几类内部具有不连续性的高阶微分算子的自共轭性与耗散性及其谱分析
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摘要
近年来内部具有不连续性的微分算子问题,微分方程与边界条件中带特征参数的微分算子引起了越来越多的数学、物理工作者的关注.许多实际的物理问题都可以转化为内部具有不连续性的微分算子问题,在工程技术领域中,一些偏微分方程经过分离变量法可以转化为边界条件中带特征参数的微分算子问题,且有些问题需要转换为高阶的来进行处理,因此对具有转移条件及边界条件带特征参数的高阶微分算子的自共轭性及其谱的研究是非常重要的.耗散算子是算子理论中非常重要的一类非自共轭算子,也有着很强的应用背景.本文主要研究了内部具有不连续性及边界条件带特征参数的四阶与高阶微分算子的自共轭性与谱分析,以及内部具有不连续性的四阶耗散算子的特征函数与相伴函数的完备性问题,其中内部的不连续性由问题中的转移条件来刻画.
文章首先研究了内部具有不连续性的高阶微分算子,研究工作包括两部分.第一部分研究了具有转移条件2n阶微分算子的自共轭性,当边界条件与转移条件的系数均由矩阵给定,且满足一定条件时,根据自共轭微分算子的定义,利用矩阵表示的方法将问题简化,进而证明算子是自共轭的,其全部特征值都是实的,对应于不同特征值的特征函数是正交的.第二部分研究了具有转移条件的2n阶微分算子自共轭的充要条件,给出一般情形的边界条件与转移条件,且其系数矩阵都是复矩阵.将此问题放在一个与转移条件相关的Hilbert空间中进行处理,利用微分算子的一般理论,得到了这类算子为自共轭的充要条件,要求转移条件的系数矩阵行列式值相同(不等于零).
其次,我们研究了一类在工程技术领域中有着广泛应用的边界条件带特征参数且内部具有不连续性的四阶微分算子问题.当算子的两个边界条件带特征参数时,在适当的Hilbert空问中定义一个与特征参数相关的线性算子,使得所考虑的依赖于特征参数的不连续四阶微分算子与此算子的特征值相同,即把问题转化为研究这个新的Hilbert空间中算子特征值与特征函数的问题,证明了算子是自共轭的,所有的特征值都是实的,且对应于不同特征值的特征函数在对应内积的意义下是正交的；对于四个边界条件都带特征参数的不连续四阶微分算子,给定转移条件及一般化的带特征参数的边界条件,利用边界条件与转移条件的系数构造两个四阶矩阵,利用自共轭算子的一般理论得到算子是自共轭的充要条件,并通过构造微分方程的基本解,得到确定特征值的整函数,并证明结论：若复数λ是算子的特征值当且仅当此整函数等于零；证明了算子仅有点谱,进而得到算子的格林函数,且此格林函数与通常边界条件情形下的格林函数是不同的.
然后,我们研究了一类2n阶微分算子,具有转移条件,边界条件,及其n个带特征参数的边界条件,我们巧妙的利用带特征参数边界条件的系数构造出n个二阶行列式,并由此二阶行列式的值定义算子的一个新内积,进而定义与特征参数相关的算子A,将问题转化为研究算子A的特征值问题.在转移条件的矩阵满足一定条件的情形下证明了算子A是自共轭的,得到判断特征值的整函数,且有结论：所研究问题的特征值恰好是整函数detΦ(1,λ)的零点；最后证明算子A只有点谱.
最后,我们研究了一类不连续的四阶耗散算子A,给定边界条件与转移条件,其中正则端点a的边界条件是通常的情形,而奇异端点b的边界条件要求满足一定的条件,利用耗散算子的定义,在所构造的具有特殊内积的Hilbert空间中证明算子A是耗散算子,且没有实的特征值；进而得到确定算子特征值的整函数△(λ)；为了得到算子A的逆算子,我们通过计算确定出算子A的格林函数,且证明零不是算子A的特征值；最后利用Livsic定理证明了算子A的特征函数与相伴函数在H中是的完备的,且有无穷多个特征值.
全文共分为七部分：一、本文所研究问题的背景与本文的主要结果；二、具有转移条件高阶微分算子的自共轭性问题；三、具有转移条件高阶微分算子自共轭的充要条件；四、具有转移条件及两个边界条件带特征参数的四阶微分算子的自共轭性问题；五、具有转移条件及四个边界条件带特征参数的四阶微分算子自共轭的充要条件及其特征函数的完备性；六、具有转移条件及边界条件带特征参数的2n阶微分算子的自共轭性及其特征函数的完备性；七、具有转移条件的四阶耗散算子及其特征函数与相伴函数的完备性.
In recent years, more and more mathematical and physical researchers pay attention to a class of differential operators with discontinuity in the interior point, and the differential operator problem for the differential equation and boundary conditions with eigenparameter. A lot of actual physical problem can be transformed for the interior discontinuity differ-ential operator problems, in the fields of engineering and technology, some partial differential equations by the method of separation of variables can be transformed into the boundary conditions with eigenparameter of the differential operator problem, and some problem need to be converted to high order case to handle, so it is very important to investigate the self-adjointness and spectrum of the high order differential operators with transmission conditions and eigenparameter boundary conditions. Dissi-pative operator is a class of very important nonself-adjoint operator in operator theory, and also has a strong application background. In this paper, we investigate the self-adjointness and spectrum analysis for the fourth-order and high order differential operators with discontinuity in the interior point and with eigenparameter-dependent boundary conditions, and the completeness of eigenfunctions and associated functions for dis-continuous fourth-order dissipative operator, the interior discontinuity is characterized by the transmission conditions of the problem.
First, we investigate a class of high order differential operator with discontinuity at an interior point in the interval, it contains two parts. In the first part, we investigate the self-adjointness of2nth order differential operator with transmission conditions, while the coefficients of boundary conditions and transmission conditions are determined by matrix, and sat-isfy certain conditions, by the definition of self-adjoint differential operator, and by matrix representation to simplify the problem, we prove the opera-tor is self-adjoint, and all eigenvalues are real, to the different eigenvalues, the corresponding eigenfunctions are orthogonal. In the second part, we investigate the necessary and sufficient conditions of the self-adjointness for2nth order differential operator with transmission conditions, give the gen-eral boundary conditions and transmission conditions, and the coefficient matrix are complex. We deal with them in a Hilbert space associated with the transmission conditions, by the general theory of differential operator, we obtain the necessary and sufficient conditions of self-adjointness, and require the values of the determinants are equal to(not equal to zero).
