Spectral shift function in strong magnetic fields

Abstract

Abstract. The three-dimensional Schrödinger operator H with constant magnetic field of strength b > 0 is considered under the assumption that the electric potential V ∈ L 1 (R 3 ) admits certain power-like estimates at infinity. The asymptotic behavior as b → ∞ of the spectral shift function ξ(E; H, H 0 ) is studied for the pair of operators (H, H 0 ) at the energies E = Eb + λ, E > 0 and λ ∈ R being fixed. Two asymptotic regimes are distinguished. In the first regime, called asymptotics far from the Landau levels, we pick E/2 ∈ Z + and λ ∈ R; then the main term is always of order √ b, and is independent of λ. In the second asymptotic regime, called asymptotics near a Landau level, we choose E = 2q 0 , q 0 ∈ Z + , and λ = 0; in this case the leading term of the SSF could be of order b orThe main object of investigation in the present paper is the spectral shift function (SSF) for the three-dimensional Schrödinger operator with constant magnetic field, perturbed by an electric potential that decays sufficiently fast at infinity. We recall the abstract setting in which the SSF for a pair of selfadjoint operators occurs. First, let T 0 and T be two selfadjoint operators acting in the same Hilbert space, and let T −T 0 ∈ S 1 , where S 1 denotes the trace class. Then there exists a unique function ξ(·; T , T 0 ) ∈ L 1 (R) such that the Lifshits-Kreȋn trace formula

Resonances and Spectral Shift Function near the Landau levels

“…-According to Proposition 2.6 of [7], for V satisfying (1.1) with m ⊥ > 0 and m 3 > 1 there exists C > 0 such that H ε has no embedded eigenvalues in R + \ (2bN+] − εC, εC[ ). Then, following the proof of Theorem 2 (i) (or see proof of Proposition 2.5 of [7]), we have only to check that ε 0 > 0 can be chosen independently of λ ∈ R + \ 2bN such that for any ε ε 0 and λ ∈ R + \ 2bN, I + εT V (λ) is invertible when V is non-negative. For negative V , we have to choose ε 0 > 0 such that for any ε ε 0 and…”

An integer-valued version of the Birman—Krein formula

“…[2,1,11,15,29,28,14,13,8,4,19,16,17]); its properties were reviewed and proven in a systematic fashion in [25]. For λ < inf σ ess (A), both projections E A (λ), E B (λ) have finite rank and so by (2.8) we have…”

Lifshitz Tails in Constant Magnetic Fields

“…26) form the so-called angular-momentum orthogonal basis of P q L 2 (R 2 ), q ∈ Z + (see [8] or [3,Section 9]). Here…”