Schrödinger operators with a nonspherical radiation condition

Abstract

The Schrodinger operators with potentials p(x) which do not necessarily converge to a constant at infinity will be discussed. The potential p(x) = *i/|*|, x = (x\, x 2 ,--, x n ) G I^> * s an example. The radiation condition associated with such Schrόdinger operators is shown to have the form Vw -iyfk(vR)u = small at infinity, where R = R(x, λ) is a solution of the eikonal equation \VR\ 2 = 1 -p(x)/λ. This radiation condition is "nonspherical" in the sense that v^ is not proportional to the vector 3c = x/\x\ in general. The limiting absorption principle will be obtained using a priori estimates for the radiation condition.

Generalized Fourier Transform for Schrödinger Operators with Potentials of Order Zero

“…Earlier work in this direction has been done by SaitoÄ [19], who establishes a radiation condition and a limiting absorption principle for a class of Schro dinger operators that contains those satisfying (1.1), and by Herbst [8], who investigates the spectral and scattering theory for Schro dinger operators with homogeneous potentials of degree zero.…”

“…K Instead of Lemma B.4 we may use the uniqueness results of SaitoÄ [19]. Since we want to show how to treat this problem as a Schro dinger equation with a short-range potential we present Lemma B.4 for the reader's convenience.…”

Energy concentration and explicit Sommerfeld radiation condition for the electromagnetic Helmholtz equation

“…Moreover, there is a strong connection between this topic and the limiting absorption principle for Schrödinger operators, to which a lot of research has been devoted (see, for example, [6,15,17,18,1,2,14,29,[21][22][23]30,16,4,7,31,26,28,33]). Let us now give a brief picture about the recent advances on Sommerfeld radiation.…”

“…In 1987, Saito considers [31] more general long range potentials and proves the existence of a unique solution of Eq. (1.1) for n(x) = λ(1 +p(x)) + Q(x) with a nonspherical radiation condition…”

“…In order to get the limiting absorption principle for the electromagnetic Helmholtz equation with long range potentials that in particular include those which are homogeneous of degree zero, we will follow Saito [31]. Let us consider the equation…”