Convergence to the Møller Wave‐Matrix

Abstract

The asymptotic behaviour of a quantum mechanical wave-packet scattered by a potential will be considered. *The packet comes in towards the scatterer from a great distance and then recedes again after the collision, so it is only weakly affected by the potential during most of its history. Advantage can be taken of this fact by using a timedependent representation of the Hilbert space such that packets unaffected by the potential correspond to motionless elements of the representing functionspace. This is the so-called "interaction picture" of the process and is constructed as follows: Let the Hamiltonian H be the closure (which exists and is self-adjoint by [3]) of the sum K + V, where, with a proper choice of units,

A note on Cook’s wave-matrix theorem

“…In Theorem III we also generalize to dimension n ^ 3. Cook's results are used by Ikebe [4] in showing S-W* W-, the "S-matrix", to be unitary with Y + = Y_ and in showing the expected connection of the positive part of the spectrum of H with scattering theory under considerably more stringent conditions upon V. Our n -3 existence result II for W ± also includes that of Jauch & Zinnes ([5], p. 566), who assume V{x) = C\x\~β with 1 < β < 3/2, and that of Hack [3], who replaces +||F|| γ < +oo for some τe [2,3) (u, R n ) and +||w|| r =/ r (w, Z) 6 + ) and _||u|| r = / r (w, Dϊ) for specified real 6 > 0.…”

“…With sr = Aj and defining [Fu] Since 2 ^ 7 < n in the last part of the lemma, this only applies when dimension n ^ 3. From the crucial (2.10) and (2.11) (Corollary 2 and 1 of Cook's Lemma 2), the other arguments of Cook's paper [2] apply without other change and yeild all the conculsions of our following Theorems II and III, except for the unstated by Cook orthogonality of each H eigenvector in X to Y ± , which is an easy consequence of W ± H=HW ± and hence H= W£HW ± and the reduction of H by Y ± . This is our new version of Cook's theorem, the special case here 7 = 2 being exactly Cook's statement.…”

“…Assuming that Ve L 2 (R n ) for n = 3, Cook [2] has shown that the unique existent (see Theorem I following) self-adjoint extension H of H Q has the unitary operator /2~p there will do so under Cook's assumptions. In Theorem III we also generalize to dimension n ^ 3.…”

Scattering and reflection positivity in relativistic Euclidean quantum mechanics

“…A sufficient condition for the convergence of this integral is the Cook condition [7], which exploits the unitarity of the time evolution operator…”

Schrödinger operators in the twentieth century

“…Fundamental work was done by Jauch, 133 Cook, 51 Rosenblum, 221 Kato, 149 Birman, 33 and Birman and Krein. 35 The basic result for positive spectrum for ''short-range'' potentials is: 172 Agmon and Hörmander, 6 and Hörmander.…”