The spectrum of an elliptic operator of second order

Under minimal requirements on the coefficients and the boundary of the domain it is proved that the spectrum of the first boundary-value problem for an elliptic operator of second order always lies in the half-plane $\lambda'\le\operatorname{Re}\lambda$, where $\lambda'$ is the leading eigenvalue to which there corresponds a nonnegative eigenfunction. On the line $\operatorname{Re}\lambda=\lambda'$, there are no other points of the spectrum.