[Paper Review] Sharp Spectral Asymptotics for Operators with Irregular Coefficients. V. Multidimensional Schroedinger operator with a strong magnetic field. Non-Full-rank case

Victor Ivrii|arXiv (Cornell University)|Oct 16, 2005

Spectral Theory in Mathematical Physics13 references3 citations

TL;DR

This paper establishes sharp spectral asymptotics for the multidimensional Schrödinger operator with a strong magnetic field under non-full-rank conditions, where the magnetic field intensity matrix has constant rank $2r < d$ and nullity $q = d - 2r \geq 1$. It derives remainder estimates of order $O(h^{1-d} + \mu^r h^{1-r-q})$, improving upon prior results through refined microlocal analysis and multiscale assumptions on the magnetic field strengths.

ABSTRACT

Sharp spectral asymptotics for multidimensional Schroedinger operators with the strong magnetic field are derived under rather weak smoothness conditions. I assume that magnetic intensity matrix has constant defect r&gt;0 at each point. In comparison with version 1 of 5.5 year ago this version contains more results (we also study some degenerations), improvements and some minor corrections.

Motivation & Objective

• Derive sharp spectral asymptotics for the multidimensional Schrödinger operator with a strong magnetic field when the magnetic field intensity matrix has constant but not full rank.
• Address the non-full-rank case ($q = d - 2r \geq 1$) where the kernel of the magnetic field matrix is nontrivial, leading to a $q$-dimensional degeneracy in the dynamics.
• Improve the remainder estimate to $O(h^{1-d} + \mu^r h^{1-r-q})$ by introducing multiscale assumptions on the magnetic field eigenvalues $f_p$.
• Analyze degenerations and non-degenerate regimes, particularly distinguishing the cases $q=1$, $q=2$, and $q \geq 3$, and their impact on spectral estimates.
• Establish conditions under which microhyperbolicity and non-degeneracy ensure optimal remainder bounds, even in the presence of singularities or vanishing field components.

Proposed method

• Use a coordinate system adapted to the integrable foliation defined by the kernel of the magnetic field matrix $F_{jk}$, reducing the problem to a $q$-dimensional base and $2r$-dimensional fiber.
• Apply gauge transformations to eliminate vector potentials $V_j$ along the kernel directions ($j=1,\dots,q$), simplifying the operator to a form depending only on $x^\perp$.
• Employ multiscale analysis to group the magnetic field strengths $f_p$ into different magnitudes, enabling the treatment of degenerations and hierarchical structures.
• Apply $\mu^{-1}h$-pseudo-differential operator techniques in the $r$-dimensional fiber directions, combined with $h$-pseudo-differential methods in the $q$-dimensional base.
• Use canonical forms of the magnetic Schrödinger operator to reduce the spectral problem to a family of lower-dimensional operators, allowing asymptotic analysis.
• Introduce and apply microhyperbolicity and non-degeneracy conditions in the $z$-variables (fiber coordinates) to control remainder terms, especially in the $q=1,2$ cases.

Experimental results

Research questions

• RQ1 What is the sharp spectral asymptotics for the Schrödinger operator with a strong magnetic field when the magnetic field intensity matrix has constant but not full rank?
• RQ2 How does the remainder estimate depend on the dimension $d$, the rank $2r$ of the magnetic field, and the nullity $q = d - 2r$?
• RQ3 What improvements in remainder estimates are achievable through multiscale assumptions on the magnetic field eigenvalues $f_p$?
• RQ4 How do degenerations in the magnetic field (e.g., vanishing $f_p$) affect the spectral asymptotics and remainder bounds?
• RQ5 Under what conditions—microhyperbolicity or non-degeneracy—can the remainder estimate be bounded as $O(\mu^r h^{1+r-d})$ even in the $q=1,2$ cases?

Key findings

• The sharp spectral asymptotics are derived with a remainder estimate of order $O(h^{1-d} + \mu^r h^{1-r-q})$ for the $d$-dimensional Schrödinger operator with a strong magnetic field and non-full-rank magnetic intensity matrix.
• The remainder estimate improves significantly in the $q \geq 2$ case due to the absence of microhyperbolicity assumptions, while $q=1$ requires additional non-degeneracy control.
• For $q=1$, the remainder estimate is $O(\mu^r h^{1/2 - d})$ under the $(l,\sigma) = (2,0)$ condition, reflecting a more delicate behavior due to lower-dimensional degeneracy.
• In the superstrong magnetic field regime ($\mu \gtrsim h^{-2}$), the spectral projector satisfies $e(x,y,0) = O(\mu^{-\infty})$, indicating localization to the lowest Landau level.
• When $f_j \asymp \gamma(x)$ with $\gamma(x) = \mathrm{dist}(x,Y)$ and $Y$ a $\mathscr{C}^{2,1}$ manifold of codimension 3, the asymptotics scale as $h^{-d}\bar{\gamma}^{3-d}$ with $\bar{\gamma} = 1/(\mu h)$, and the remainder is $O(\mu^{-2}h^{-1-d})$ for $q \geq 3$.
• Under microhyperbolicity and non-degeneracy assumptions, the remainder estimate matches the non-degenerate case: $O(\mu^r h^{1+r-d})$ for $q \geq 3$, and $O(\mu^r h^{1/2 + r - d})$ for $q=1$.