A note on Cook’s wave-matrix theorem
F. H. Brownell · Pacific Journal of Mathematics · 1962
Introduction.Consider the linear operator H o defined by ± co.Moreover, W ± H = HW ± , the range spaces Y ± = W ± X reduce H, and each H eigenvector is orthogonal to Y ± .In Theorem II bellow we give a significant sharpening of these results by weakening the restrictions upon V at oo.Thus, with arbitrary p > 0, any function of the form C\ Jc| -χp over \x\ Ξg b will qualify under our assumptions (the Coulomb case Clxl" 1 thus being borderline), while only such of form C|x|~3 /2 ~p there will do so under Cook's assumptions.In Theorem III we also generalize to dimension n ^ 3. Cook's results are used by Ikebe [4] in showing S-W* W-, the "S-matrix", to be unitary with Y + = Y_ and in showing the expected connection of the positive part of the spectrum of H with scattering theory under considerably more stringent conditions upon V. Our n -3 existence result II for W ± also includes that of Jauch & Zinnes ([5], p. 566), who assume V{x) = C\x\~β with 1 0.* 2 Statements* As notation for our theorems, denoteD£={xe R n \ |x|^6}