Eigenvalues Of Singular Nonlocal Problems In The Limit-circle Case

The spectral theory for ordinary differe.ntial operat.or originates from Fourier method which is used to solve solid heat conduction mode,l.In 1863,Sturm and Liouville gener-alized J.Fourier’s separation variable method.thereafter,the.theory of Sturm-Liouville(S-L)problem was formed.In 1910,H.Weyl extended the classical S-L problem to the singular S-L problem which is defined in the infinite interval and pioneered the theory of the singular S-L problem.In 1925,the quantum mechanics theory was established,and the Schrodinger equation became the ma.thematical language which is used to describe the state of microscopic particles.And the Schrodinger equation is exactly an extension of the singular S-L theory in three dimensions.As a result,many mathematicians and physicists began to focus on the S-L problem.Classical spectral theory mainly deals with differential equations and spectral problems that derived from various definite solutions problem.The study of definite solutions problem about,non-local differential equations are also already started as early as in the 1960s[9].Non-local problems came from a va-riety of phenomena such as reaction diffusion process,non-local mechanics and non-local quantum mechanics.It is well known that the spectral theory of linear problem is an important means and tools to understand and solve nonlinear problems.But the study of spectral theory for non-local linear equations is just at.the beginning stage.Although non-local S-L differential equation is a generalization of the classical S-L differential equa-tion in mathematical form,the generation of non-local terms leads to essential changes in the properties of the spectrum at this time.In this paper,we study the eigenvalue of singular non-local problems in limit-cir-cle case.The eigenvalue problem of singular non-local is transformed into the eigenvalue problem of regular non-local by doing the transformation.Because there is no singularity this reduces the difficulty of the study.Sccondly,the eigenvalue of non-local problem corresponds to the eigenvalue of a compact self-adjoint operator through the Green func-tion.It is found that the eigenvalue problem of non-local has only a countable number of real eigenvalues.Furthermore,we obtain the characteristic function of eigenvalues for non-local problem in terms of a modified Green function.Finally,we prove that the eigenvalues of the non-local problem have lower bounds.The judgment condition of the eigenvalue multiplicity is given.The specific position of the eigenvalue is obtained by us-ing the properties of the characteristic function at—∞ and the Rouche’s theorem.The main arrangements a.re as follows:In the first chapt.er,Prolegomenon.This cha.pter ma.inly introduces the background,actual source and current situation of the research.In the second chapter,we give the preparatory knowledge of the singular Sturm-Liouville problem.In the third chapter,We mainly study the properties of the eigenvalues of singular nonlocal problems in limit-circle case,including the lower bound,the multiplicities and the specific distribution location of eigenvalues.First,we transform the singular non-local prolblcem into a regular non-local problem by using two transformations.Thus,the singular nonlocal problem is transformed into a,regular non-local problem.Secondly,the eigenvalue of non-local problem corresponds to the eigenvalue of a,compact self-adjoint operator through the Green function.It,is found that the eigenvalue problem ofnon-local has only a countable number of real eigenvalues.Furthermore,we obtain the characteristic function of eigenvalues for non-local problem in terms of a modified Green function.Finally,wo prove that the eigenvalues of the non-local problem have lower bounds.The judgment condition of the eigenvalue multiplicity is given.The specific position of the eigenvalue is obtained by using the properties of the characteristic function at —∞ and the Rouche’s theorem.