[Paper Review] Resonances and Spectral Shift Function Singularities for Magnetic Quantum Hamiltonians

Jean‐François Bony, Vincent Bruneau+1 more|arXiv (Cornell University)|May 15, 2012

Spectral Theory in Mathematical Physics46 references3 citations

TL;DR

This paper investigates the asymptotic behavior of the spectral shift function (SSF) and resonances for 3D magnetic Schrödinger operators with constant magnetic fields. It establishes that the SSF exhibits singularities at Landau levels, with the leading asymptotic term expressed via compact Berezin–Toeplitz operators, and links these singularities to the accumulation of resonances near the same thresholds, proving infinite resonance counts in complementary sectors and deriving their asymptotic density using the same operators.

ABSTRACT

In this survey article we consider the operator pair $(H,H_0)$ where $H_0$ is the shifted 3D Schrödinger operator with constant magnetic field, $H : = H_0 + V$, and $V$ is a short-range electric potential of a fixed sign. We describe the asymptotic behavior of the Krein spectral shift function (SSF) $ξ(E; H,H_0)$ as the energy $E$ approaches the Landau levels $2bq$, $q \in {\mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. The main asymptotic term of $ξ(E; H,H_0)$ as $E o 2bq$ with a fixed $q \in {\mathbb Z}_+$ is written in the terms of appropriate compact Berezin-Toeplitz operators. Further, we investigate the relation between the threshold singularities of the SSF and the accumulation of resonances at the Landau levels. We establish the existence of resonance free sectors adjoining any given Landau level and prove that the number of the resonances in the complementary sectors is infinite. Finally, we obtain the main asymptotic term of the local resonance counting function near an arbitrary fixed Landau level; this main asymptotic term is again expressed via the Berezin-Toeplitz operators which govern the asymptotics of the SSF at the Landau levels.

Motivation & Objective

• To analyze the asymptotic behavior of the Krein spectral shift function (SSF) near the Landau levels for magnetic Schrödinger operators with short-range potentials.
• To establish a connection between the threshold singularities of the SSF and the accumulation of resonances at the Landau levels.
• To derive the main asymptotic term of the local resonance counting function near any fixed Landau level.
• To extend the results to Pauli and Dirac operators with non-constant magnetic fields.

Proposed method

• The SSF is analyzed using the Lifshits–Krein trace formula and the Birman–Krein formula, linking it to scattering phases and eigenvalue counting.
• The main asymptotic term of the SSF near a Landau level $2bq$ is expressed in terms of compact Berezin–Toeplitz operators associated with the spectral projection onto the $q$-th Landau level.
• Resonances are defined as poles of the meromorphic continuation of the resolvent $(H - z)^{-1}$ to an infinitely sheeted Riemann surface.
• The existence of resonance-free sectors adjacent to each Landau level is proven, while the complementary sectors contain infinitely many resonances.
• The local resonance counting function near $2bq$ is asymptotically characterized using the same Berezin–Toeplitz operators that govern the SSF singularities.
• The analysis relies on spectral decomposition in cylindrical coordinates and the mini-max principle to estimate eigenvalue counts in the reduced 1D Schrödinger operators.

Experimental results

Research questions

• RQ1 How does the spectral shift function behave asymptotically as energy approaches a Landau level?
• RQ2 What is the precise relation between the singularities of the SSF and the accumulation of resonances at Landau levels?
• RQ3 Do resonance-free sectors exist near each Landau level, and how many resonances accumulate in the complementary regions?
• RQ4 Can the asymptotic density of resonances near a Landau level be expressed in terms of the same operators that govern the SSF singularities?

Key findings

• The main asymptotic term of the SSF as $E \to 2bq$ is given by a trace of a compact Berezin–Toeplitz operator associated with the $q$-th Landau level projection.
• The number of resonances in a sector complementary to a resonance-free region near $2bq$ is infinite, confirming accumulation at the Landau level.
• The local resonance counting function near $2bq$ has a main asymptotic term expressed via the same Berezin–Toeplitz operators that govern the SSF singularities.
• The generalized Levinson formula is derived as a consequence of the SSF asymptotics, linking eigenvalue counting to spectral shifts.
• For axisymmetric, non-positive potentials satisfying decay conditions, the operator $H$ has infinitely many embedded eigenvalues in the essential spectrum, accumulating at each Landau level.
• The asymptotic behavior of the SSF and resonance counting function is symmetric with respect to the real axis, consistent with the analytic structure of the resolvent.