Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields

Akira Iwatsuka; Hideo Tamura

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 2, page 479-515
  • ISSN: 0373-0956

Abstract

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This article studies the asymptotic behavior of the number N ( λ ) of the negative eigenvalues < - λ as λ → + 0 of the two dimensional Pauli operators with electric potential V ( x ) decaying at ∞ and with nonconstant magnetic field b ( x ) , which is assumed to be bounded or to decay at ∞ . In particular, it is shown that N ( λ ) = ( 1 / 2 π ) ∫ V ( x ) > λ b ( x ) d x ( 1 + o ( 1 ) ) , when V ( x ) decays faster than b ( x ) under some additional conditions.

How to cite

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Iwatsuka, Akira, and Tamura, Hideo. "Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields." Annales de l'institut Fourier 48.2 (1998): 479-515. <http://eudml.org/doc/75290>.

@article{Iwatsuka1998,
abstract = {This article studies the asymptotic behavior of the number $N(\lambda )$ of the negative eigenvalues $&lt; -\lambda $ as $\lambda \rightarrow +0$ of the two dimensional Pauli operators with electric potential $V(x)$ decaying at $\infty $ and with nonconstant magnetic field $b(x)$, which is assumed to be bounded or to decay at $\infty $. In particular, it is shown that $N(\lambda ) = (1/2\pi ) \int _\{V(x)&gt;\lambda \}b(x)dx(1+o(1))$, when $V(x)$ decays faster than $b(x)$ under some additional conditions.},
author = {Iwatsuka, Akira, Tamura, Hideo},
journal = {Annales de l'institut Fourier},
keywords = {Pauli operator; negative eigenvalues; magnetic fields; asymptotic distribution; asymptotic distribution of negative eigenvalues},
language = {eng},
number = {2},
pages = {479-515},
publisher = {Association des Annales de l'Institut Fourier},
title = {Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields},
url = {http://eudml.org/doc/75290},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Iwatsuka, Akira
AU - Tamura, Hideo
TI - Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 2
SP - 479
EP - 515
AB - This article studies the asymptotic behavior of the number $N(\lambda )$ of the negative eigenvalues $&lt; -\lambda $ as $\lambda \rightarrow +0$ of the two dimensional Pauli operators with electric potential $V(x)$ decaying at $\infty $ and with nonconstant magnetic field $b(x)$, which is assumed to be bounded or to decay at $\infty $. In particular, it is shown that $N(\lambda ) = (1/2\pi ) \int _{V(x)&gt;\lambda }b(x)dx(1+o(1))$, when $V(x)$ decays faster than $b(x)$ under some additional conditions.
LA - eng
KW - Pauli operator; negative eigenvalues; magnetic fields; asymptotic distribution; asymptotic distribution of negative eigenvalues
UR - http://eudml.org/doc/75290
ER -

References

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  2. [2] J. AVRON, I. HERBST and B. SIMON, Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J., 45 (1978), 847-883. Zbl0399.35029MR80k:35054
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  5. [5] L. ERDÖS, Magnetic Lieb-Thirring inequalities and stochastic oscillatory integrals, Operator Theory, Advances and Applications, 78 (1994), Birkhäuser Verlag, 127-134. Zbl0833.35117MR96i:81068
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  7. [7] L. ERDÖS and J. P. SOLOVEJ, Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields. I. Non-asymptotic Lieb-Thirring type estimate, preprint, 1996. 
  8. [8] L. ERDÖS and J. P. SOLOVEJ, Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields. II. Leading order asymptotic estimates, Commun. Math. Phys., 188 (1997), 599-656. Zbl0909.47052
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  10. [10] A. IWATSUKA and H. TAMURA, Asymptotic distribution of eigenvalues for Pauli operators with nonconstant magnetic fields, preprint, 1997 (to be published in Duke Math. J.). Zbl0932.35159MR99e:35167
  11. [11] A. MOHAMED and G. D. RAIKOV, On the spectral theory of the Schrödinger operator with electromagnetic potential, Pseudo-differential Calculus and Mathematical Physics, Adv. Partial Differ. Eq., Academic Press, 5 (1994), 298-390. Zbl0813.35065MR96e:35122
  12. [12] I. SHIGEKAWA, Spectral properties of Schrödinger operators with magnetic fields for a spin 1/2 particle, J. Func. Anal., 101 (1991), 255-285. Zbl0742.47002MR93g:35101
  13. [13] A. V. SOBOLEV, Asymptotic behavior of the energy levels of a quantum particle in a homogeneous magnetic field, perturbed by a decreasing electric field I, J. Soviet Math., 35 (1986), 2201-2211. Zbl0643.35028
  14. [14] A. V. SOBOLEV, On the Lieb-Thirring estimates for the Pauli operator, Duke Math. J., 82 (1996), 607-637. Zbl0882.47056MR97e:81030
  15. [15] H. TAMURA, Asymptotic distribution of eigenvalues for Schrödinger operators with homogeneous magnetic fields, Osaka J. Math., 25 (1988), 633-647. Zbl0731.35073MR90c:35159

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