两个四阶奇异微分算子积的自伴性

On the Self-adjointness of Product of Two 4th-order Singular Differential Operators

本文在区间[a,∞)上研究由具有任意亏指数的对称常微分算式ly:=y(4)-(py′)′+qy生成的两个四阶奇型微分算子Li(i=1,2)的积L2L1的自伴性.在0∈Π(L0(l))及l2在L2[0,∞)中是部分分离的假设条件下,借助实参数解对自共轭域的描述定理,获得两个四阶微分算子乘积自伴的充要条件,同时证明若L1和L2自伴,则L=L2L1自伴的充要条件是L1=L2,其中-∞<a<∞,2≤d≤4,Π(L0(l))是l在L2[a,∞)中产生的最小算子L0(l)的正则型域.

In this paper,the self-adjointness of product L2L1 of two 4th-order singular differential operators Li（i=1,2）generated by the symmetric ordinary differential expressionly=y（4）-（py′）′＋qy on[a,∞）,with arbitrary deficiency indices is studied.Under the assumption that the product l2 is partially separated in L^2[a,∞）and 0∈Π（L0（l））,by means of the theorem of description for self-adjoint domains in terms of certain solutions for realλ,we obtain a necessary and sufficient condition for the self-adjointness of product of two 4th-order differential operators and prove that if both L1 and L2are self-adjoint,then L =L2L1is selfadjoint if and only if L1=L2,where-∞〈a〈∞,2≤d≤4,Π（L0（l））is the regularity domain of the minimal operator L0（l）generated by lin L^2[a,∞）.