Accurate Spectral Asymptotics for periodic operators

Victor Ivrii

Journées équations aux dérivées partielles (1999)

  • page 1-11
  • ISSN: 0752-0360

Abstract

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Asymptotics with sharp remainder estimates are recovered for number 𝐍 ( τ ) of eigenvalues of operator A ( x , D ) - t W ( x , x ) crossing level E as t runs from 0 to τ , τ → ∞ . Here A is periodic matrix operator, matrix W is positive, periodic with respect to first copy of x and decaying as second copy of x goes to infinity, E either belongs to a spectral gap of A or is one its ends. These problems are first treated in papers of M. Sh. Birman, M. Sh. Birman-A. Laptev and M. Sh. Birman-T. Suslina.

How to cite

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Ivrii, Victor. "Accurate Spectral Asymptotics for periodic operators." Journées équations aux dérivées partielles (1999): 1-11. <http://eudml.org/doc/93382>.

@article{Ivrii1999,
abstract = {Asymptotics with sharp remainder estimates are recovered for number $\{\mathbf \{N\}\}(\tau )$ of eigenvalues of operator $A(x,D)-tW(x,x)$ crossing level $E$ as $t$ runs from $0$ to $\tau $, $\tau \rightarrow \infty $. Here $A$ is periodic matrix operator, matrix $W$ is positive, periodic with respect to first copy of $x$ and decaying as second copy of $x$ goes to infinity, $E$ either belongs to a spectral gap of $A$ or is one its ends. These problems are first treated in papers of M. Sh. Birman, M. Sh. Birman-A. Laptev and M. Sh. Birman-T. Suslina.},
author = {Ivrii, Victor},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-11},
publisher = {Université de Nantes},
title = {Accurate Spectral Asymptotics for periodic operators},
url = {http://eudml.org/doc/93382},
year = {1999},
}

TY - JOUR
AU - Ivrii, Victor
TI - Accurate Spectral Asymptotics for periodic operators
JO - Journées équations aux dérivées partielles
PY - 1999
PB - Université de Nantes
SP - 1
EP - 11
AB - Asymptotics with sharp remainder estimates are recovered for number ${\mathbf {N}}(\tau )$ of eigenvalues of operator $A(x,D)-tW(x,x)$ crossing level $E$ as $t$ runs from $0$ to $\tau $, $\tau \rightarrow \infty $. Here $A$ is periodic matrix operator, matrix $W$ is positive, periodic with respect to first copy of $x$ and decaying as second copy of $x$ goes to infinity, $E$ either belongs to a spectral gap of $A$ or is one its ends. These problems are first treated in papers of M. Sh. Birman, M. Sh. Birman-A. Laptev and M. Sh. Birman-T. Suslina.
LA - eng
UR - http://eudml.org/doc/93382
ER -

References

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  1. [B1] M. BIRMAN. The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. I. Regularperturbations. Boundary value problems, Schrödinger operators, deformation quantization, Math. Top., 8, Akademie Verlag, Berlin, 1995, pp. 334-352. Zbl0848.47032MR97d:47055
  2. [B2] M. BIRMAN. The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential. St. Petersburg Math. J., 8 (1997), no. 1, pp. 1-14. Zbl0866.35087MR97h:47047
  3. [B3] M. BIRMAN. Discrete spectrum in the gaps of the perturbed periodic Schrödinger operator. II. Non-regular perturbations. St. Petersburg Math. J., 9 (1998), no. 6, pp. 1073-1095. Zbl0911.35082MR99h:47054
  4. [BL1] M. BIRMAN, A. LAPTEV. The negative discrete spectrum of a two-dimensional Schrödinger operator. Comm. Pure Appl. Math., 49 (1996), no. 9, pp. 967-997. Zbl0864.35080MR97i:35131
  5. [BL2] M. BIRMAN, A. LAPTEV. «Non-standard» spectral asymptotics for a two-dimensional Schrödinger operator. Centre de Recherches Mathematiques, CRM Proceedings and Lecture Notes, 12 (1997), pp. 9-16. Zbl0910.35086MR1479234
  6. [BLS] M. BIRMAN, A. LAPTEV, T. SUSLINA. Discrete spectrum of the twodimensional periodic elliptic second order operator perturbed by a decreasing potential. I. Semiinfinite gap (in preparation). Zbl1070.47041
  7. [BS] M. BIRMAN, T. SUSLINA. Birman, Suslina. Discrete spectrum of the twodimensional periodic elliptic second order operator perturbed by a decreasing potential. II. Internal gaps (in preparation). Zbl1070.47041
  8. [Ivr1] V. IVRII. Microlocal Analysis and Precise Spectral Asymptotics. Springer-Verlag, SMM, 1998, 731+15 pp. Zbl0906.35003MR99e:58193
  9. [Ivr2] V. IVRII. Accurate Spectral Asymptotics for Neumann Laplacian in domains with cusps (to appear in Applicable Analysis). 
  10. [JMS] V. JAKŠIĆ, S. MOLČANOV and B. SIMON. Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps. J. Func. Anal., 106, (1992), pp. 59-79. Zbl0783.35040MR93f:35165
  11. [Sol1] M. SOLOMYAK. On the negative discrete spectrum of the operator —ΔN —αV for a class of unbounded domains in Rd, CRM Proceedings and Lecture Notes, Centre de Recherches Mathematiques, 12, (1997), pp. 283-296. Zbl0888.35075MR98i:35138

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