Nodal Domains for One- and Two-Dimensional Elliptic Differential Equations

Abstract

Concerning the Schrödinger equation $-\Delta u+q(x)u=0,x=(x_1,x_2)\in {\mathbb{R}}^2$, a condition for $q$ is formulated to ensure that there exists a nodal domain being a circle. Similarly, conditions for the Sturm–Liouville equation are obtained for the existence of a pair of conjugate points.