Spectral representation for Schrödinger operators with long-range potentials

On solutions of the Schrödinger equation with radiation conditions at infinity: the long-range case

“…This result was first proven in [6] for 6 > 1/2 and for a suitably chosen sequence \Xn\ -)> oo. These technical restrictions have been independently overcome in [9] and [12].…”

“…It follows from (3.8) and (3.9) that relation (3.6) holds for some sequence Tn -> oo which implies the equality 6 ).…”

“…More precisely, we need the sharp form of the limiting absorption principle [I], [8], [10] and the asymptotics [6], [9] of the function ({H -A d= zO)-1 /)^) as \x\ -^ oo.…”

“…is any function satisfying (3.1) up to a term Oda-l" 1 "' 6 ), e > 0, as \x\ ->• oo and estimates (3.2). In fact, one can construct by iterations (see for example [12]) an explicit function <l>(a;,A) satisfying these conditions but, in general, this construction requires assumption (2.8) for |a[ > 3.…”

“…up to a term O^x^- 6 ), e > 0, as \x\ -^ oo and estimates (3.2 ), the scattering matrix can be expressed in terms of the operators G±(A) denned by (3.14), (3.16). Indeed, it follows from (4.1) that S{\)a = Tr^A-^J^G^a, <?…”