Second, we investigate a class of discontinuous fourth order differ-ential operator with eigenparameter-dependent boundary conditions prob-lem which has a wide range of applications in the fields of engineering and technology. While two boundary conditions with eigenparameter, in a suitable Hilbert space, we define a linear operator associated with eigen-parameter, such that the eigenvalue of this problem is equal to the discon-tinuous fourth-order differential operator with eigenparameter-dependent problems, i.e. we transform this problem to study the eigenvalues and eigenfunctions in this new Hilbert space, we prove the operator is self- adjoint, all eigenvalues are real, to the different eigenvalues, the corre-sponding eigenfunctions are orthogonal in the sense of corresponding inner product; to the discontinuous fourth order differential operator with four eigenparameter-dependent boundary conditions, we give the general trans-mission conditions and eigenparameter-dependent boundary conditions, by the coefficients of boundary conditions and transmission conditions we con-struct two fourth order matrix, by the general theory of self-adjoint oper-ator, we obtain the necessary and sufficient conditions of self-adjointness, then we construct basic solutions, and obtain the entire function which concern the eigenvalues, and have the conclusion:the complex number λ is an eigenvalue of the operator if and only if the entire function is equal to zero; then, we prove the operator has only point spectrum, and obtain the Green's function of the operator, and it is different from the case of general boundary conditions.
Then, we investigate a class of2nth-order differential operator with transmission conditions, boundary conditions and n eigenparameter-dependent boundary conditions, in the instances of coefficients of eigenparameter-dependent boundary conditions, we ingeniously construct n second-order determinants, and define inner product by the values of these second-order determinants, and define the operator associated with eigenparame-ter, transform the problem into investigate the eigenvalues of operator A, while the boundary conditions and transmission conditions satisfy some conditions, we prove the self-adjointness of the operator, obtain the entire function which concern the eigenvalues and we have the conclusion:the eigenvalues of the problem coincide with the zeros of the entire function detΦ(1,λ), and last we obtain the operator has only point spectrum.
In the last part of this paper, we investigate a class of discontinu-ous fourth-order dissipative operator, we give the boundary conditions and transmission conditions, where the regular point a satisfy general boundary conditions, the coefficients of boundary conditions for the singular point b satisfy certain conditions, by definition of dissipative operator, we prove this operator are dissipative in the Hilbert space with special inner prod-uct, and have no real eigenvalues; then we obtain the entire function Δ(λ) which concerned the eigenvalue; to obtain the inverse of operator A, we determine the Green's function by calculation, and prove zero is not the eigenvalue of A; in the last, by the Livsic theorem, we prove the eigenfunc-tions and associated functions of operator A are complete in the space H, and has infinitely many eigenvalues.
This paper contains seven parts. The first part:an introduction of the background we investigate and main results we obtain in this paper. The second part:the self-adjointness of high order differential operator with transmission conditions. The third part:the necessary and sufficient conditions of self-adjoint high order differential operator with transmission conditions. The fourth part:the self-adjointness of fourth order differential operator with transmission conditions and two eigenparameter-dependent boundary conditions. The fifth part:the the necessary and sufficient con-ditions and completeness of eigenfunction for fourth order differential op-erator with transmission conditions and four eigenparameter-dependent boundary conditions. The sixth part:the self-adjointness and complete-ness of eigenfunctions for2nth-order differential operator with transmission conditions and eigenparameter-dependent boundary conditions. The last part:the discontinuous fourth order dissipative operator and the complete-ness of the eigenfunctions and associated functions.
文章首先研究了内部具有不连续性的高阶微分算子,研究工作包括两部分.第一部分研究了具有转移条件2n阶微分算子的自共轭性,当边界条件与转移条件的系数均由矩阵给定,且满足一定条件时,根据自共轭微分算子的定义,利用矩阵表示的方法将问题简化,进而证明算子是自共轭的,其全部特征值都是实的,对应于不同特征值的特征函数是正交的.第二部分研究了具有转移条件的2n阶微分算子自共轭的充要条件,给出一般情形的边界条件与转移条件,且其系数矩阵都是复矩阵.将此问题放在一个与转移条件相关的Hilbert空间中进行处理,利用微分算子的一般理论,得到了这类算子为自共轭的充要条件,要求转移条件的系数矩阵行列式值相同(不等于零).
其次,我们研究了一类在工程技术领域中有着广泛应用的边界条件带特征参数且内部具有不连续性的四阶微分算子问题.当算子的两个边界条件带特征参数时,在适当的Hilbert空问中定义一个与特征参数相关的线性算子,使得所考虑的依赖于特征参数的不连续四阶微分算子与此算子的特征值相同,即把问题转化为研究这个新的Hilbert空间中算子特征值与特征函数的问题,证明了算子是自共轭的,所有的特征值都是实的,且对应于不同特征值的特征函数在对应内积的意义下是正交的；对于四个边界条件都带特征参数的不连续四阶微分算子,给定转移条件及一般化的带特征参数的边界条件,利用边界条件与转移条件的系数构造两个四阶矩阵,利用自共轭算子的一般理论得到算子是自共轭的充要条件,并通过构造微分方程的基本解,得到确定特征值的整函数,并证明结论：若复数λ是算子的特征值当且仅当此整函数等于零；证明了算子仅有点谱,进而得到算子的格林函数,且此格林函数与通常边界条件情形下的格林函数是不同的.
然后,我们研究了一类2n阶微分算子,具有转移条件,边界条件,及其n个带特征参数的边界条件,我们巧妙的利用带特征参数边界条件的系数构造出n个二阶行列式,并由此二阶行列式的值定义算子的一个新内积,进而定义与特征参数相关的算子A,将问题转化为研究算子A的特征值问题.在转移条件的矩阵满足一定条件的情形下证明了算子A是自共轭的,得到判断特征值的整函数,且有结论：所研究问题的特征值恰好是整函数detΦ(1,λ)的零点；最后证明算子A只有点谱.
最后,我们研究了一类不连续的四阶耗散算子A,给定边界条件与转移条件,其中正则端点a的边界条件是通常的情形,而奇异端点b的边界条件要求满足一定的条件,利用耗散算子的定义,在所构造的具有特殊内积的Hilbert空间中证明算子A是耗散算子,且没有实的特征值；进而得到确定算子特征值的整函数△(λ)；为了得到算子A的逆算子,我们通过计算确定出算子A的格林函数,且证明零不是算子A的特征值；最后利用Livsic定理证明了算子A的特征函数与相伴函数在H中是的完备的,且有无穷多个特征值.
