by the spectral theory of ordinary differential operator mainly include: the direct problem—self-adjoint characterization of boundary conditions,asymptotic analysis of eigenfunctions,properties and estimations of eigenvalues,trace formulae,etc;the inverse problem—the uniqueness of unknown elements of the system determined by the spectral data,reconstruction algorithm and stability of the inverse problem.A quantum graph is made up of geometry,differential equations on edges,and matching conditions on vertices of a graph.Differential operators on graphs can be used to simulate the motion of quantum particles confined to some low-dimensional structures,which is a rapidly developing new field in modern mathematical physics.The Lasso graph is a quantum graph with a loop,consisting of a ring and a ray(line)segment intersecting it at a point.In this paper,eigenvalue asymptotic expressions and trace formulas of three kinds of differential operators on Lasso graphs are obtained.In Chapter 1,we introduce regularized trace formulae and quantum graphs,summarize research works for differential operators on graphs,and give the main work in this paper.In Chapter 2,we give the trace formula of the Sturm-Liouville differential operator with delays on Lasso graphs.In Chapter 3,the trace formula of the integral-differential operator on Lasso graphs is presented.In Chapter 4,we deal with the trace formula of the differential operator with energydependent potential on Lasso graphs.