The Weyl Classification Of Nonlocal Singular Second Order Differential Equations

As an extremely important branch of subject in modern mathematics:the differential equation was born in the 17th century.With the development of all aspects differential equations have been widely used in natural fields such as engineering and astronomy,as well as in social fields such as finance,insurance and economy.As a subject widely used in many fields,the ordinary differential equation not only promotes each other with the growth processes of physics,mechanics,etc.,but also influences and urges each other with astronomy.Many theories and methods of ordinary differential equations are not only applied to natural categories,but also are increasingly applied to various categories of contemporary society.We can anticipate that the theories and applications of differential equations will be indispensable and they will permeate into all aspects of our society and life in the future.As a basic theory of modern mathematics,the spectral theory of linear operators is not only an important part in functional analysis,but also an important component of the tlheoretical system of operators.In Hilbert space,the theories about bounded opera-tors and unbounded self-conjugate operators are relatively complete,when we apply these theories to differential equations or integral equations,many extremely important prob-lems in contemporary mathematics are solved.The use of linear operator theory to study such problems has led to the generation of differential operators and integral operators.In fact,quite a lot of problems not only in engineering and physics,but,also in modern scientific and technological life can be finally attributed to the problems of differential operators or integral operators.For example,the final result may be to transform the problems into spectral problems of differential operators,cigenvalue problems or prob-lems related to eigenfunctions,etc.In addition,people also find that spectral analysis of differential operators is a.relatively basic mathematical tool to solve many quantum mechanics problems.The spectral theory of differential operators has important applications not only in mathematics,but also in classical quantum mechanics.As early as 1910,German math-ematician Weyl extended the study of Sturm-Liouville problem from finite interval to infinite interval.Wevl discussed the square integrable solutions of the second order sin-gular Sturm-Liouville equation with real coefficients.The Weyl cla.ssificat.ion,i.e.the differential equation belongs to the limit-point case or limit-circle case of this kind of'equation is given.The research on this problem makes Sturm-Liouville theory develop rapidly and enter a,brand-new growth process.The distinction between the limit-point case and limit-circle case of differential expressions is closely related to the study of spec-trum of differential operators,because the former is the basis for studying the spectrum information.By studying the relationship between them,a concrete solution can be pro-vided for the problem of spectral decomposition of operators.And the research in this aspect has now been extended to the deficiency index problems of general n-order singular diferential operators.so that this research field shows a broader prospect.The Sturm-Liouville problem with nonlocal boundary conditions has been studied in[32.45].The model of differential equation with nonlocal terms appears in the fields of reaction diffusion process,quantum mechanics,etc.Simple models have been studied by many researchers,e.g.[13.25]Spectral problems,especially eigenvalue problems,of nonlocal equations with different boundary conditions on regular intervals have been stucdie.d,and some good results have been obtained[3.40.41].The present paper is concerneed with the Weyl classification of second order singular Sturm-Liouville equations with nonlocal point potentials.The definitions of limit--point(-circle)case for these equations are given and sufficient and necessary conditions for equations belonging to limit-point case are obtained.Furthermore,the number of square integrable solutions of' such equations for A on the real axis is studied and correspond-ing sufficient and necessarv conditions are.obtained.The conclusions illustrate that the situations are essentially different from that of the classical local equations.The main research work of this paper is as follows:In the seeond chapter the Weyl classification of classical singular second order differ-ential equations is given by the number of square integrable solutions,and some important related property theorems are also given.In addition,we also provide several basic and commonly usecd methods to distinguish the limit-point(-circle.)case,especially a sufficient and neecssary condition for the limit-point case of the classical second order differential equation.In the third chapter,since classical singular second order differe,ntial equations call only belong to limit-point case or limit-circle case.we will study the number of square integrable solutiongshut of noulocal singular second order differential equations for these two cases.and the defmition that nonlocal singular second order differential equations belong to limit-point or limit-circle case are given.That is to say,we give the Weyl classification of nonlocal singular second order differential equations or differential expressions.On this basis,a necessary and sufficient condition for nonlocal singular differential equations belonging to limit-point case is further obtained.In the fourth chapter,we discuss the solutions in L2[0,?)of nonlocal singular second order equations for ?(?)R.Since it is easy for the limit-circle case,we focus on the limit-point case,where we find the differences between nonlocal problems and classical local problems.And we give a quantitative description of this.