# Accurate Spectral Asymptotics for periodic operators

Journées équations aux dérivées partielles (1999)

- page 1-11
- ISSN: 0752-0360

## Access Full Article

top## Abstract

top## How to cite

topIvrii, 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

top- [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
- [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
- [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
- [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
- [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
- [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
- [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
- [Ivr1] V. IVRII. Microlocal Analysis and Precise Spectral Asymptotics. Springer-Verlag, SMM, 1998, 731+15 pp. Zbl0906.35003MR99e:58193
- [Ivr2] V. IVRII. Accurate Spectral Asymptotics for Neumann Laplacian in domains with cusps (to appear in Applicable Analysis).
- [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
- [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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.