Nonuniform right definiteness (Q797116)

The multiparameter eigenvalue problem \[ W_ m({\underset \tilde{} \lambda})x_ m=x_ m,\quad W_ m({\underset \tilde{} \lambda})=T_ m+\sum^{k}_{n=1}\lambda_ nV_{mn},\quad m=1,...,k, \] where \({\underset \tilde{} \lambda}\in {\mathbb{C}}^ k\), \(x_ m\) is a nonzero element of the separable Hilbert space \(H_ m\), \(andT_ m\) and \(V_{mn}\) are compact symmetric is studied. Various properties, including existence and uniqueness, of \({\underset \tilde{} \lambda}={\underset \tilde{} \lambda}^ i\in {\mathbb{C}}^ k\) for which the \(i_ mth\) greatest eigenvalue of \(W_ m(\lambda^ i)\) equals one are poved. ''Right definiteness'' is assumed, which means positivity of the determinant with (m,n)th entry \((y_ m,V_{mn}y_ m)\) for all nonzero \(y_ m\in H_ m\), \(m=1...k\). This gives a ''Klein oscillation theorem'' for systems of o.d.e. satisfying a definiteness condition that is usefully weaker than in previous such results. An expansion theorem in terms of the corresponding eigenvectors \(x^ i_ m\) is also given, thereby connecting the abstract oscillation theory with a result of Atkinson.