SPECTRAL THEORY OF 2-POINT DIFFERENTIAL-OPERATORS DETERMINED BY -D2 .2. ANALYSIS OF CASES

In Part I the necessary tools are developed for the complete resolution of the spectral theory for all linear two-point differential operators L in L2[0, 1] determined by τ = -D2 and by boundary values B1, B2. The development emphasizes six numerical parameters defined in terms of the coefficients of B1, B2. In this paper we state and prove all results pertaining to the spectrum σ(L) of L, to the algebraic multiplicities of the eigenvalues λ ε{lunate} σ(L) and the ascents of the operators λI - L, to the boundedness of the family of all finite sums of the projections associated with L, to the denseness of the generalized eigenfunctions, and to the existence of bases consisting of generalized eigenfunctions; we utilize the tools developed in Part I to establish these results. In particular, this paper provides either the explicit or the asymptotic form for the eigenvalues of L, provides explicit numbers for the algebraic multiplicities and ascents, provides the explicit or the asymptotic form for the projections, and provides explicit bounds on the family of all finite sums of these projections or shows that the family is unbounded. © 1990.