一类不定Sturm-Liouville算子的谱

The Spectrum of Indefinite Sturm-Liouville Operator Problems

研究了一类权函数和首项系数均有转点并且边界条件中含有谱参数的不连续Sturm-Liouville问题,即由于权函数和首项系数均变号而产生的"不定问题."由于边界条件依赖于谱参数λ,由其确定的线性算子会随λ不同而不同.为此,构造了一个与边值问题相关联但不依赖于谱参数λ的Hilbert空间H和新算子A,使得所考虑的依赖于谱参数的不定的不连续S-L问题与算子A的特征值相同,即把原问题的特征值问题转化为研究定义在Hilbert空间H上的新算子A的特征值问题.进而证明了所考虑边值问题的特征值都是实的,且对应于不同特征值的特征函数相互正交.

A class of discontinuous Sturm-Liouville problems with spectral parameter dependent boundary conditions and indefinite weight function and indefinite leading coefficient at an interior point,i.e.＂the indefinite S-L problem＂are investigated.Because the boundary condition depends on the spectral parameter λ,the operator also depends on the spectral parameter λ.A self-adjoint linear operator A related to the boundary-value problem and independent on the spectral parameter λ is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A.It shows that the eigenvalues of A are real,and the eigenfunctions corresponding to different eigenvalues are orthogonal.