全文共分为七部分：一、本文所研究问题的背景与本文的主要结果；二、具有转移条件高阶微分算子的自共轭性问题；三、具有转移条件高阶微分算子自共轭的充要条件；四、具有转移条件及两个边界条件带特征参数的四阶微分算子的自共轭性问题；五、具有转移条件及四个边界条件带特征参数的四阶微分算子自共轭的充要条件及其特征函数的完备性；六、具有转移条件及边界条件带特征参数的2n阶微分算子的自共轭性及其特征函数的完备性；七、具有转移条件的四阶耗散算子及其特征函数与相伴函数的完备性.
In recent years, more and more mathematical and physical researchers pay attention to a class of differential operators with discontinuity in the interior point, and the differential operator problem for the differential equation and boundary conditions with eigenparameter. A lot of actual physical problem can be transformed for the interior discontinuity differ-ential operator problems, in the fields of engineering and technology, some partial differential equations by the method of separation of variables can be transformed into the boundary conditions with eigenparameter of the differential operator problem, and some problem need to be converted to high order case to handle, so it is very important to investigate the self-adjointness and spectrum of the high order differential operators with transmission conditions and eigenparameter boundary conditions. Dissi-pative operator is a class of very important nonself-adjoint operator in operator theory, and also has a strong application background. In this paper, we investigate the self-adjointness and spectrum analysis for the fourth-order and high order differential operators with discontinuity in the interior point and with eigenparameter-dependent boundary conditions, and the completeness of eigenfunctions and associated functions for dis-continuous fourth-order dissipative operator, the interior discontinuity is characterized by the transmission conditions of the problem.
First, we investigate a class of high order differential operator with discontinuity at an interior point in the interval, it contains two parts. In the first part, we investigate the self-adjointness of2nth order differential operator with transmission conditions, while the coefficients of boundary conditions and transmission conditions are determined by matrix, and sat-isfy certain conditions, by the definition of self-adjoint differential operator, and by matrix representation to simplify the problem, we prove the opera-tor is self-adjoint, and all eigenvalues are real, to the different eigenvalues, the corresponding eigenfunctions are orthogonal. In the second part, we investigate the necessary and sufficient conditions of the self-adjointness for2nth order differential operator with transmission conditions, give the gen-eral boundary conditions and transmission conditions, and the coefficient matrix are complex. We deal with them in a Hilbert space associated with the transmission conditions, by the general theory of differential operator, we obtain the necessary and sufficient conditions of self-adjointness, and require the values of the determinants are equal to(not equal to zero).
Second, we investigate a class of discontinuous fourth order differ-ential operator with eigenparameter-dependent boundary conditions prob-lem which has a wide range of applications in the fields of engineering and technology. While two boundary conditions with eigenparameter, in a suitable Hilbert space, we define a linear operator associated with eigen-parameter, such that the eigenvalue of this problem is equal to the discon-tinuous fourth-order differential operator with eigenparameter-dependent problems, i.e. we transform this problem to study the eigenvalues and eigenfunctions in this new Hilbert space, we prove the operator is self- adjoint, all eigenvalues are real, to the different eigenvalues, the corre-sponding eigenfunctions are orthogonal in the sense of corresponding inner product; to the discontinuous fourth order differential operator with four eigenparameter-dependent boundary conditions, we give the general trans-mission conditions and eigenparameter-dependent boundary conditions, by the coefficients of boundary conditions and transmission conditions we con-struct two fourth order matrix, by the general theory of self-adjoint oper-ator, we obtain the necessary and sufficient conditions of self-adjointness, then we construct basic solutions, and obtain the entire function which concern the eigenvalues, and have the conclusion:the complex number λ is an eigenvalue of the operator if and only if the entire function is equal to zero; then, we prove the operator has only point spectrum, and obtain the Green's function of the operator, and it is different from the case of general boundary conditions.
Then, we investigate a class of2nth-order differential operator with transmission conditions, boundary conditions and n eigenparameter-dependent boundary conditions, in the instances of coefficients of eigenparameter-dependent boundary conditions, we ingeniously construct n second-order determinants, and define inner product by the values of these second-order determinants, and define the operator associated with eigenparame-ter, transform the problem into investigate the eigenvalues of operator A, while the boundary conditions and transmission conditions satisfy some conditions, we prove the self-adjointness of the operator, obtain the entire function which concern the eigenvalues and we have the conclusion:the eigenvalues of the problem coincide with the zeros of the entire function detΦ(1,λ), and last we obtain the operator has only point spectrum.
In the last part of this paper, we investigate a class of discontinu-ous fourth-order dissipative operator, we give the boundary conditions and transmission conditions, where the regular point a satisfy general boundary conditions, the coefficients of boundary conditions for the singular point b satisfy certain conditions, by definition of dissipative operator, we prove this operator are dissipative in the Hilbert space with special inner prod-uct, and have no real eigenvalues; then we obtain the entire function Δ(λ) which concerned the eigenvalue; to obtain the inverse of operator A, we determine the Green's function by calculation, and prove zero is not the eigenvalue of A; in the last, by the Livsic theorem, we prove the eigenfunc-tions and associated functions of operator A are complete in the space H, and has infinitely many eigenvalues.
This paper contains seven parts. The first part:an introduction of the background we investigate and main results we obtain in this paper. The second part:the self-adjointness of high order differential operator with transmission conditions. The third part:the necessary and sufficient conditions of self-adjoint high order differential operator with transmission conditions. The fourth part:the self-adjointness of fourth order differential operator with transmission conditions and two eigenparameter-dependent boundary conditions. The fifth part:the the necessary and sufficient con-ditions and completeness of eigenfunction for fourth order differential op-erator with transmission conditions and four eigenparameter-dependent boundary conditions. The sixth part:the self-adjointness and complete-ness of eigenfunctions for2nth-order differential operator with transmission conditions and eigenparameter-dependent boundary conditions. The last part:the discontinuous fourth order dissipative operator and the complete-ness of the eigenfunctions and associated functions.
引文
[1]安建业,孙炯,On the self-adjointness of the product operators of two mth-order differential operators on [0, ∞), Acta Math. Sincia, English Series,2004,20(5):793-802.
