一类四阶与六阶微分算子积的自伴性
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摘要
讨论了一类四阶正则对称微分算式D~((4))+1与一类六阶正则对称微分算式D~((6))+1生成的两个微分算子L_i(i=1,2)的乘积L_2L_1的自伴性问题。在常型情况下,通过构造矩阵G,进一步得到矩阵S=Q~(-1)G,其中Q为微分算子的Lagrange双线性型矩阵。利用矩阵运算和微分算子的基本理论,得到了积算子L_2L_1为自伴算子时的边条件应满足的一个充要条件为CS(a) A*=DS(b) B*,这与两个同阶的对称微分算式生成的微分算子L_i(i=1,2)的乘积L_2L_1为自伴算子的充要条件是AQ~(-1)C*=BQ~(-1)D*这个结论极为相似,这一结果为进一步给出一般的两类不同偶数阶微分算子乘积自伴性的充要条件提供了新的思路。
The self-adjointness of product L_2L_1 of two differential operators L_i(i = 1,2) generated by one class of 4th-order regular symmetric differential expression D~((4))+ 1 and 6th-order regular symmetric differential expression D~((6))+ 1 is discussed. In the case of the normal type,by constructing the matrix G,the matrix S = Q~(-1) G is further obtained,where Q is the Lagrange bilinear matrix of differential operators,and also obtained the sufficient and necessary condition for the self-adjointness of product L_2L_1 is CS(a) A*= DS(b) B*with the theory of operators and matrix operations. This is very similar to the necessary and sufficient condition to be a self-adjointness operator for the product of the differential operator L_i(i = 1,2)generated by the two same order symmetric differential expression is AQ~(-1) C*= BQ~(-1) D*. This interesting result provides a new way to give the necessary and sufficient conditions for the product self-adjoint properties of two different even-order differential operators.
The self-adjointness of product L_2L_1 of two differential operators L_i(i = 1,2) generated by one class of 4th-order regular symmetric differential expression D~((4))+ 1 and 6th-order regular symmetric differential expression D~((6))+ 1 is discussed. In the case of the normal type,by constructing the matrix G,the matrix S = Q~(-1) G is further obtained,where Q is the Lagrange bilinear matrix of differential operators,and also obtained the sufficient and necessary condition for the self-adjointness of product L_2L_1 is CS(a) A*= DS(b) B*with the theory of operators and matrix operations. This is very similar to the necessary and sufficient condition to be a self-adjointness operator for the product of the differential operator L_i(i = 1,2)generated by the two same order symmetric differential expression is AQ~(-1) C*= BQ~(-1) D*. This interesting result provides a new way to give the necessary and sufficient conditions for the product self-adjoint properties of two different even-order differential operators.
引文
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[15]林秋红.K-单全正交矩阵及其性质[J].高师理科学刊,2016,36(9):21-25.
[2]林秋红.具两奇异端点的J-对称微分算子的J-自伴域[J].中北大学学报:自然科学版,2018,39(5):489-494.
[3]高雪,许美珍.两区间二维Sturm-Liouville向量微分算子的自伴扩张[J].内蒙古工业大学学报,2018,37(2):81-89.
[4]边学军.二阶自伴微分算子方幂的自伴性[J].内蒙古大学学报:自然科学版,1996,27(1):1-10.
[5]曹之江,孙炯.Edmunds D.E.二阶微分算子积的自伴性[J].数学学报,1999,42(4):649-654.
[6]张新艳,王万义,杨秋霞.一类高阶微分算子积的自伴性[J].内蒙古大学学报:自然科学版,2006,37(5):484-490.
[7]杨传富,黄振友,杨孝平.两个四阶微分算子积的自伴性[J].数学进展,2004,33(3):291-302.
[8]杨传富,杨孝平,黄振友.M个微分算子乘积的自伴边界条件[J].数学年刊,2006,27A(3):313-324.
[9]张新艳,王万义,杨秋霞.三个二阶微分算子积的自伴性[J].内蒙古师范大学学报:自然科学,2008,36(1):35-42.
[10]张新艳,王万义.三个高阶极限点型微分算子积的自伴性[J].内蒙古大学学报:自然科学版,2008,37(5):599-603.
[11]郑召文,刘宝圣.三个极限圆型Hamolton算子乘积的自伴性[J].应用数学学报,2014,37(5):769-785.
[12]玉林,王桂霞,王万义.一类二阶与四阶微分算子积的自伴性[J].数学的实践与认识,2016,46(18):218-222.
[13]曹之江.常微分算子[M].上海:上海科技出版社,1986.
[14]CODDINGTON E A.The spectral representation o ordinary self-adjoint differential operators[J].Ann of Math,1954,60(1):192-211.
[15]林秋红.K-单全正交矩阵及其性质[J].高师理科学刊,2016,36(9):21-25.