Discreteness conditions for the spectrum of ordinary differential operators

Abstract

It is well known (61 that the expression M determines a minimal operator T, in the weighted Hilbert space Li(O, co) which is closed, symmetric, densely defined and has self-adjoint extensions. All self-adjoint extensions of T0 are known [6] to have the same essential spectrum. (Although these results are established in [6] only for the case w(f) = 1 and under stronger hypotheses on the coefficients pj, they can be extended to the results mentioned above by the same methods.) Here we are interested in finding conditions on w andpj which ensure that the spectrum of every self-adjoint extension of r,, is discrete, i.e., the essential spectrum is empty. Such conditions have been found by many authors including Berkowitz [l], Brinck (21, Friedrichs [4,5], Glazman [6j, Ismagilov [ll], Hinton [9], Hinton and Lewis [8, lo], Lewis [12], MtillerPfeiffer [18], Molchanov [17], Read [ 191, Rollins [2Oj, Tkachenko (see 161). This list is not intended to be comprehensive-the literature on this problem is voluminous.