自共轭微分算子边界条件的分类及其标准型
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摘要
本文主要研究自共轭微分算子边界条件的分类及其标准型。边界条件,作为微分算子定义的组成部分,对于微分算子的研究具有重要的意义。我们知道对于实参数解给出的自共轭公式,其自共轭边界条件的系数矩阵满足AEA~*=BEB~*,并且自共轭边界条件可分为严格分离型边界条件、完全耦合型边界条件和混合型边界条件。我们运用矩阵分块运算以及行变换、列变换对边界条件矩阵A,B性质的影响,分别对矩阵秩的不同情况进行了详细的计算,结合边界条件的三种不同分类给出了n=4,n=6的实参数解描述的高阶微分算子自共轭边界条件的各种标准型。进一步地,我们把上述结果类似地推广到一般的高阶情况。
In this paper, the classifications of boundary conditions of the self-adjoint differential operators and it's canonical form are studied. As a part of definition of differential operators, boundary conditions are very important for the research of differential operators. We knew that gives regarding the real parameter solution from the self-adjoint formula, for it's self-adjoint boundary conditions, that the coefficients matrices are satisfied equation AEA* = BEB*. And the self-adjoint boundary conditions may divide into two classifications, namely separated, coupled, and mixed. We utilize the matrix piecemeal operation as well as a line of transformation, the row transform pair boundary condition influence matrix A, B nature , has carried on the detailed computation separately to the matrix rank's different situation, unified the boundary condition three kind of different classifications to give the n=4, n=6 real parameter solution description higher order derivative operator from self-adjoint boundary condition each kind of standard type. Further, we similarly promote above result to the general higher order situation.
In this paper, the classifications of boundary conditions of the self-adjoint differential operators and it's canonical form are studied. As a part of definition of differential operators, boundary conditions are very important for the research of differential operators. We knew that gives regarding the real parameter solution from the self-adjoint formula, for it's self-adjoint boundary conditions, that the coefficients matrices are satisfied equation AEA* = BEB*. And the self-adjoint boundary conditions may divide into two classifications, namely separated, coupled, and mixed. We utilize the matrix piecemeal operation as well as a line of transformation, the row transform pair boundary condition influence matrix A, B nature , has carried on the detailed computation separately to the matrix rank's different situation, unified the boundary condition three kind of different classifications to give the n=4, n=6 real parameter solution description higher order derivative operator from self-adjoint boundary condition each kind of standard type. Further, we similarly promote above result to the general higher order situation.
引文
[1]A.P.Wang,J.Sun and A.Zettl,Characterization of Domains of Self-Adjoint Ordinary Differentiol Operators,pre-print.
[2]A.P.Wang,J.Sun and A.Zettl,The Classification of Self-Adjoint Boundary Conditions:Separated,Coupled,And Mixed,pre-print.
[3]A.P.Wang,J.Sun,The self-adjoint extensions or singular differential operators with a real regularity point,Journal of Spectral Mathematics and its Applications,2006.
[4]A.Zettl,Adjoint linear differential operators,Proc.Amer.Math.Soc.,1965.
[5]A.Zettl,Sturm-Lioville Theory,American Mathematical Society,2005
[6]曹之江,常微分算子,上海科学技术出版社,1987.
[7]曹之江,自伴常微分算子的解析描述,内蒙古大学学报,第18第3期,1987.
[8]曹之江,高阶极限圆型微分算子的自共轭扩张,数学进展,1985.
[9]D.E.Edmunds,J.Sun,Embedding theorems and the spectra of certain differential operators,Proc.R.Soc.London A,1991,434:643-656.
[10]E.A.Conddington and N.Levinson,Theory of Ordinary Differential Equitions,McGraw-Hill,New York/London/Toronto,1955.
[11]Edmunds D E,Kufner A,Sun Jiong,Extension of functions in weighted Sobolev spaces,Rend.Accad.Naz.XL Mere.Mat.1990°,108(16)17:327-339.
[12]Edmunds D E,Sun Jiong,Approximation and entropy numbers of sobolev embedding over domains with finite measures,Quart.J.Math.Oxford,1990,(41)2:385-394.
[13]J.Sun,On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices.Acta Math.Sincia,New Series,1986,2(2):152-167.
[14]J.Sun,On the Self-adjoint Extensions of Symmetric Ordinary Differential Operators with Middle Deficiency Indices,Acta Mathematica Sinica,New Series,Vol.2,No.2.
[15]M.A.Naimark,王志成译,线性微分算子,科学出版社,1964.
[16]N.Dunford and J.T.Schwartz,linear Operators,v.2,Wiley,New York,1963.
