Characterization of Eigenvalues in Spectral Gap for Singular Differential Operators

Abstract

The spectral properties for $n$ order differential operators are considered. When given a spectral gap $(a,b)$ of the minimal operator $T_0$ with deficiency index $r$, arbitrary $m$ points ${\beta }_i(i=\mathrm{1,2},\dots ,m)$ in $(a,b)$, and a positive integer function $p$ such that ${\sum }_{i=1}^{m}p({\beta }_i)\le r$, $T_0$ has a self-adjoint extension $\tilde{T}$ such that each ${\beta }_i(i=\mathrm{1,2},\dots ,m)$ is an eigenvalue of $\tilde{T}$ with multiplicity at least $p({\beta }_i)$.