On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

Abstract

We consider the three-dimensional Schrödinger operators H 0 and H ± where H 0 = (i∇+A) 2 −b, A is a magnetic potential generating a constant magnetic field of strength b > 0, and H ± = H 0 ± V where V ≥ 0 decays fast enough at infinity. Then, A. Pushnitski's representation of the spectral shift function (SSF) for the pair of operators H ± , H 0 is well defined for energies E = 2qb, q ∈ Z + . We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels 2qb, q ∈ Z + . Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity. Résumé. On considère les opérateurs de Schrödinger tridimensionnelsAlors, la représentation obtenue par A. Pushnitski de la fonction du décalage spectral pour les opérateurs H ± , H 0 est bien définie pour desénergies E = 2qb, q ∈ Z + . Onétudie le comportement du représentant associé de la classe d'équivalence déterminée par la fonction du décalage spectral, au voisinage des niveaux de Landau 2bq, q ∈ Z + . En réduisant l'analyseà l'investigation de l'asymptotique des valeurs propres d'une famille d'opérateurs de Toeplitz compacts, onétablit une relation entre le type des singularités de la fonction du décalage spectral aux niveaux de Landau et la vitesse de la décroissance de Và l'infini.

Asymptotics Near ±m of the Spectral Shift Function for Dirac Operators with Non-Constant Magnetic Fields

“…The relation between the behaviour of the spectral shift function near λ = +m and near λ = −m is explained in Remark 6.15 by using the charge conjugation symmetry. These results are consistent with the results of [27] (where Pauli operators with non-constant magnetic fields are considered) and [12] (where Schrödinger operators with constant magnetic field are considered). Part of the interest of this work relies on the fact that we were able to exhibit a non-trivial class of matrix potentials V satisfying (1.1), even though H 0 is not a bounded perturbation of the free Dirac operator.…”

Resonances and Spectral Shift Function near the Landau levels

“…At last, in Section 5, we represent the derivative of the spectral shift function near the Landau levels as a sum of a harmonic measure related to the resonances and the imaginary part of a holomorphic function (see Theorem 3). Such a representation justifies the Breit-Wigner approximation, implies a trace formula, and for a special class of V sufficiently slowly decaying with respect to the variables perpendicular to the magnetic field, allows us to estimate the remainder in the asymptotic relations obtained in [11].…”

“…(see [30] for q = 0, and the proof of Proposition 5.3 of [11] for any q ∈ N), then for any s > 0 we have…”

“…It is well known that H can have infinite negative discrete spectrum and, for some special V , it can have infinitely many embedded eigenvalues below each Landau level (see [1], [23] or [24]). On the other hand, it is shown in [11] that in the case of sign-definite V , the spectral shift function (SSF) for the operator pair (H, H 0 ) has a singularity at each Landau level. Therefore, it is natural to expect that there could be accumulation of the resonances of the operator H near the Landau levels.…”

The Spectral Density of a Difference of Spectral Projections

“…[7,Lemma 3.1]). In the proof of Proposition 11.1 we have already seen that Z 0 (λ) is a rank two operator which is Hölder continuous in λ > 0.…”