Time dependent approach to scattering from impurities in a crystal

Abstract

A time dependent scattering theory for a quantum mechanical particle moving in an infinite, three dimensional crystal with impurity is given. It is shown that the Hamiltonian for the particle in the crystal without impurity has only absolutely continuous spectrum. The domain of the resulting wave operators is therefore the entire Hubert space.

Uniform Sobolev inequalities and absolute continuity of periodic operators

“…In [22], the first named author proved that given a, b ∈ R d with |a| = 1 and a, b = 0, there exists δ ∈ R such that (1.2) ψ q ≤ C H ρ ψ p , for any ψ ∈ C ∞ (T d ) and any ρ ∈ R with |ρ| ≥ 2, where C is a constant independent of ψ and ρ, and 1 < p < 2 < q satisfy (1.3) 1 p + 1 q = 1, 1 p − 1 q = 2 d , i.e., p = 2d/(d + 2) and q = 2d/(d − 2). Using inequality (1.2) and L. Thomas's original approach [35], Shen established the absolute continuity of the spectrum of the Schrödinger operator −∆ + V (x) on L 2 (R d ), under the condition that the potential V is real, periodic and V ∈ L d/2 loc (R d ) [22]. In the case d ≥ 5, this improved the earlier results by Reed-Simon [21] and Birman-Suslina [4] , the Morrey class as well as the Kato class were carried out in [22], [24], [25].…”

“…For the Schrödinger operator −∆ + V (x) on R 3 with real periodic potential V , L. Thomas [35] proved that the spectrum of the operator is absolutely continuous if V ∈ L 2 loc (R 3 ). This was done by using the analytic extension in the so-called quasi-momentum k and the resolvent estimates for a family of operators of the form (D+k) 2 +V on L 2 (R d /Γ).…”

Energetic and Dynamic Properties of a Quantum Particle in a Spatially Random Magnetic Field with Constant Correlations along one Direction

“…For Hamiltonians on L 2 (Ê d ) with (rather general) d -periodic scalar potentials (only), the non-existence of flat bands has been proven several decades ago [66,54,69,15]. One class of UMF's, for which (2.9) was proven for all n ∈ AE 0 , concerns certain UMF's of a definite sign and is due to Iwatsuka himself.…”

“…, but each member of which is -periodic and hence has only absolutely continuous spectrum [66,54,69]. The dynamical characterisation of scattering states in Hilbert space by the RAGEtheorem [13,68] therefore implies (for the present situation of one dimension and without singular continuous spectrum) the second of the following two equalities …”

Absence of eigenvalues for the generalized two-dimensional periodic Dirac operator

“…For the proof of Theorem 1.1, we apply Thomas' method (originating from [42] and used for checking the absence of eigenvalues in the spectrum of periodic elliptic differential operators). With the help of this method, in this section we reduce Theorem 1.1 to Theorem 1.2.…”