[2]B.P. Allahverdiev, A nonself-adjoint singular Sturm-Liouville problem with a spectral parameter in the boundary condition, Math.Nach.,2005,278(7-8):743-755.
[3]B.P. Allahverdiev, A dissipative singular Sturm-Liouville problem with a spectral parameter in the boundary condition, J. Math. Anal.,2006,316:510-524.
[4]B.P. Allahverdiev, Dissipative Sturm-Liouville operators in limit-point case, Acta Appli-candae Mathematicae,2005,86:237-248.
[5]B.P. Allahverdiev, Dissipative Sturm-Liouville operators with nonseparated boundary con-ditions, Monatsh. Math.,2003,140:1-17.
[6]B.P. Allahverdiev, A. Canoglu, Spectral analysis of dissipative Schrodinger operators. Proc. Royal Soc. Edinburgh A.,1997,127:1113-1121.
[7]Altnisikn, M. Kadakal, Eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problems with eigenparameter-dependent boundary conditions, Acta Math Hungarica, 2004,102:159-175.
[8]G.D. Birkhoff, On the asymptotic character of solution of the certain linear differential equations containting parameter, Trans. Amer, Soc.,1908,9:219-231.
[9]P.A. Binding, P.J. Browne, B.A. Watson, Sturm-Liouville problems with boundary condi-tions rationally dependent on the eigenparameter II, Journal of Computational and Applied Mathematics,2002,148:147-169.
[10]P.A. Binding, P.J. Browne, B.A. Watson, Transformations between Sturm-Liouville prob-lems with eigenvalue dependent and independent boundary conditions, Bull London Math Soc,2001,33:749-757.
[11]E. Bairamov, A.M. Krall, Dissipative operators generated by the Sturm-Liouville differen-tial expression in the Weyl limit circle case, J. Math. Anal. Appl.,2001,254:178-190.
[12]E. Bairamov, E. Ugurlu, The determinants of dissipative Sturm-Liouville operators with transmission conditions, Math. and Computer Modelling,2011,53:805-813.
[13]D. Buschmann, G. Stolz, J. Weidmann, One-dimensional Schrodinger operators with local point interactions. Reine Angew Math,1995,467:169-186.
[14]E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York,1955.
[15]曹之江,孙炯,D. E. Edmunds, On self-adjointness of the product of two 2-order differential operators, Acta Math. Sincia, English Series,1999,15(3):375-386.
[16]曹之江,孙炯,D.E. Edmunds,二阶微分算子积的自伴性,数学学报,1999,42(4)：649-654.
[17]曹之江,常微分算子,上海科技出版社,上海,1987.
[18]曹之江,自伴常微分算子的解析描述,内蒙古大学学报(自然科学版),1987,18(3)：393-401.
[19]曹之江,孙烔,微分算子文集,内蒙古大学出版社,呼和浩特,1992.
[20]W. J. Code. Sturm-Liouville problems with eigenparameter dependent boundary conditions, Saskatoon:University of Saskatchewan,2003.
[21]M.G. Crandail, R.S. Philips, On the extension problem for dissipative operators, Journal of Functional Analysis,1968,2:147-176.
[22]陈金设,微分算子特征值的一种数值解法与对称算子自共轭扩张的边值空间理论(博士学位论文),内蒙古大学,2009.
[23]M. Demirci, Z. Akdogan, O. Mukhtarov, Asymptotic Behavior of Eigenvalues and Eigen-functions of one Discontinuous Boundary-value Problem. International Journal of Compu-tational Cognition,2004,2(3):101-113.
[24]N. Dunford and J.T. Schwartz, Linear Operators, v. II, Wiley, New York,1963.
[25]D.E. Edmunds, W.D. Evans, Spectral theory and differentail operators, Oxford University Press, Oxford.1987.
[26]W.D. Evans, E.I. Sobhy, Boundary conditions for general ordinary differential operators and their adjoints. Proceedings of the Royal Society of Edinburgh,1990,114 A:99-117.
[27]W.N. Everitt, V. K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric differ-ential expressions I:the general theory. Nieuw Archief voor Wiskunde,1976,34(3):1-48.
[28]W.N. Everitt, V.K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric dif-ferential expressions II:The odd order case, Nieuw Archief voor Wiskunde,1976,34(3): 109-145.
[29]W.N. Everitt, I.W. Knowles, T.T. Read, Limit-point and limit-circle criteria for Sturm-Liouville equations with intermittently negative principal coefficients, Proc. Roy. Soc. Ed-inburgh Sect.A,1986,103:215-228.
[30]W.N. Everitt, L. Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators. Mathematical Surveys and Monographs 61. American Math. Soc.,1999.
[31]W.N. Everitt, L. Markus, Complex symplectic geometry with applications to ordinary differential operators. Transactions of the American Mathematical Society,1999,351(12): 4905-4945.
[32]C.T. Fulton, Parametrization of Titchmarsh's m(A)-functions in the limit circle case, Trans. Amer. Math. Soc.,1977,229:51-63.
[33]C.T. Fulton, Two-point boundary value problems with parameter contained in the bound-ary conditions. Proc. Roy. Soc. Edinburgh A,1977,77:293-308.
[34]C.T. Fulton, S. Pruess, Numerical methods for a singular eigenvalue problem with eigen-parameter in the boundary conditions, J. Math. Anal. Appl.,1979,71:431-462.
[35]M.G. Gasymov, G.Sh. Guseinov, Uniqueness theorems for inverse spectral analysis problems for Sturm-Liouville operators in the Weyl limit-circle case, Differ. Equ.,1989,25:394-402.
[36]I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operator, in:Transl. Math. Monographs, vol.18, American Math. Soc., Providence, RI,1969.
[37]G.Sh. Guseinov, H. Tuncay, The determinants of perturbation connected with a dissipative Sturm-Liouville operators, J. Math. Anal. Appl.,1995,194:39-49.
[38]I.M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Israel Program for Scientific Translations,1965.
[39]D.B. Hinton, An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary conditions, Quart J. Math., Oxford,1979,30:33-42.
[40]B. Helffer, Semi-classical analysis for the Schrodinger operators and applications, Lecture Note in Math.,1336, Springer-Verlag,1988.
[41]郝晓玲,微分算子实参数平方可积解的个数与谱的定性分析(博士学位论文),内蒙古大学,2010.
[42]A. Kuzhel, Characteristic Functions andModels of Nonselfadjoint Operators, Kluwer Aca-demic, Dordrecht,1996.
