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 ₀ Re z, where is the leading eigenvalue to which there corresponds a nonnegative eigenfunction. Apart from ₀, there are no other points of the spectrum on the straight line Re z=₀.Translated from Matematicheskie Zametki, Vol. 7, No. 4, pp. 495–502, April, 1970.The authors are grateful to V. S. Vladimirov for discussing the results of the present paper and for pointing out the proposition in 3], which made it possible to shorten the proof that the leading eigenvalue of the problem (1) is simple.