2n阶线性奇异边值问题的谱理论

Spectral Theory of Singular Boundary Value Problems of 2n-th Order Linear Differential Equations

考虑2n阶线性微分方程的奇异边值问题(-1)nu2n(t)=λa(t)u(t),0<t<1;u2k(0)=u2k(1)=0,k=0,1,…,n-1,其中λ是常数,a∈C(0,1),a(t)>0.首先证明奇异边值问题是线性自共轭全连续微分算子,然后利用线性自共轭全连续算子的谱理论给出了2n阶线性微分方程的奇异边值问题的谱.

For singular boundary value problems of 2n-th order linear differential equations （-1）^nu^2n（t）=λa（t）u（t）.0〈t〈1;u^2k（0）=u^2k（1）=0,k=0,1,…,n-1, where λ is a constant,α∈ C（0,1） ,and α（t）~O. It is proved that the singular boundary value problem is a linear self-adjoint completely continuous differential operator. The spectrum of the singular boundary value problems of the 2n-th order linear differential equations is obtained by means of the spectral theory of linear self-adjoint completely continuous operator.