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Magnetic Lieb-Thirring inequalities

Abstract

We study the generalizations of the well-known Lieb-Thirring inequality for the magnetic Schrödinger operator with nonconstant magnetic field. Our main result is the naturally expected magnetic Lieb-Thirring estimate on the moments of the negative eigenvalues for a certain class of magnetic fields (including even some unbounded ones). We develop a localization technique in path space of the stochastic Feynman-Kac representation of the heat kernel which effectively estimates the oscillatory effect due to the magnetic phase factor.

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Communicated by Ya.G. Sinai

Work supported by the NSF grant PHY90-19433 A02, by the Alfred Sloan Foundation dissertation Fellowship and by the Erwin Schrödinger Institute for Mathematical Physics in Vienna.

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Erdős, L. Magnetic Lieb-Thirring inequalities. Commun.Math. Phys. 170, 629–668 (1995). https://doi.org/10.1007/BF02099152

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  • DOI: https://doi.org/10.1007/BF02099152

Keywords

  • Magnetic Field
  • Neural Network
  • Statistical Physic
  • Complex System
  • Nonlinear Dynamics