Asymptotics of the eigenvalues of the dirichlet and neumann problems for the abstract sturm-liouville operator

Let A>0 be an unbounded self-adjoint operator in a Hilbert space H. In the Hilbert space H₁=L₂ (0, π; H) we study the spectrum of the differential equations−y″(x)+Ay=λy, y (0)=y(π)=0,−y″(x)+Ay=λy, y′(0) =y′(π)=0. We find the principal terms of the asymptotics of the functions N(λ) for these problems and we ascertain the conditions under which they are asymptotically not equivalent. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 209–212, February, 1977.