On the oscillation properties of eigenfunctions of Sturm--Liouville problem with singular coefficients

Abstract

In the paper we consider singular spectral Sturm--Liouville problem $-(p{y}^{\prime }{)}^{\prime }+(q-\lambda r)y=0$, $(U-1)y^{\vee }+i(U+1)y^{\wedge }=0$, where function $p\in L_{\infty }[0,1]$ is uniformly positive, generalized functions $q,r\in W_2^{-1}[0,1]$ are real-valued and unitary matrix $U\in {\mathbb{C}}^{2\times 2}$ is diagonal. The main goal is to prove that well-known (for smooth case) facts about number and distribution of zeros of eigenfunctions hold in general case.

arXiv comment7 pages