On manifolds of self-adjoint elliptic operators with multiple eigenvalues (Q2754854)

The author considers the following class of problems. Suppose we have a family \(G\) of self-adjoint elliptic operators on a compact manifold \(\Omega\). Let \(\nu_2,\nu_3,\ldots ,\nu_n\) be nonnegative integers and let \(G_\nu\) be a subset of \(G\) consisting of operators which have \(\nu_k\) eigenvalues of the multiplicity \(k\) for \(k=2,\ldots ,n\). It is shown for various specific situations that typically \(G_\nu\) has a finite codimension in \(G\); for some cases the codimension is calculated explicitly. In particular, the author studies in details the case where \(G\) is the set \(\{g+p:\;p\in C^3(\Omega)\), \(p(x)>0\}\), \(g\) is a fixed elliptic operator. A similar problem is considered for the manifold of eigenfunctions \(Q=\{ (p,\lambda ,y): gy+py-\lambda y=0\}\).