[43]M. Kadakal, O.Sh. Mukhtarov, Discontinuous Sturm-Liouville problems containing eigen-parameter in the boundary conditions, Acta Math. Sinica,2006,22:1519-1528.
[44]M. Kadakal,O.Sh. Mukhtarov, F.S. Muhtarov, Some spectral problems of Sturm-Liouville problem with transmission conditions, Math. Meth. Appl. Sci.,2007,30:1719-1738.
[45]M. Kobayashi, Eigenvalues of discontinuous Sturm-Liouville problems with symmmetric potentials, Computers Math Applic,1989,18(4):357-364.
[46]M. Kadakal, O.Sh. Mukhtarov, Sturm-Liouville problems with discontinuities at two points, Comput. Math. Appl.,2007,54:1367-1379.
[47]M. V. Keldysh, On the completeness of the eigenfunctions of some classes of non self-adjoint linear operators, Soviet Math. Dokl.77 (1951)11-14.
[48]Y. Kamimura, A criterion for the complete continuity of a 2n-th order differential operator with complex coefficients, Proc. Royal Soc. Edinburgh A,1990,116:161-176.
[49]A.M. Krall, A. Zettl, Singular self-adjoint Sturm-Liouville problems II:interior singular points, SIAM J. Math. Anal.,1988,19:1135-1141.
[50]I. Knowles, Dissipative Sturm-Liouville operators, Proc. Roy. Soc. Edinburgh Sect. A, 1981,88:329-343.
[51]I. Knowles, On the boundary conditions characterizing J-selfadjoint extensions of J-symmetric operators, J. Differential Equations,1981,40:193-216.
[52]M.G. Krein, On the indeterminant case the Sturm-Liouville boundary problems in the interval (0,+∞), Izv. Akad. Nuak SSSR Ser. Mat.,1952,16:293-324.
[53]A.V. Likov, Yu A. Mikhalilov, The Theory of Heat and Mass Transfer Qosenergaizdat, 1963 (Russian).
[54]V.A. Marchenko, Expansion in eigenfunctions of non-selfadjoint singular second order dif-ferential operators, Amer. Math. Soc. Transl.(Ser.2),1963,25:77-130.
[55]M. Moller, B. Zinsou, Spectral asymptotics of self-adjoint fourth order Differential operators with eigenvalue parameter dependent boundary conditions, Complex Analysis and Operator Theory,2012(6):799-818.
[56]M.A. Naimark, Linear Differential Operators, Ungar, New York,1968.
[57]V.A. Naimarko, Expansion in eigenfunctions of non-self-adjoint singular second order dif-ferential operators, Amer. Math. Soc. Transl. Ser.,1963,225:77-130.
[58]B.S. Pavlov, Spectral analysis of a dissipative singular Schroodinger operator in terms of a functional model, Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fundam. Napravleniya,1991, 65:95-163.(in Russian); English transl. in:Partial Differential Equations 8, Encyc. Math. Sci.,1996,65:87-163.
[59]G. Pontrelli, D.F. Monte, Mass diffusion through two-layer porous media:an application to the drug-eluting stent, International Journal of Heat and Mass Transfer,2007,50:3658-3669.
[60]R.S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Transactions of the American Mathematical Society,1959 90(2):193-254.
[61]D. Race, On the location of the essential spectra and regularity fields of complex Sturm-Liouville operators, Proc. royal Soc. Edinburgh,1980,85A:1-14.
[62]D. Race, On the essential spectra of linear 2nth order differential operators with complex coefficients, Proc. Royal Soc. Edinburgh,1982,92A:65-74.
[63]A. Soffer, Multichannel nonlinear scattering for integrable equations. J. Differential Equa-tion,1992,98:376-390.
[64]A.R. Sims, Secondary conditions for linear differential operators of the second order, J. Math.,1957,6:247-285.
[65]A.A. Shkalikov, Boundary value problems for ordinary differential equations with a param-eter in boundary conditions, Trudy Sem. Petrovsk,1983,9:190-229(Russian).
[66]孙炯,安建业, On self-adjiontness of products of two differential operators on [a, b], Anal. Diff. Equations,1998,18(1):48-55.
[67]孙炯,On the self-adjoint extensions of symmetric ordinary differential operators with mid-dle deficiency indices, Acta Math. Sincia, New Series,1986,2(2):152-167.
[68]孙炯,王忠,线性算子的谱分析,北京：科学出版社,2005.
[69]孙炯,王爱平,A. Zettl, Continuous Spectrum and Square-Integrable Solutions of Differ-ential Operators with Intermediate Deficiency Index, Journal of Functional Analysis,2008, 255:3229-3248.
[70]孙炯,王爱平,A. Zettl, Two-interval Sturm-Liouville operators in direct sum spaces with inner product multiples, Results Math.,2007,50:155-168.
[71]孙炯,王爱平,Sturm-Liouville operators with interface conditions, The Progress of Re-search for Math., Mech., Phy. and High New Tech., Science Press, Beijing,2008,12:513-516.
[72]孙炯,王万义,微分算子的自共轭域和谱分析-微分算子研究在内蒙古大学三十年,内蒙古大学学报(自然科学版),2009,40(4)：469-485.
[73]索建青,两区间微分算子自伴域的实参数解刻画及谱的离散性(博士学位论文),内蒙古大学,2012.
[74]E.C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford,1946.
[75]E. Tune, O.Sh. Mukhtarov, Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions, Appl. Math. Comput.,2004,157:347-355.
[76]E.C. Titchmarsh, Eigenfunction expansions associated with second-order differential equa-tions, Part I, second edition, Oxford,1962.
[77]A.N. Tikhonov, A.A. Samarskii, Equations of Mathematical Physics, Dover Publications Inc., New York,1990.
[78]N.N. Voitovich, B.Z. Katsenelbaum, A.N. Sivov, Generalized method of eigenvibration in the theory of diffraction, Nauka, Moskov,1997(Russian).
[79]王爱平,孙炯,郝晓玲,姚斯琴,Completeness of eigenfunctions of Sturm-Liouville prob-lems with transmission conditions, Methods and Applications of Analysis,2009,16(3): 299-312.
[80]王爱平,孙炯,A. Zettl, Characterization of domains of self-Adjoint ordinary differential operators, Journal of Differential Equations,2009,246:1600-1622.
