Spectral and Scattering Theory for Schrödinger Operators with Cartesian Anisotropy

Abstract

We study the spectral and scattering theory of some n-dimensional anisotropic Schrödinger operators. The characteristic of the potentials is that they admit limits at infinity separately for each variable. We give a detailed analysis of the spectrum: the essential spectrum, the thresholds, a Mourre estimate, a limiting absorption principle and the absence of singularly continuous spectrum. Then the asymptotic completeness is proved and a precise description of the asymptotic states is obtained in terms of a suitable family of asymptotic operators. §1. IntroductionIn this paper we shall be interested in the spectral and scattering theory of some anisotropic Schrödinger operators H = −∆ + V in the Hilbert space L 2 (IR n ). A general theory for highly anisotropic potentials is still lacking, but various partial approaches are already well developed. The most famous one, and best achieved, is with no doubt the N-body problem (see [16]

Minimal Escape Velocities for Unitary Evolution Groups

“…In other words, the state e −iHt ϕ propagates to infinity or "flees the origin" [4] with a velocity at least equal to υ min . Let us point out that the hypotheses of the previous estimate are easily fulfilled by Schrödinger operators with very general N-body potentials or cartesian potentials [17]. In the case where V is a two-body potential, the relation (1) is similar to some results obtained in [6].…”

“…The minimal escape velocity is one variant of the generically called minimal velocity estimates, which are a key ingredient in the proof of asymptotic completeness for various models in quantum mechanics. We refer for example to [19], [21], [9] and [3] for their importance in the N-body problem, and to [11] and [17] for their use in some other anisotropic situations.…”

“…Secondly, let us observe that hypothesis ii) of Proposition 3 is fulfilled if the operator Q −1/2η (H)Aη(H) Q −1/2 is bounded and if the value of b is chosen equal to its norm. But this new condition is quite standard and can be easily proved with some commutators calculations (statement i) of Lemma 6.2 of [17] may help). The other requirements of Proposition 3 are then also easily checked.…”

Spectral and propagation results for magnetic Schrödinger operators; A C∗-algebraic framework

“…In [1] and [41], C * -algebras are used in order to get a Mourre estimate, which is basic for obtaining useful resolvent estimates, finer spectral properties and scattering theory. Such developments require usually more detailed informations about the models under study and cannot be done for magnetic operators in the very general setting in which we will be placed below.…”

Time delay for one-dimensional quantum systems with steplike potentials

“…The existence of f r (for E > V r ) requires the assumption that µ > 1 in (11) has a unique solution provided that µ > n + 1 in (11). By using the bounds for the functions f (j) r (0 ≤ j ≤ n), one can also verify a posteriori, proceeding again recursively, that these functions are indeed the derivatives of the Jost solution (using a theorem permitting the interchange of the integral with the derivatives with respect to κ in the occuring Volterra equations, e.g., Lemma 2 in Chapter XIV of [47]).…”