*(English)*Zbl 0716.35052

The paper deals with the following problem.

Let A denote a differential operator on a non-compact Riemannian manifold M, and assume that A defines an unbounded operator in the Hilbert space ${L}^{2}\left(M\right)$. Assume that for some complex $\lambda $ we know a solution u of the equation $Au=\lambda u$, when can we conclude that $\lambda $ is in the spectrum $\sigma $ (A) of the operator A in ${L}^{2}\left(M\right)?$ After an introduction with the main definitions and some preliminary lemmata, one considers weight estimates and decay of Green functions; then the third section deals with uniform properly supported pseudo-differential operators and structural inverse operators; and the last section contains some results related to the spectral properties of uniformly elliptic operators on manifolds of subexponential growth.

##### MSC:

35P05 | General topics in linear spectral theory of PDE |

35J15 | Second order elliptic equations, general |

35S05 | General theory of pseudodifferential operators |