二阶自伴微分算子方幂的自伴性

On Selfadjointness of Power of 2-nd Order Selfadjoint Differential Operators

算子方幂的研究，以往主要致力于对亏指数的探讨，而有关算子方幂的自伴性，据知还无人涉足．研究了对于二阶的自伴边条件下由对称微分算式ｌ（ｙ）＝－ｙ＂＋ｑ（ｘ）ｙ所生成的算子Ｌ（包括正则和奇异的情形），讨论其方幂算子Ｌ２＝ＬＬ的自伴性．先在［０，π］上考虑正则的暴算子Ｌ２的自伴性，然后推广到［０，∞）上奇异的情形．

On this paper following results on self adjointness of product L1 L2 of two 2-order self adjoint differential operators L1, L2, as well as the power L2 of 2-order self-adjoint operator L are obtained:Theorem 1 The product L1L2 of two 2-order selfadjoint differential operators is selfadjoint if and only if L1=L2.Theorem 2 The power L2 of 2-order differential operator L is selfadjoint if L is selfadjoint.These results are proved in ordinary case and partly proved in singular case.