Beyond the classical Weyl and Colin de Verdière’s formulas for Schrödinger operators with polynomial magnetic and electric fields

Mitya Boyarchenko[1]; Sergei Levendorski[2]

  • [1] University of Chicago Department of Mathematics Chicago, IL 60637 (USA)
  • [2] University of Texas Department of Economics Austin, TX (USA)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 6, page 1827-1901
  • ISSN: 0373-0956

Abstract

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We present a pair of conjectural formulas that compute the leading term of the spectral asymptotics of a Schrödinger operator on L 2 ( ℝ n ) with quasi-homogeneous polynomial magnetic and electric fields. 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. By using the direct variational method, we 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 many “irregular” cases, with different types of degeneration of potentials.

How to cite

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Boyarchenko, Mitya, and Levendorski, Sergei. "Beyond the classical Weyl and Colin de Verdière’s formulas for Schrödinger operators with polynomial magnetic and electric fields." Annales de l’institut Fourier 56.6 (2006): 1827-1901. <http://eudml.org/doc/10193>.

@article{Boyarchenko2006,
abstract = {We present a pair of conjectural formulas that compute 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. 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. By using the direct variational method, we 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 many “irregular” cases, with different types of degeneration of potentials.},
affiliation = {University of Chicago Department of Mathematics Chicago, IL 60637 (USA); University of Texas Department of Economics Austin, TX (USA)},
author = {Boyarchenko, Mitya, Levendorski, Sergei},
journal = {Annales de l’institut Fourier},
keywords = {Schrödinger operators; spectral asymptotics; orbit method; nilpotent Lie algebras},
language = {eng},
number = {6},
pages = {1827-1901},
publisher = {Association des Annales de l’institut Fourier},
title = {Beyond the classical Weyl and Colin de Verdière’s formulas for Schrödinger operators with polynomial magnetic and electric fields},
url = {http://eudml.org/doc/10193},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Boyarchenko, Mitya
AU - Levendorski, Sergei
TI - Beyond the classical Weyl and Colin de Verdière’s formulas for Schrödinger operators with polynomial magnetic and electric fields
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 6
SP - 1827
EP - 1901
AB - We present a pair of conjectural formulas that compute 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. 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. By using the direct variational method, we 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 many “irregular” cases, with different types of degeneration of potentials.
LA - eng
KW - Schrödinger operators; spectral asymptotics; orbit method; nilpotent Lie algebras
UR - http://eudml.org/doc/10193
ER -

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