一类四阶与六阶微分算子积的自伴性

Self-adjointness of Class 4th-order and 6th-order Differential Operator Products

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Li(i=1,2)的乘积L2L1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-1G,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L2L1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Li(i=1,2)的乘积L2L1为自伴算子的充要条件是AQ-1C*=BQ-1D*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 L2L1 of two differential operators Li(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 L2L1 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 Li(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.