Beyond the classical Weyl and Colin de Verdière's formulas for Schrödinger operators with polynomial magnetic and electric fields
type: Item
instance of: scholarly article
title: Beyond the classical Weyl and Colin de Verdière's formulas for Schrödinger operators with polynomial magnetic and electric fields
title: Generalizations of the classical Weyl and Colin de Verdière's formulas and the orbit method
zbMATH Open document ID: 1127.35028
zbMATH Open document ID: 1378.35207
author: Mitya Boyarchenko
author: Sergei Levendorskiĭ
published in: Proceedings of the National Academy of Sciences
published in: Annales de l’institut Fourier
publication date: 2007-04-24T00:00:00Z
publication date: 2007-06-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0406071
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10193
review text: The goal of this paper is to write down a conjectural formula for the leading term of the spectral asymptotics of a Schrödinger operator on \(L^2(\mathbb{R}^n)\) with quasi-homogeneous polynomial magnetic and electric fields \(B\) and \(V\). The construction is based on the orbit method due to Kirillov. It makes sense for any nilpotent Lie algebra and is related to the geometry of coadjoint orbits, as well as to the growth properties of certain ``algebraic integrals'', studied by Nilsson. The precise version of the conjecture uses the function \(\Psi^*\) constructed by Helffer and Mohamed: \[ \Psi^*(x):= \sum_\alpha|\partial^\alpha V(x)|^{1/(|\alpha|+ 2)}+ \sum_{\alpha,j,k}|\partial^\alpha B_{jk}(x)|^{1/(|\alpha|+ 2)}, \] compared with the function \[ \Phi^*(x):= \sum_\alpha |\partial^\alpha V(x)|^{1/2}+ \sum_{\alpha, j,k} |\partial^\alpha B_{jk}(x)|^{1/2}. \] By using the direct variational method, the authors prove that the formulas give the correct answer not only in the ``regular'' cases where the classical formulas of Weyl or Colin de Verdière are applicable, but in may ``irregular'' cases, such as for the Schrödinger operator in 2D with magnetic tensor \(B(x)= x^k_1 x^\ell_2\) \((k\geq \ell\geq 1)\) and zero electric potential.
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 58J37
zbMATH DE Number: 5145730
zbMATH DE Number: 5169142
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: spectral asymptotics
zbMATH Keywords: orbit method
zbMATH Keywords: nilpotent Lie algebras
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
OpenAlex ID: W1607502981
OpenAlex ID: W2132918467
arXiv ID: math/0406071
Wikidata QID: Q33936915
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DOI: 10.5802/AIF.2229
DOI: 10.1073/PNAS.0500444102
description: scientific article; zbMATH DE number 5169142
alternative label: Generalizations of the classical Weyl and Colin de Verdière's formulas and the orbit method