[81]王爱平,孙炯,A.Zettl, The classification of self-adjoint boundary conditions:Separated, coupled and mixed, Journal of Functional Analysis,2008,255:1554-1573.
[82]王爱平,孙炯,A.Zettl, The classification of self-adjoint boundary conditions of differential operators with two singular endpoints, J. Math. Anal. Appl.,2011,378:493-506.
[83]王爱平,关于Weidmann猜想及具有转移条件微分算子的研究(博士学位论文),内蒙古大学,2006.
[84]H. Weyl, Uber gewohnliche differentialgleichungen mit singularityten und kiezugehorigen entwicklungen willkurlicher funktionen, Math. Ann.,1910,68:220-269.
[85]J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary condi-tion, Math. Z.,1973,133:301-312.
[86]J. Weidmann, Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, Springer-Verlag, Berlin,1980.
[87]J. Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathe-matics 1258, Springer-Verlag, Berlin,1987.
[88]王万义,微分算子的辛结构与一类微分算子的谱分析(博士学位论文),内蒙古大学,2002.
[89]王万义,孙炯,高阶常型微分算子自伴域的辛几何刻画,应用数学,2003,16(1)：17-22.
[90]王万义,孙炯,Complex J-symplectic geometry characterization for J-symmetric extensions of J-symmetric differential operators, Advances in Mathematics,2003,32(4):481-484.
[91]王忠,H. Wu, The completeness of eigenfunctions and perturbation connected with Sturm-Liouville operators. J. System and Complexity,2006,19:527-537.
[92]王忠,H. Wu, Dissipative non-self-adjoint Sturm-Liouville operators and completeness of their eigenfunctions, J. Math. Anal. Appl.,2012,394:1-12.
[93]王志敬,宋岱才,直和空间上对称微分算子自共轭域的辛几何刻画(Ⅰ),石油化工高等学校学报,2008,21(1)：92-95.
[94]杨传富,杨孝平,黄振友,m个微分算式乘积的自伴边界条件,数学年刊2006,27A(3)：313324.
[95]杨传富,黄振友,杨孝平,2n阶微分算子乘积自伴的充分必要条件,数学物理学报,2006,26A(6)：953-962.
[96]杨传富,黄振友,杨孝平,微分算子的对称扩张及Friedrichs扩张的辛几何刻画,数学学报,2006,49(2)：421-430.
[97]杨传富,杨孝平,An interior inverse problem for Sturm-Liouville operator with discontin-uous conditions, Appl.Math.Letters,2009,22:1315-1319.
[98]S.Yakubov, Completeness of Root Functions of Regular Differential Operators. New York, Longman, Scientific Technical,1994.
[99]S. Yakubov, Y. Yakubov, Differential Operator Equations (Ordinary and Partial Differen-tial Equations), Chapman and Hall/CRC, Boca Raton 2000.
[100]A. Zettl, Adjoint linear differential operators, Proc. Amer. Math. Soc.,1965,16:1239-1241.
[101]A. Zettl, Sturm-Liouville Theory, American Mathematical Society, Mathematical Surveys and Monographs,2005,121.
[102]A.Zettle, Sturm-Liouville Problem. Spectral Theory and Computatinal Methods of Sturm-Liouville Problem, Marcel Dekker, Berlin,1997,1-104.
[2]B.P. Allahverdiev, A nonself-adjoint singular Sturm-Liouville problem with a spectral parameter in the boundary condition, Math.Nach.,2005,278(7-8):743-755.
[3]B.P. Allahverdiev, A dissipative singular Sturm-Liouville problem with a spectral parameter in the boundary condition, J. Math. Anal.,2006,316:510-524.
[4]B.P. Allahverdiev, Dissipative Sturm-Liouville operators in limit-point case, Acta Appli-candae Mathematicae,2005,86:237-248.
[5]B.P. Allahverdiev, Dissipative Sturm-Liouville operators with nonseparated boundary con-ditions, Monatsh. Math.,2003,140:1-17.
[6]B.P. Allahverdiev, A. Canoglu, Spectral analysis of dissipative Schrodinger operators. Proc. Royal Soc. Edinburgh A.,1997,127:1113-1121.
[7]Altnisikn, M. Kadakal, Eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problems with eigenparameter-dependent boundary conditions, Acta Math Hungarica, 2004,102:159-175.
[8]G.D. Birkhoff, On the asymptotic character of solution of the certain linear differential equations containting parameter, Trans. Amer, Soc.,1908,9:219-231.
[9]P.A. Binding, P.J. Browne, B.A. Watson, Sturm-Liouville problems with boundary condi-tions rationally dependent on the eigenparameter II, Journal of Computational and Applied Mathematics,2002,148:147-169.
[10]P.A. Binding, P.J. Browne, B.A. Watson, Transformations between Sturm-Liouville prob-lems with eigenvalue dependent and independent boundary conditions, Bull London Math Soc,2001,33:749-757.
[11]E. Bairamov, A.M. Krall, Dissipative operators generated by the Sturm-Liouville differen-tial expression in the Weyl limit circle case, J. Math. Anal. Appl.,2001,254:178-190.
[12]E. Bairamov, E. Ugurlu, The determinants of dissipative Sturm-Liouville operators with transmission conditions, Math. and Computer Modelling,2011,53:805-813.
[13]D. Buschmann, G. Stolz, J. Weidmann, One-dimensional Schrodinger operators with local point interactions. Reine Angew Math,1995,467:169-186.
[14]E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York,1955.
[15]曹之江,孙炯,D. E. Edmunds, On self-adjointness of the product of two 2-order differential operators, Acta Math. Sincia, English Series,1999,15(3):375-386.
[16]曹之江,孙炯,D.E. Edmunds,二阶微分算子积的自伴性,数学学报,1999,42(4)：649-654.
[17]曹之江,常微分算子,上海科技出版社,上海,1987.
[18]曹之江,自伴常微分算子的解析描述,内蒙古大学学报(自然科学版),1987,18(3)：393-401.
[19]曹之江,孙烔,微分算子文集,内蒙古大学出版社,呼和浩特,1992.
[20]W. J. Code. Sturm-Liouville problems with eigenparameter dependent boundary conditions, Saskatoon:University of Saskatchewan,2003.
[21]M.G. Crandail, R.S. Philips, On the extension problem for dissipative operators, Journal of Functional Analysis,1968,2:147-176.
[22]陈金设,微分算子特征值的一种数值解法与对称算子自共轭扩张的边值空间理论(博士学位论文),内蒙古大学,2009.
