Spectrum of an elliptic equation

It is shown that the spectrum for the first boundary-value problem for a second-order elliptic equation always lies in the half-plane $\lambda_0\leqslant\mathrm{Re}\,z$, where $\lambda_0$ is the leading eigenvalue to which there corresponds a nonnegative eigenfunction. Apart from $\lambda_0$, there are no other points of the spectrum on the straight line $\mathrm{Re}\,z=\lambda_0$.