On the relative completeness of the generalized eigenvectors of elliptic eigenvalue problems with indefinite weight functions (Q1062254)

In the bounded domain \(\Omega \subset {\mathbb{R}}^ N\) (N\(\geq 1)\) the linear elliptic eigenvalue problem \[ (*)\quad {\mathcal L}u=\lambda mu\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega, \] with respect to the real-valued (possibly indefinite) weight function \(m\in L^{\infty}(\Omega)\), \(m\not\equiv 0\), is considered. Here \({\mathcal L}\) is a (not necessarily selfadjoint) second order strongly uniformly elliptic differential operator for which the maximum principle holds. Denoting by L the realization of \({\mathcal L}\) and the boundary conditions in the complex Hilbert space \({\mathcal H}=L^ 2(\Omega)\), and by M the multiplication operator induced by m, the following is proved: (i) The eigenvalue problem \(Lu=\lambda Mu\) in \({\mathcal H}\) has a discrete spectrum, and for arbitrary \(0<\epsilon <\pi /2\) all the eigenvalues \(\lambda\), expect possibly a finite number of them, lie in the two sectors \(G^+_{\epsilon}=\{\zeta \in {\mathbb{C}}:-\epsilon <\arg \zeta <\epsilon \},G^-_{\epsilon}=\{\zeta \in {\mathbb{C}}:\pi -\epsilon <\arg \zeta <\pi +\epsilon \}.\) (ii) The system of generalized eigenvectors of \(L^{-1}M\) is complete in \(\overline{R(L^{-1}M)}.\) (iii) (*) has infinitely many eigenvalues in \(G^+_{\epsilon}\) iff m is positive on a set of positive measure. These statements complete the results of \textit{P. Hess}, \textit{T. Kato}: Commun. Partial Differ. Equations 5, 999-1030 (1980; Zbl 0477.35075), concerning the principal eigenvalue of (*), and extend those of \textit{S. Agmon}: Commun. Pure Appl. Math. 15, 119-147 (1962; Zbl 0109.327), which hold for the standard case \(m=1\). They are obtained as an application of a generalization of an abstract result by \textit{M. V. Keldysh}, Dokl. Akad. Nauk SSSR, n. Ser. 77, 11-14 (1951; Zbl 0045.394).