Riesz Basis And Spectral Properties For Certain Sturm-Liouville Problems

As we all know,the Sturm-Liouville problem,which originated from the solid heat conduction model,is one of the essential models in different physics fields,such as quantum mechanics,magnetostatics,and fluid mechanics.These problems frequently arise due to employing the method of separation of variables to tackle classical partial differential equations in physics.For instance,the Schrodinger equation is a partial differential equation,but in the case of spherically symmetric potentials such as the Coulomb potential,the standard method of separation of variables transforms the equation to a sequence of ordinary differential equations,such as Sturm-Liouville equation.Based on the wide application of Sturm-Liouville problems in various fields,this thesis will investigate several types of regular Sturm-Liouville problems with separated boundary conditions.These Sturm-Liouville problems cover the following cases:Sturm-Liouville problems with boundary conditions and transmission conditions independent of the eigenparameter,Sturm-Liouville problems with boundary conditions dependent rationally on the eigenparameter,Sturm-Liouville problems with transmission(interface)conditions dependent rationally on the eigenparameter,and Sturm-Liouville problems with transmission conditions dependent rationally on the eigenparameter,where all boundary conditions dependent linearly on the eigenparameter.In Chapter 1,we first introduce the research background and current status of the Sturm-Liouville problems.It then describes an overview of the research on a Riesz basis.Finally,it outlines the research content and the results obtained in this thesis.In Chapter 2,we consider a class of regular Sturm-Liouville problems with separated boundary conditions such that the left and right conditions do not contain the eigenparameter.We define the Prüfer transformation for this type of Sturm-Liouville problem and thoroughly investigate the spectral properties of such problems.We conclude that the considered problem has an infinity of eigenvalues,λ0,λ1,λ2,…,which are real and tend to+∞.Furthermore,we provide the oscillation properties of the normalized eigenfunctions corresponding to these eigenvalues.Finally,we give two examples to verify our main result.In Chapter 3,we study two classes of regular Sturm-Liouville problems on the finite interval[a,b]with separated boundary conditions,where the left boundary condition does not contain the eigenparameter or dependent rationally on the eigenparameter,and right boundary conditions dependent rationally on the eigenparameter.Our research establishes some of the problem’s spectral properties under consideration.We prove that the problems have precisely countably many eigenvalues,λ1,λ2,…,which are real and tend to +∞.Furthermore,we demonstrate that the system of weak eigenfunctions of such problems forms a Riesz basis in the appropriate Hilbert space.In Chapter 4,we study a class of discontinuous Sturm-Liouville problems with separated boundary conditions,where the boundary conditions are independent of the eigenparameter and the transmission conditions are dependent rationally on the eigenparameter.Our objective is to reveal important spectral properties of this type of problem.Until then,the Riesz basis property of a system of weak eigenfunctions for such problems has not been discussed.To achieve this,we transform the problem under consideration into an eigenvalue problem for a suitable integral equation using suitable integral transformations,defining the concept of a weak solution through this equation.Subsequently,we introduce some compact operators to reduce this integral equation to the appropriate operator-pencil equation and prove that this operator-pencil is self-adjoint and positively definite for sufficiently large negative values of the parameter.Finally,we demonstrate that the spectrum of the problem under consideration is discrete,and the system of corresponding weak eigenfunctions forms a Riesz basis in the appropriate Hilbert space.Additionally,we develop the corresponding Rayleigh-Ritz formula and find lower bounds estimates for the eigenvalues.In Chapter 5,we consider two classes of discontinuous Sturm-Liouville problems with separated boundary conditions,where the left boundary condition is independent of the eigenparameter or dependent linearly on the eigenparameter,the right boundary condition dependent linearly on the eigenparameter,and the transmission condition is dependent rationally on the eigenparameter.Until now,the Riesz basis properties of the system of weak eigenfunctions for these problems have also not been discussed.Compared to the problem considered in Chapter 4,since the boundary conditions dependent linearly on the eigenparameter,we must introduce two new parameters when transforming the considered problem into the corresponding integral equation.Then,using a method similar to Chapter 4,i.e.,defining a weak solution to the problem under consideration and introducing the necessary compact operators,the integral equation is transformed into an operator-pencil equation,which is self-adjoint and positive definite for large enough negative parameters.We also prove that the spectrum of such problems is discrete,that the system of weak eigenfunctions forms a Riesz basis in an appropriate Hilbert space,and give lower bound estimates for the eigenvalues.In Chapter 6,we summarize the main research contents of this thesis and look ahead to possible future works.