
doi: 10.2307/2371929
in a direct manner, namely, in terms which do not postulate the delicate knowledge needed for the determination of spectral forms (densities). The latter, which concern "normalizations" (Hellinger), depend on sharp estimates of asymptotic behavior; estimates which are not at all, or not easily, available in many cases in which the location of the set formed by the continuous spectrum and the cluster points of the point spectrum (that is, by the set representing the " essential " portion of the spectrum) can be determined a priori. The method will consist of appropriate adaptations of that indicated in [5] for the particular case in which the f(x) in (1) is a lattice potential (a case in which the range, 0 < x < oo, of (1) becomes replaced by oo < x < oo ). The possibility of such a direct approach is due to the fact that the spectrum itself can be defined without an involvement of the Hilbert-Hellinger theory of spectral forms (the points of increase of the latter define the spectrum itself). Needless to say, what has a spectrum, or a spectral form, is not (1) itself, where f(x) is any given real-valued continuous function, but is represented by (1) and the boundary condition together. On the other hand, the boundary condition is two-fold: for x= oo, it is assigned by the (L2) -requirement of Hilbert's space, whereas for x = 0 it is any homogeneous, linear assignment,
Ordinary differential equations
Ordinary differential equations
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