Non-real Eigenvalues Of Indefinite Problems

The classical Sturm-Liouville problem,that is the generally Sturm-Liouville problem with weight function,originated from the problem of solving the sol-id heat conduction model by using the Fourier method at the beginning of the last century.In 1836,C.Sturm and J.Liouville generalized this method,thereafter,the theory of Sturm-Liouville problem was formed.This theory is the basis of heat conduction problems,oscillate problems,microwave transmis-.sion problems and the microscopic particles state problems in quantum,and it has been widely applied to the mathematical physics,quantum mechanics,engineering and many other fields.With the development of the unbounded operator theory,spectral analysis and the application of practical problems,a new kind of boundary value problems,known as the nonlocal boundary value problem was generated.This problem has been appeared in voltage-driven electrical systems,population dynamics,processes with conserved first inte-gral and nonlocal problems with convective terms,and the nonlocal operators generated by these problems also appeared in quantum mechanics and reac-tion diffusion.Hence,the spectrum of Sturm-Liouville problem with weighted function and nonlocal Sturm-Liouville problem needs to be comprehensively analyzed and the research methods should be explored.The research of spectral theory for regular right-definite,left-definite and indefinite Sturm-Liouville problems and the regular right-definite p-Laplacian problem has been relatively complete,such as the spectral decomposition of self-adjoint operators,comparison theorem,oscillate theory,Prufer transfor-mation,distribution,asymptotic and differentiability of eigenvalues,a priori upper and lower bounds and existence of non-real eigenvalues,etc.In contrast to the study on regular problems,the singular Sturm-Liouville problems,es-pecially the singular indefinite Sturm-Liouville problem and regular indefinite p-Laplacian problem,were so far only studied in vary special situations,so the properties of the spectral for these problems were not systematic and perfect.The research tools of spectral theory for these problems were not as perfect as the regular problems.Since the diversity and complexity of biomathematics models,population dynamic systems,transport models and microwave propagation problems,the nonlocal indefinite Sturm-Liouville problem has not only attracted a lot of attention in recent years,but also played an important role in reaction-diffusion equation and quantum-mechanical theory.The nature of the spectrum for nonlocal operator has changed substantially because of the nonlocal term.Therefore,the research of spectral for nonlocal Sturm-Liouville problems will farther improve the operator theory and become the basis of solving nonlocal problems.In this dissertation,two kinds of spectral problems are researched.The first kind involves a priori bounds and existence of non-real eigenvalues for singular indefinite Sturm-Liouville differential operator and one dimensional p-Laplacian operator with indefinite weight function.The other kind deals with the spectral problems of nonlocal indefinite Sturm-Liouville differential operators,including the change rule of(non-)left-definiteness for the local case under the nonlocal perturbations,the finiteness and the estimate of non-real eigenvalues for nonlocal regular indefinite problem and a priori bounds and non-existence of non-real eigenvalues for the nonlocal singular indefinite prob-lem.Firstly,the non-real eigenvalues of singular indefinite Sturm-Liouville prob-lem will be considered.For the Sturm-Liouville boundary value problem with weighted function,the problem is called right-definite if the weighted functions don't change signs,otherwise the problem is right-indefinite.The spectral the-ory of the right-definite Sturm-Liouville problem has been accomplished,but the spectral structure of indefinite problem is quite different from and morc complicated than that of right-definite problem.For example,the real eigen-values are unbounded from both below and above of indefinite Sturm-Liouville boundary value problem.Moreover,the non-real eigenvalues may admit and it's also the essential difference between indefinite and right-definite problems.The research,of indefinite Sturm-Liouville problem was noticed by Hilb,Richardson,Bocher,Haupt at the beginning of the last century.In 1915 and 1918,when Haupt and Richardson generalized the vibration theory to indefi-nite equation,they found that the non-real eigenvalues might possible existence and Richardson considered the indefinite Sturm-Liouville problems of Richard-son equation with Dirichlet boundary value conditions in which the weighted function is the sign function.Turyn in 1980,Mingarelli in 1982,Atkinson,Jabon and Fleckinger,Mingarelli in 1984,Binding,Volkmer in 1996 and many-other scholars have made profound and meticulous research on the indefinite problems.Mingarelli in 1986,Kong,Moller,Wu and Zettl in 2003 and Zettl in 2005 proposed the open problems of a priori bounds and the sufficient con-dition for the existence of non-real eigenvalues.Recently,this open problems have been solved by Xie and Qi in 2013,Qi and Chen in 2014,Qi,Xie and Chen in 2016 for the regular indefinite Sturm-Liouville problems.For the singular indefinite case,Behrndt,Katatbeh and Trunk in 2009 have given sufficient conditions for the existence of non-real eigenvalues of the singular indefinite Sturm-Liouville problems with limit-point type endpoints and the weight func-tion being the sign function.What's more,Behrndt,Philipp,Trunk in 2013,Behrndt,Gsell,Schmitz,Trunk in 2017 and Behrndt,Schmitz,Trunk in 2016,2018 and 2019 evaluated the upper bounds of non-real eigenvalues,with real,imaginary parts and absolute values of the indefinite Sturm-Liouville problem-s under the conditions that the potential and weight functions satisfied some conditions.It's worth noting that the above research and conclusions of the indefinite Sturm-Liouville problems are only under the limit-point type