[23]M. Demirci, Z. Akdogan, O. Mukhtarov, Asymptotic Behavior of Eigenvalues and Eigen-functions of one Discontinuous Boundary-value Problem. International Journal of Compu-tational Cognition,2004,2(3):101-113.
[24]N. Dunford and J.T. Schwartz, Linear Operators, v. II, Wiley, New York,1963.
[25]D.E. Edmunds, W.D. Evans, Spectral theory and differentail operators, Oxford University Press, Oxford.1987.
[26]W.D. Evans, E.I. Sobhy, Boundary conditions for general ordinary differential operators and their adjoints. Proceedings of the Royal Society of Edinburgh,1990,114 A:99-117.
[27]W.N. Everitt, V. K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric differ-ential expressions I:the general theory. Nieuw Archief voor Wiskunde,1976,34(3):1-48.
[28]W.N. Everitt, V.K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric dif-ferential expressions II:The odd order case, Nieuw Archief voor Wiskunde,1976,34(3): 109-145.
[29]W.N. Everitt, I.W. Knowles, T.T. Read, Limit-point and limit-circle criteria for Sturm-Liouville equations with intermittently negative principal coefficients, Proc. Roy. Soc. Ed-inburgh Sect.A,1986,103:215-228.
[30]W.N. Everitt, L. Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators. Mathematical Surveys and Monographs 61. American Math. Soc.,1999.
[31]W.N. Everitt, L. Markus, Complex symplectic geometry with applications to ordinary differential operators. Transactions of the American Mathematical Society,1999,351(12): 4905-4945.
[32]C.T. Fulton, Parametrization of Titchmarsh's m(A)-functions in the limit circle case, Trans. Amer. Math. Soc.,1977,229:51-63.
[33]C.T. Fulton, Two-point boundary value problems with parameter contained in the bound-ary conditions. Proc. Roy. Soc. Edinburgh A,1977,77:293-308.
[34]C.T. Fulton, S. Pruess, Numerical methods for a singular eigenvalue problem with eigen-parameter in the boundary conditions, J. Math. Anal. Appl.,1979,71:431-462.
[35]M.G. Gasymov, G.Sh. Guseinov, Uniqueness theorems for inverse spectral analysis problems for Sturm-Liouville operators in the Weyl limit-circle case, Differ. Equ.,1989,25:394-402.
[36]I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operator, in:Transl. Math. Monographs, vol.18, American Math. Soc., Providence, RI,1969.
[37]G.Sh. Guseinov, H. Tuncay, The determinants of perturbation connected with a dissipative Sturm-Liouville operators, J. Math. Anal. Appl.,1995,194:39-49.
[38]I.M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Israel Program for Scientific Translations,1965.
[39]D.B. Hinton, An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary conditions, Quart J. Math., Oxford,1979,30:33-42.
[40]B. Helffer, Semi-classical analysis for the Schrodinger operators and applications, Lecture Note in Math.,1336, Springer-Verlag,1988.
[41]郝晓玲,微分算子实参数平方可积解的个数与谱的定性分析(博士学位论文),内蒙古大学,2010.
[42]A. Kuzhel, Characteristic Functions andModels of Nonselfadjoint Operators, Kluwer Aca-demic, Dordrecht,1996.
[43]M. Kadakal, O.Sh. Mukhtarov, Discontinuous Sturm-Liouville problems containing eigen-parameter in the boundary conditions, Acta Math. Sinica,2006,22:1519-1528.
[44]M. Kadakal,O.Sh. Mukhtarov, F.S. Muhtarov, Some spectral problems of Sturm-Liouville problem with transmission conditions, Math. Meth. Appl. Sci.,2007,30:1719-1738.
[45]M. Kobayashi, Eigenvalues of discontinuous Sturm-Liouville problems with symmmetric potentials, Computers Math Applic,1989,18(4):357-364.
[46]M. Kadakal, O.Sh. Mukhtarov, Sturm-Liouville problems with discontinuities at two points, Comput. Math. Appl.,2007,54:1367-1379.
[47]M. V. Keldysh, On the completeness of the eigenfunctions of some classes of non self-adjoint linear operators, Soviet Math. Dokl.77 (1951)11-14.
[48]Y. Kamimura, A criterion for the complete continuity of a 2n-th order differential operator with complex coefficients, Proc. Royal Soc. Edinburgh A,1990,116:161-176.
[49]A.M. Krall, A. Zettl, Singular self-adjoint Sturm-Liouville problems II:interior singular points, SIAM J. Math. Anal.,1988,19:1135-1141.
[50]I. Knowles, Dissipative Sturm-Liouville operators, Proc. Roy. Soc. Edinburgh Sect. A, 1981,88:329-343.
[51]I. Knowles, On the boundary conditions characterizing J-selfadjoint extensions of J-symmetric operators, J. Differential Equations,1981,40:193-216.
[52]M.G. Krein, On the indeterminant case the Sturm-Liouville boundary problems in the interval (0,+∞), Izv. Akad. Nuak SSSR Ser. Mat.,1952,16:293-324.
[53]A.V. Likov, Yu A. Mikhalilov, The Theory of Heat and Mass Transfer Qosenergaizdat, 1963 (Russian).
[54]V.A. Marchenko, Expansion in eigenfunctions of non-selfadjoint singular second order dif-ferential operators, Amer. Math. Soc. Transl.(Ser.2),1963,25:77-130.
[55]M. Moller, B. Zinsou, Spectral asymptotics of self-adjoint fourth order Differential operators with eigenvalue parameter dependent boundary conditions, Complex Analysis and Operator Theory,2012(6):799-818.
[56]M.A. Naimark, Linear Differential Operators, Ungar, New York,1968.
[57]V.A. Naimarko, Expansion in eigenfunctions of non-self-adjoint singular second order dif-ferential operators, Amer. Math. Soc. Transl. Ser.,1963,225:77-130.
[58]B.S. Pavlov, Spectral analysis of a dissipative singular Schroodinger operator in terms of a functional model, Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fundam. Napravleniya,1991, 65:95-163.(in Russian); English transl. in:Partial Differential Equations 8, Encyc. Math. Sci.,1996,65:87-163.
[59]G. Pontrelli, D.F. Monte, Mass diffusion through two-layer porous media:an application to the drug-eluting stent, International Journal of Heat and Mass Transfer,2007,50:3658-3669.
