Boundary-value problem with almost oscillatory spectrum (Q941971)

Let \(L=d^n/dx^n+\sum_{k=0}^{n-1}p_k(x)d^k/dx^k\) be an operator with coefficients \(p_k\in L^1[a,b]\). Denote by \(G(x,s)\) the Green function of the boundary-value problem \(Ly=0\); \(l_{jc_j}y=0\), \(j=\overline{1,r}\), \(a\leq c_1\leq\dots\leq c_r\leq b\); \(y^{\nu_i}(a_i)=0\), \(\nu_i=\overline{0,k_i-1}\), \(i=\overline{1,m}\), \(a\leq a_1<\dots<a_m\leq b\). Let \(\sigma(x)=(-1)^{r-j}\), \(c_j<x<c_{j+1}\), \(j=\overline{0,r}\), \(c_0=a\), \(c_{r+1}=b\); \(w(x)=(x-a_1)^{k_1}\dotsm(x-a_m)^{k_m}\) for \(r<n\) and \(w(x)=1\) for \(r=n\). Set \(K(x,s)=G(x,s)\sigma(s)/w(x)\). Sufficient conditions are given for the kernel \(K(x,s)\) to be (strongly) sign regular in the sense of [\textit{A. Yu. Levin} and \textit{G. D. Stepanov}, Sib. Math. Zh. 17, No. 3, 606--626 (1976; Zbl 0337.45003)]. These conditions ensure certain oscillatory properties of eigenvalues and eigenfunctions of the equation \(Ly=\lambda q(\cdot)y\), where \(q\in L^1[a,b]\) and \((-1)^rq(x)w(x)>0\) a.e. on \([a,b]\).