Low Energy Asymptotics of the Spectral Shift Function for Pauli Operators with Nonconstant Magnetic Fields

Abstract

We consider the 3D Pauli operator with nonconstant magnetic field B of constant direction, perturbed by a symmetric matrix-valued electric potential V whose coefficients decay fast enough at infinity. We investigate the low-energy asymptotics of the corresponding spectral shift function. As a corollary, for generic negative V , we obtain a generalized Levinson formula, relating the low-energy asymptotics of the eigenvalue counting function and of the scattering phase of the perturbed operator.

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

“…The estimates of Proposition 6.10 are consistent with the ones of [27,Cor. 3.6], where the corresponding situation for magnetic Pauli operators is considered.…”

“…Our work is closely related to [27] where G. D. Raikov treats a similar issue in the case of magnetic Pauli operators. It can also be considered as a complement of [33], where general properties of the spectrum of Dirac operators with variable magnetic fields of constant direction and matrix perturbations are determined.…”

A simple criterion for the existence of nonreal eigenvalues for a class of 2D and 3D Pauli operators

“…and if there exists an integrated density of states for the operator P − 2 (see [29] definition (3.11)), then by [29, Lemma 3.3],…”

“…(i) The class of admissible magnetic fields described above is essentially the one introduced in [28,29]. We refer to these papers for more details and examples of admissible magnetic fields.…”

Levinson’s Theorem: An Index Theorem in Scattering Theory

“…Beside the extensions already presented in Sections 6 and 8, others are appealing. For example, it would certainly be interesting to recast the generalized Levinson's theorem exhibited in [44,55] in our framework. Another challenging extension would be to find out the suitable algebraic framework for dealing with scattering systems described in a two-Hilbert spaces setting.…”