实参数解的个数对微分算子特征值分布的影响

Influence of the Number of Real-Parameter Solutions on the Distribution of Eigenvalues of Differential Operators

研究了一类具有中间亏指数(m,m)的奇异对称常微分算子谱的性质.通过微分算子自共轭域的结构分析,证明了若对任何λ∈(μ1,μ2),方程τy=λy存在m个线性无关的L2-解.则由τ生成的最小算子T0的任何自共扩张A的特征值在区间(μ1,μ2)中是无处稠密的.

The spectral property of a class of singular symmetric ordinary differential operators with middle indices （m,m） is investigated. By means of analyzing the construction of self-adjoint domains of differential operators,it is proved that if for every λ∈（μ1,μ2） ,there exist m linearly indepent L^2-solutions of （τ-λ）y=0. Then the eigenvalues of every self-adjoint extension A of minimal operator To generated by differential expression τ are nowhere dense in interval （μ1 ,μ2）.