end-points instead of the limit-circle type endpoints.Hence,this dissertation will study the bounds and existence of non-real eigenvalues for singular indefinite Sturm-Liouville problem with limit-circle type non-oscillation endpoints firstly.Moreover,as to the one dimensional indefinite p-Laplacian eigenvalue problem,the evaluation of the upper bounds of non-real eigenvalues is obtained.As far as the researching methods in this part are concerned,this paper propose the hypothesis of the coefficient functions to study the non-real eigen-values of singular indefinite Sturm-Liouville problem and hence obtained the Sturm-Liouville differential equation is limit-circle type non-oscillation at the endpoints.The main ingredient and innovation of this part is the asymptotic behaviors of eigenfunctions at the end points and the equivalence of boundary conditions constructed by the principal solutions of Sturm-Liouville differential equation with different eigenvalue,which obtained with the aids of principal and non-principal solutions and measure theory.Then the upper bounds and the sufficient condition for the existence of non-real eigenvalues are obtained with the analysis theory,oscillate theory,comparison theorem and the PT symmetry conditions in quantum mechanics.Furthermore,for the one di-mensional indefinite p-Laplacian problem,the latest research methods dealing with the non-real eigenvalues of regular indefinite problems by Kikonko and Mingarelli in 2016 are applied to this one dimensional indefinite p-Laplacian eigenvalue problem.The bounded variation function and Ganelius lemma as well as the Sobolev theory and pure analysis methods are used as tools to evaluate a priori upper bounds on the imaginary part and the absolute values of the non-real eigenvalues.Secondly,this dissertation will consider the eigenvalues problems of nonlo-cal indefinite Sturm-Liouville differential equation with a nonlocal point inter-ference potential function in quantum mechanics and equipped with nonlocal boundary condition.For the nonlocal differential equation,the research has been started in the 1960s and have been widely studied in reaction-diffusion processes,nonlocal mechanics and quantum physics.As a matter of fact,the effect of nonlocal problems are more pronounced in natural ecosystems.For example,the movement of biological populations is not on a small scale but on a large scale.In recent years,nonlocal problems gradually begin to emerge in bionomics,fracture mechanics,peridynarmics theory,machine learning,medi-cal image theory since the diversity and complexity of species population and ecological dynamic systems.In particular,the Laplacian operator with frac-tional can be regarded as a spectral form of nonlocal diffusion model.In 2011,Qi and Chen have been studied the fractional problems from nonlocal con-tinuum mechanics.The Lyapunov inequality,eigenvalues and corresponding extremal problems were studied in 2019.Albeverio,Hryniv,Nizhnik in 2007 and Nizhnik in 2009 and 2010 studied the inverse spectral problems for this nonlocal problem.However,there has been no spectral theory on the nonlocal indefinite problems as perfect as that of Sturm-Liouville operators.Therefore,the left-definiteness and non-left-definiteness,the upper and lower bounds,the finiteness and the non-existence of non-real eigenvalues for the nonlocal regular and singular indefinite Sturm-Liouville problems axe considered in this part.As far as the research methods are concerned,the general research meth-ods of the operator theory are not work for the nonlocal problems because of the Sturm-Liouville differential equation and boundary condition contain nonlocal terms.So the theory of quadratic form of operator are applied to the research of spectral theory for the nonlocal indefinite problems and this is also an innovation in the study of nonlocal indefinite problems.Then the left-definiteness and non-left-definiteness for the nonlocal Sturm-Liouville problems are obtained under the perturbation of nonlocal term provided by Green func-tion,perturbation theory and the classical Sturm-Liouville theory.What's more,the methods on the finiteness of non-real eigenvalues for the regular indefinite problems by Mingarelli in 1982 and Zettl in 2005 are applied to the non-local indefinite problems and the finiteness of non-real eigenvalues for this problem are given.Furthermore,the methods dealing with the regu-lar indefinite Sturm-Liouville problems are applied to the nonlocal indefinite Sturm-Liouville boundary value problems.The evaluation of the upper and lower bounds on the imaginary and real parts as well as the absolute values of the non-real eigenvalues are obtained with the nonlocal separation and cou-pled boundary conditions.Finally,a priori upper bounds and the sufficient condition for the non-existence of non-real eigenvalues for this nonlocal sin-gular indefinite problems are obtained by optimizing the above methods of the singular indefinite problems and the latest methods of singular indefinite problems by Behrndt,Schmitz and Trunkin in 2009.This dissertation is divided into five chapters.The first chapter is the introduction part,where the the background,the present research and main conclusions are presented.The second and third chapters are include in the first half part of this paper.In Chapter 2,a priori upper bounds and the sufficient condition for the existence of non-real eigenvalues for the singular in-definite Sturm-Liouville problem are presented.The estimation on the upper bounds of non-real eigenvalues for the one dimensional indefinite p-Laplacian problem is given in Chapter 3.The second half consists of the forth and fifth chapters.Chapter 4 provides the change rule of(non-)left-definiteness,the upper and lower bounds and the finiteness of non-real eigenvalues for the non-local regular indefinite Sturm-Liouville problem.In the last chapter,the upper bounds evaluation as well as the sufficient condition for the non-existence of non-real eigenvalues for the nonlocal singular indefinite Sturm-Liouville prob-lem are presented.