[60]R.S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Transactions of the American Mathematical Society,1959 90(2):193-254.
[61]D. Race, On the location of the essential spectra and regularity fields of complex Sturm-Liouville operators, Proc. royal Soc. Edinburgh,1980,85A:1-14.
[62]D. Race, On the essential spectra of linear 2nth order differential operators with complex coefficients, Proc. Royal Soc. Edinburgh,1982,92A:65-74.
[63]A. Soffer, Multichannel nonlinear scattering for integrable equations. J. Differential Equa-tion,1992,98:376-390.
[64]A.R. Sims, Secondary conditions for linear differential operators of the second order, J. Math.,1957,6:247-285.
[65]A.A. Shkalikov, Boundary value problems for ordinary differential equations with a param-eter in boundary conditions, Trudy Sem. Petrovsk,1983,9:190-229(Russian).
[66]孙炯,安建业, On self-adjiontness of products of two differential operators on [a, b], Anal. Diff. Equations,1998,18(1):48-55.
[67]孙炯,On the self-adjoint extensions of symmetric ordinary differential operators with mid-dle deficiency indices, Acta Math. Sincia, New Series,1986,2(2):152-167.
[68]孙炯,王忠,线性算子的谱分析,北京：科学出版社,2005.
[69]孙炯,王爱平,A. Zettl, Continuous Spectrum and Square-Integrable Solutions of Differ-ential Operators with Intermediate Deficiency Index, Journal of Functional Analysis,2008, 255:3229-3248.
[70]孙炯,王爱平,A. Zettl, Two-interval Sturm-Liouville operators in direct sum spaces with inner product multiples, Results Math.,2007,50:155-168.
[71]孙炯,王爱平,Sturm-Liouville operators with interface conditions, The Progress of Re-search for Math., Mech., Phy. and High New Tech., Science Press, Beijing,2008,12:513-516.
[72]孙炯,王万义,微分算子的自共轭域和谱分析-微分算子研究在内蒙古大学三十年,内蒙古大学学报(自然科学版),2009,40(4)：469-485.
[73]索建青,两区间微分算子自伴域的实参数解刻画及谱的离散性(博士学位论文),内蒙古大学,2012.
[74]E.C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford,1946.
[75]E. Tune, O.Sh. Mukhtarov, Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions, Appl. Math. Comput.,2004,157:347-355.
[76]E.C. Titchmarsh, Eigenfunction expansions associated with second-order differential equa-tions, Part I, second edition, Oxford,1962.
[77]A.N. Tikhonov, A.A. Samarskii, Equations of Mathematical Physics, Dover Publications Inc., New York,1990.
[78]N.N. Voitovich, B.Z. Katsenelbaum, A.N. Sivov, Generalized method of eigenvibration in the theory of diffraction, Nauka, Moskov,1997(Russian).
[79]王爱平,孙炯,郝晓玲,姚斯琴,Completeness of eigenfunctions of Sturm-Liouville prob-lems with transmission conditions, Methods and Applications of Analysis,2009,16(3): 299-312.
[80]王爱平,孙炯,A. Zettl, Characterization of domains of self-Adjoint ordinary differential operators, Journal of Differential Equations,2009,246:1600-1622.
[81]王爱平,孙炯,A.Zettl, The classification of self-adjoint boundary conditions:Separated, coupled and mixed, Journal of Functional Analysis,2008,255:1554-1573.
[82]王爱平,孙炯,A.Zettl, The classification of self-adjoint boundary conditions of differential operators with two singular endpoints, J. Math. Anal. Appl.,2011,378:493-506.
[83]王爱平,关于Weidmann猜想及具有转移条件微分算子的研究(博士学位论文),内蒙古大学,2006.
[84]H. Weyl, Uber gewohnliche differentialgleichungen mit singularityten und kiezugehorigen entwicklungen willkurlicher funktionen, Math. Ann.,1910,68:220-269.
[85]J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary condi-tion, Math. Z.,1973,133:301-312.
[86]J. Weidmann, Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, Springer-Verlag, Berlin,1980.
[87]J. Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathe-matics 1258, Springer-Verlag, Berlin,1987.
[88]王万义,微分算子的辛结构与一类微分算子的谱分析(博士学位论文),内蒙古大学,2002.
[89]王万义,孙炯,高阶常型微分算子自伴域的辛几何刻画,应用数学,2003,16(1)：17-22.
[90]王万义,孙炯,Complex J-symplectic geometry characterization for J-symmetric extensions of J-symmetric differential operators, Advances in Mathematics,2003,32(4):481-484.
[91]王忠,H. Wu, The completeness of eigenfunctions and perturbation connected with Sturm-Liouville operators. J. System and Complexity,2006,19:527-537.
[92]王忠,H. Wu, Dissipative non-self-adjoint Sturm-Liouville operators and completeness of their eigenfunctions, J. Math. Anal. Appl.,2012,394:1-12.
[93]王志敬,宋岱才,直和空间上对称微分算子自共轭域的辛几何刻画(Ⅰ),石油化工高等学校学报,2008,21(1)：92-95.
[94]杨传富,杨孝平,黄振友,m个微分算式乘积的自伴边界条件,数学年刊2006,27A(3)：313324.
[95]杨传富,黄振友,杨孝平,2n阶微分算子乘积自伴的充分必要条件,数学物理学报,2006,26A(6)：953-962.
[96]杨传富,黄振友,杨孝平,微分算子的对称扩张及Friedrichs扩张的辛几何刻画,数学学报,2006,49(2)：421-430.
[97]杨传富,杨孝平,An interior inverse problem for Sturm-Liouville operator with discontin-uous conditions, Appl.Math.Letters,2009,22:1315-1319.
[98]S.Yakubov, Completeness of Root Functions of Regular Differential Operators. New York, Longman, Scientific Technical,1994.
[99]S. Yakubov, Y. Yakubov, Differential Operator Equations (Ordinary and Partial Differen-tial Equations), Chapman and Hall/CRC, Boca Raton 2000.
[100]A. Zettl, Adjoint linear differential operators, Proc. Amer. Math. Soc.,1965,16:1239-1241.
[101]A. Zettl, Sturm-Liouville Theory, American Mathematical Society, Mathematical Surveys and Monographs,2005,121.
[102]A.Zettle, Sturm-Liouville Problem. Spectral Theory and Computatinal Methods of Sturm-Liouville Problem, Marcel Dekker, Berlin,1997,1-104.