Accurate Spectral Asymptotics for periodic operators
Journées équations aux dérivées partielles (1999), article no. 5, 11 p.

Asymptotics with sharp remainder estimates are recovered for number 𝐍(τ) of eigenvalues of operator A(x,D)-tW(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.

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Victor Ivrii. Accurate Spectral Asymptotics for periodic operators. Journées équations aux dérivées partielles (1999), article  no. 5, 11 p. https://proceedings.centre-mersenne.org/item/JEDP_1999____A5_0/

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | Zbl

[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). | Zbl

[Ivr1] V. Ivrii. Microlocal Analysis and Precise Spectral Asymptotics. Springer-Verlag, SMM, 1998, 731+15 pp. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl