Absence of high-energy spectral concentration for Dirac systems with divergent potentials
It is known that one-dimensional Dirac systems with potentials q which tend to −∞ (or ∞) at infinity, such that 1/q is of bounded variation, have a purely absolutely continuous spectrum covering the whole real line. We show that, for the system on a half-line, there are no local maxima of the spectral density (points of spectral concentration) above some value of the spectral parameter if q satisfies certain additional regularity conditions. These conditions admit thrice-differentiable potentials of power or exponential growth. The eventual sign of the derivative of the spectral density depends on the boundary condition imposed at the regular end-point.
Bounded solutions and absolute continuity of Sturm-Liouville operators
This work is a contribution to the theory of the absolutely continuous spectrum of one-dimensional Schriidinger operators, i.e., Sturm-Liouville operators generated by differential expressions of the form ru = --u” + qu. In proofs of absolute continuity for such operators and also for the closely related Dirac systems always asymptotic properties of solutions of zu = lu for real or complex values of 1 have played an important role, see, for example, [18, 19, 8, 4, 9, lo]. In these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity. Somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory, i.e., results of the following form were proved: Let the solutions of zu = lu satisfy some specific asymptotic property for A varying in some subset of R (no or almost no a priori assumptions on q), then operators generated by z are absolutely continuous in this set. A first result of this type was given by Carmona [3], who proved that local uniform boundedness of solutions together with their derivatives in some open A-set implies pure absolute continuity in this set. Here we will mostly rely on the concept of (non-).&or&racy of solutions, which was shown by Gilbert and Pearson [7, 12, 61 to be a powerful tool in the investigation of the continuous spectrum of Sturm-Liouville operators. In Section 2 we give some consequences of their results, whi_ch are useful in proofs of absolute continuity. These consequences range from non-existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022-247x/92 $5.00
Conserved Quantities of a Class of Nonlinear Difference-Difference Equations
Higher-order conserved quantities of a class of nonlinear difference-difference equations which are related to the discrete-time Toda equation are obtained using the Miura transformations.