On the Completeness of the System of Root Vectors of the Sturm–Liouville Operator with General Boundary Conditions

We study general boundary value problems with nondegenerate characteristic determinant $\Delta(\lambda)$ for the Sturm–Liouville equation on the interval $[0,1]$. Necessary and sufficient conditions for the completeness of root vectors are obtained in terms of the potential. In particular, it is shown that if $\Delta(\lambda)\ne\mathrm{const}$, $q(\cdot)\in C^k[0,1]$ for some $k\ge 0$, and $q^{(k)}(0)\ne(-1)^kq^{(k)}(1)$, then the system of root vectors is complete and minimal in $L^p[0,1]$ for $p\in[1,\infty)$.