[17]孙炯,王忠,常微分算子谱的定性分析,数学进展,1995,24(5);406-422.
[18]孙炯、王忠,线性算子的谱分析,科学出版社,2005.
[19]王爱萍,博士论文
[20]W.D.Evans and E.I.sobhy,Boundary conditions for general ordinary differential operators and their adjoints.Proceedings of the Royal Society of Edinburgh,1990.
[21]W.N.Everitt and A.Zettl,Generalized symmetric ordinary differential expressions I:The general theory.Nierw Archief voor Wiskunde(3),27,(1979).
[22]W.N.Everitt and F.Veuman,A concept of adjointness and symmetry of differential expressions based on the generalized Lagrange identity and Green's formula,Lecture Notes in Math.1032,Springer- Verlag,Berlin(1983),161-169.
[23]夏道行,严绍宗著,线性算子谱理论,[M].北京,科学出版社,1987
[24]袁小平,混合自伴边界条件下的正则Sturm-Liouville问题,内蒙古大学学报(自然科学版),第21卷第1期,1990.
[25]Z.J.Cao,On self-adjoint domains of second order differential operators in limit-cirele case.Acta Math.Sinica,New series,1985.
[26]Z.J.Cao And J.Sun,Self-adjoint operators generated by symmetric quasi-differential expressions.Acta Sci.Natur,Univ,NerMongol,1986.
[2]A.P.Wang,J.Sun and A.Zettl,The Classification of Self-Adjoint Boundary Conditions:Separated,Coupled,And Mixed,pre-print.
[3]A.P.Wang,J.Sun,The self-adjoint extensions or singular differential operators with a real regularity point,Journal of Spectral Mathematics and its Applications,2006.
[4]A.Zettl,Adjoint linear differential operators,Proc.Amer.Math.Soc.,1965.
[5]A.Zettl,Sturm-Lioville Theory,American Mathematical Society,2005
[6]曹之江,常微分算子,上海科学技术出版社,1987.
[7]曹之江,自伴常微分算子的解析描述,内蒙古大学学报,第18第3期,1987.
[8]曹之江,高阶极限圆型微分算子的自共轭扩张,数学进展,1985.
[9]D.E.Edmunds,J.Sun,Embedding theorems and the spectra of certain differential operators,Proc.R.Soc.London A,1991,434:643-656.
[10]E.A.Conddington and N.Levinson,Theory of Ordinary Differential Equitions,McGraw-Hill,New York/London/Toronto,1955.
[11]Edmunds D E,Kufner A,Sun Jiong,Extension of functions in weighted Sobolev spaces,Rend.Accad.Naz.XL Mere.Mat.1990°,108(16)17:327-339.
[12]Edmunds D E,Sun Jiong,Approximation and entropy numbers of sobolev embedding over domains with finite measures,Quart.J.Math.Oxford,1990,(41)2:385-394.
[13]J.Sun,On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices.Acta Math.Sincia,New Series,1986,2(2):152-167.
[14]J.Sun,On the Self-adjoint Extensions of Symmetric Ordinary Differential Operators with Middle Deficiency Indices,Acta Mathematica Sinica,New Series,Vol.2,No.2.
[15]M.A.Naimark,王志成译,线性微分算子,科学出版社,1964.
[16]N.Dunford and J.T.Schwartz,linear Operators,v.2,Wiley,New York,1963.
[17]孙炯,王忠,常微分算子谱的定性分析,数学进展,1995,24(5);406-422.
[18]孙炯、王忠,线性算子的谱分析,科学出版社,2005.
[19]王爱萍,博士论文
[20]W.D.Evans and E.I.sobhy,Boundary conditions for general ordinary differential operators and their adjoints.Proceedings of the Royal Society of Edinburgh,1990.
[21]W.N.Everitt and A.Zettl,Generalized symmetric ordinary differential expressions I:The general theory.Nierw Archief voor Wiskunde(3),27,(1979).
[22]W.N.Everitt and F.Veuman,A concept of adjointness and symmetry of differential expressions based on the generalized Lagrange identity and Green's formula,Lecture Notes in Math.1032,Springer- Verlag,Berlin(1983),161-169.
[23]夏道行,严绍宗著,线性算子谱理论,[M].北京,科学出版社,1987
[24]袁小平,混合自伴边界条件下的正则Sturm-Liouville问题,内蒙古大学学报(自然科学版),第21卷第1期,1990.
[25]Z.J.Cao,On self-adjoint domains of second order differential operators in limit-cirele case.Acta Math.Sinica,New series,1985.
[26]Z.J.Cao And J.Sun,Self-adjoint operators generated by symmetric quasi-differential expressions.Acta Sci.Natur,Univ,NerMongol,1986.