Subordinacy and spectral analysis of Schrödinger operators
This thesis is concerned with the spectral analysis of Schrödinger operators with central potentials, and some related aspects of scattering theory. After an introductory discussion on the aims of the thesis and its relation to existing work, the background mathematical material required for subsequent developments is presented in Chapter II. The theory of subordinacy, which relates the absolutely continuous, singular continuous and discrete parts of the spectrum to the relative asymptotic behaviour of solutions of the radial Schrödinger equation, is established in Chapter III for the case where L 2 = -d + V(r) /�dr2� is regular at 0 and limit point at infinity. In Chapter IV, it is shown that the general eigenfunction expansion theory of Weyl-Kodaira can be simplified for a Schrödinger operation in L2 (O,∞) whenever the corresponding operator on any finite interval containing the origin has singular spectrum and the potential is integrable at infinity; an incidental outcome is an extension of the theory of subordinacy to include cases where L is singular at both ends of the interval (O,~). The simplified expansion theory enables the class of potentials for which the usual phase shift formula for the scattering operator holds to be extended in Chapter V, so as to include more singular behaviour at the origin than any previously considered. Using this result, it is shown that a Schrödinger operator exists for which the theory is asymptotically complete and the scattering amplitude is a discontinuous function of energy. Chapter VI is concerned with the inductive construction of potentials having singular continuous spectrum; there is a particular emphasis on the generation of singular continuous measures from sequences of absolutely continuous measures, and some improvements to existing results and relevant examples are presented. The thesis is concluded with a brief indication of some outstanding problems, and suggestions for further research.