Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory

The flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs

“…13 (vi) is a special case of Theorem 3.1 on the regularity of generalized eigenfunctions which w e shall state and prove in Section 3.…”

“…13 Under these conditions H is self-adjoint on the domain of H 0 . The absolutely continuous part of the spectrum is 0 1).…”

“…We turn now to the proof of the estimates (11)(12)(13)(14)(15)(16). Recalling that (x t) = ( e ;iH0t out )(x) and, since r commutes with the free time evolution, r (x t) = ( e ;iH0t r out )(x), we can write …”

“…(ii) Let f(x) be a bounded continuous function on IR 3 For the proofs of (i), (ii), (iii) and (iv) see Ikebe 13 , for the proof of (v) see Povzner 18 . Since we will use similar reasoning, we will brie y repeat Ikebe's proof of the existence of continuous solutions of Equation (31) starting from Lemma 3.3.…”

A representation formula related to Schrödinger operators

“…The three dimension analogue was proven true by T.Ikebe [6], a member of Kato's school, in 1960. Later his result was extended by Thor to the higher dimension case [10].…”