Multiplicities of the eigenvalues of the Schrödinger equation in any dimension (Q1104477)

We consider the Schrödinger equation on a bounded region \(D\subseteq {\mathbb{R}}^ d\) with Dirichlet boundary conditions \(-\Delta u+qu=\lambda u\), \(u|_{\partial D}=0\); here q is an element in \(L^{\infty}_{{\mathbb{R}}}(D)\) and the boundary \(\partial D\) of D is sufficiently smooth. We prove that the set Q of all potentials q in \(L^{\infty}_{{\mathbb{R}}}(D)\) which have the property that the equations above have at least one multiple eigenvalue is of first category.